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<h1 id="146">14.6 &nbsp; 编辑距离问题<a class="headerlink" href="#146" title="Permanent link">&para;</a></h1>
<p>编辑距离，也称 Levenshtein 距离，指两个字符串之间互相转换的最少修改次数，通常用于在信息检索和自然语言处理中度量两个序列的相似度。</p>
<div class="admonition question">
<p class="admonition-title">Question</p>
<p>输入两个字符串 <span class="arithmatex">\(s\)</span> 和 <span class="arithmatex">\(t\)</span> ，返回将 <span class="arithmatex">\(s\)</span> 转换为 <span class="arithmatex">\(t\)</span> 所需的最少编辑步数。</p>
<p>你可以在一个字符串中进行三种编辑操作：插入一个字符、删除一个字符、将字符替换为任意一个字符。</p>
</div>
<p>如图 14-27 所示，将 <code>kitten</code> 转换为 <code>sitting</code> 需要编辑 3 步，包括 2 次替换操作与 1 次添加操作；将 <code>hello</code> 转换为 <code>algo</code> 需要 3 步，包括 2 次替换操作和 1 次删除操作。</p>
<p><a class="glightbox" href="../edit_distance_problem.assets/edit_distance_example.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="编辑距离的示例数据" class="animation-figure" src="../edit_distance_problem.assets/edit_distance_example.png" /></a></p>
<p align="center"> 图 14-27 &nbsp; 编辑距离的示例数据 </p>

<p><strong>编辑距离问题可以很自然地用决策树模型来解释</strong>。字符串对应树节点，一轮决策（一次编辑操作）对应树的一条边。</p>
<p>如图 14-28 所示，在不限制操作的情况下，每个节点都可以派生出许多条边，每条边对应一种操作，这意味着从 <code>hello</code> 转换到 <code>algo</code> 有许多种可能的路径。</p>
<p>从决策树的角度看，本题的目标是求解节点 <code>hello</code> 和节点 <code>algo</code> 之间的最短路径。</p>
<p><a class="glightbox" href="../edit_distance_problem.assets/edit_distance_decision_tree.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="基于决策树模型表示编辑距离问题" class="animation-figure" src="../edit_distance_problem.assets/edit_distance_decision_tree.png" /></a></p>
<p align="center"> 图 14-28 &nbsp; 基于决策树模型表示编辑距离问题 </p>

<h3 id="1">1. &nbsp; 动态规划思路<a class="headerlink" href="#1" title="Permanent link">&para;</a></h3>
<p><strong>第一步：思考每轮的决策，定义状态，从而得到 <span class="arithmatex">\(dp\)</span> 表</strong></p>
<p>每一轮的决策是对字符串 <span class="arithmatex">\(s\)</span> 进行一次编辑操作。</p>
<p>我们希望在编辑操作的过程中，问题的规模逐渐缩小，这样才能构建子问题。设字符串 <span class="arithmatex">\(s\)</span> 和 <span class="arithmatex">\(t\)</span> 的长度分别为 <span class="arithmatex">\(n\)</span> 和 <span class="arithmatex">\(m\)</span> ，我们先考虑两字符串尾部的字符 <span class="arithmatex">\(s[n-1]\)</span> 和 <span class="arithmatex">\(t[m-1]\)</span> 。</p>
<ul>
<li>若 <span class="arithmatex">\(s[n-1]\)</span> 和 <span class="arithmatex">\(t[m-1]\)</span> 相同，我们可以跳过它们，直接考虑 <span class="arithmatex">\(s[n-2]\)</span> 和 <span class="arithmatex">\(t[m-2]\)</span> 。</li>
<li>若 <span class="arithmatex">\(s[n-1]\)</span> 和 <span class="arithmatex">\(t[m-1]\)</span> 不同，我们需要对 <span class="arithmatex">\(s\)</span> 进行一次编辑（插入、删除、替换），使得两字符串尾部的字符相同，从而可以跳过它们，考虑规模更小的问题。</li>
</ul>
<p>也就是说，我们在字符串 <span class="arithmatex">\(s\)</span> 中进行的每一轮决策（编辑操作），都会使得 <span class="arithmatex">\(s\)</span> 和 <span class="arithmatex">\(t\)</span> 中剩余的待匹配字符发生变化。因此，状态为当前在 <span class="arithmatex">\(s\)</span> 和 <span class="arithmatex">\(t\)</span> 中考虑的第 <span class="arithmatex">\(i\)</span> 和第 <span class="arithmatex">\(j\)</span> 个字符，记为 <span class="arithmatex">\([i, j]\)</span> 。</p>
<p>状态 <span class="arithmatex">\([i, j]\)</span> 对应的子问题：<strong>将 <span class="arithmatex">\(s\)</span> 的前 <span class="arithmatex">\(i\)</span> 个字符更改为 <span class="arithmatex">\(t\)</span> 的前 <span class="arithmatex">\(j\)</span> 个字符所需的最少编辑步数</strong>。</p>
<p>至此，得到一个尺寸为 <span class="arithmatex">\((i+1) \times (j+1)\)</span> 的二维 <span class="arithmatex">\(dp\)</span> 表。</p>
<p><strong>第二步：找出最优子结构，进而推导出状态转移方程</strong></p>
<p>考虑子问题 <span class="arithmatex">\(dp[i, j]\)</span> ，其对应的两个字符串的尾部字符为 <span class="arithmatex">\(s[i-1]\)</span> 和 <span class="arithmatex">\(t[j-1]\)</span> ，可根据不同编辑操作分为图 14-29 所示的三种情况。</p>
<ol>
<li>在 <span class="arithmatex">\(s[i-1]\)</span> 之后添加 <span class="arithmatex">\(t[j-1]\)</span> ，则剩余子问题 <span class="arithmatex">\(dp[i, j-1]\)</span> 。</li>
<li>删除 <span class="arithmatex">\(s[i-1]\)</span> ，则剩余子问题 <span class="arithmatex">\(dp[i-1, j]\)</span> 。</li>
<li>将 <span class="arithmatex">\(s[i-1]\)</span> 替换为 <span class="arithmatex">\(t[j-1]\)</span> ，则剩余子问题 <span class="arithmatex">\(dp[i-1, j-1]\)</span> 。</li>
</ol>
<p><a class="glightbox" href="../edit_distance_problem.assets/edit_distance_state_transfer.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="编辑距离的状态转移" class="animation-figure" src="../edit_distance_problem.assets/edit_distance_state_transfer.png" /></a></p>
<p align="center"> 图 14-29 &nbsp; 编辑距离的状态转移 </p>

<p>根据以上分析，可得最优子结构：<span class="arithmatex">\(dp[i, j]\)</span> 的最少编辑步数等于 <span class="arithmatex">\(dp[i, j-1]\)</span>、<span class="arithmatex">\(dp[i-1, j]\)</span>、<span class="arithmatex">\(dp[i-1, j-1]\)</span> 三者中的最少编辑步数，再加上本次的编辑步数 <span class="arithmatex">\(1\)</span> 。对应的状态转移方程为：</p>
<div class="arithmatex">\[
dp[i, j] = \min(dp[i, j-1], dp[i-1, j], dp[i-1, j-1]) + 1
\]</div>
<p>请注意，<strong>当 <span class="arithmatex">\(s[i-1]\)</span> 和 <span class="arithmatex">\(t[j-1]\)</span> 相同时，无须编辑当前字符</strong>，这种情况下的状态转移方程为：</p>
<div class="arithmatex">\[
dp[i, j] = dp[i-1, j-1]
\]</div>
<p><strong>第三步：确定边界条件和状态转移顺序</strong></p>
<p>当两字符串都为空时，编辑步数为 <span class="arithmatex">\(0\)</span> ，即 <span class="arithmatex">\(dp[0, 0] = 0\)</span> 。当 <span class="arithmatex">\(s\)</span> 为空但 <span class="arithmatex">\(t\)</span> 不为空时，最少编辑步数等于 <span class="arithmatex">\(t\)</span> 的长度，即首行 <span class="arithmatex">\(dp[0, j] = j\)</span> 。当 <span class="arithmatex">\(s\)</span> 不为空但 <span class="arithmatex">\(t\)</span> 为空时，最少编辑步数等于 <span class="arithmatex">\(s\)</span> 的长度，即首列 <span class="arithmatex">\(dp[i, 0] = i\)</span> 。</p>
<p>观察状态转移方程，解 <span class="arithmatex">\(dp[i, j]\)</span> 依赖左方、上方、左上方的解，因此通过两层循环正序遍历整个 <span class="arithmatex">\(dp\)</span> 表即可。</p>
<h3 id="2">2. &nbsp; 代码实现<a class="headerlink" href="#2" title="Permanent link">&para;</a></h3>
<div class="tabbed-set tabbed-alternate" data-tabs="1:12"><input checked="checked" id="__tabbed_1_1" name="__tabbed_1" type="radio" /><input id="__tabbed_1_2" name="__tabbed_1" type="radio" /><input id="__tabbed_1_3" name="__tabbed_1" type="radio" /><input id="__tabbed_1_4" name="__tabbed_1" type="radio" /><input id="__tabbed_1_5" name="__tabbed_1" type="radio" /><input id="__tabbed_1_6" name="__tabbed_1" type="radio" /><input id="__tabbed_1_7" name="__tabbed_1" type="radio" /><input id="__tabbed_1_8" name="__tabbed_1" type="radio" /><input id="__tabbed_1_9" name="__tabbed_1" type="radio" /><input id="__tabbed_1_10" name="__tabbed_1" type="radio" /><input id="__tabbed_1_11" name="__tabbed_1" type="radio" /><input id="__tabbed_1_12" name="__tabbed_1" type="radio" /><div class="tabbed-labels"><label for="__tabbed_1_1">Python</label><label for="__tabbed_1_2">C++</label><label for="__tabbed_1_3">Java</label><label for="__tabbed_1_4">C#</label><label for="__tabbed_1_5">Go</label><label for="__tabbed_1_6">Swift</label><label for="__tabbed_1_7">JS</label><label for="__tabbed_1_8">TS</label><label for="__tabbed_1_9">Dart</label><label for="__tabbed_1_10">Rust</label><label for="__tabbed_1_11">C</label><label for="__tabbed_1_12">Zig</label></div>
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<div class="highlight"><span class="filename">edit_distance.py</span><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a><span class="k">def</span> <span class="nf">edit_distance_dp</span><span class="p">(</span><span class="n">s</span><span class="p">:</span> <span class="nb">str</span><span class="p">,</span> <span class="n">t</span><span class="p">:</span> <span class="nb">str</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="nb">int</span><span class="p">:</span>
<a id="__codelineno-0-2" name="__codelineno-0-2" href="#__codelineno-0-2"></a><span class="w">    </span><span class="sd">&quot;&quot;&quot;编辑距离：动态规划&quot;&quot;&quot;</span>
<a id="__codelineno-0-3" name="__codelineno-0-3" href="#__codelineno-0-3"></a>    <span class="n">n</span><span class="p">,</span> <span class="n">m</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">s</span><span class="p">),</span> <span class="nb">len</span><span class="p">(</span><span class="n">t</span><span class="p">)</span>
<a id="__codelineno-0-4" name="__codelineno-0-4" href="#__codelineno-0-4"></a>    <span class="n">dp</span> <span class="o">=</span> <span class="p">[[</span><span class="mi">0</span><span class="p">]</span> <span class="o">*</span> <span class="p">(</span><span class="n">m</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> <span class="k">for</span> <span class="n">_</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)]</span>
<a id="__codelineno-0-5" name="__codelineno-0-5" href="#__codelineno-0-5"></a>    <span class="c1"># 状态转移：首行首列</span>
<a id="__codelineno-0-6" name="__codelineno-0-6" href="#__codelineno-0-6"></a>    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
<a id="__codelineno-0-7" name="__codelineno-0-7" href="#__codelineno-0-7"></a>        <span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="n">i</span>
<a id="__codelineno-0-8" name="__codelineno-0-8" href="#__codelineno-0-8"></a>    <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">m</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
<a id="__codelineno-0-9" name="__codelineno-0-9" href="#__codelineno-0-9"></a>        <span class="n">dp</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="n">j</span>
<a id="__codelineno-0-10" name="__codelineno-0-10" href="#__codelineno-0-10"></a>    <span class="c1"># 状态转移：其余行和列</span>
<a id="__codelineno-0-11" name="__codelineno-0-11" href="#__codelineno-0-11"></a>    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
<a id="__codelineno-0-12" name="__codelineno-0-12" href="#__codelineno-0-12"></a>        <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">m</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
<a id="__codelineno-0-13" name="__codelineno-0-13" href="#__codelineno-0-13"></a>            <span class="k">if</span> <span class="n">s</span><span class="p">[</span><span class="n">i</span> <span class="o">-</span> <span class="mi">1</span><span class="p">]</span> <span class="o">==</span> <span class="n">t</span><span class="p">[</span><span class="n">j</span> <span class="o">-</span> <span class="mi">1</span><span class="p">]:</span>
<a id="__codelineno-0-14" name="__codelineno-0-14" href="#__codelineno-0-14"></a>                <span class="c1"># 若两字符相等，则直接跳过此两字符</span>
<a id="__codelineno-0-15" name="__codelineno-0-15" href="#__codelineno-0-15"></a>                <span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="n">dp</span><span class="p">[</span><span class="n">i</span> <span class="o">-</span> <span class="mi">1</span><span class="p">][</span><span class="n">j</span> <span class="o">-</span> <span class="mi">1</span><span class="p">]</span>
<a id="__codelineno-0-16" name="__codelineno-0-16" href="#__codelineno-0-16"></a>            <span class="k">else</span><span class="p">:</span>
<a id="__codelineno-0-17" name="__codelineno-0-17" href="#__codelineno-0-17"></a>                <span class="c1"># 最少编辑步数 = 插入、删除、替换这三种操作的最少编辑步数 + 1</span>
<a id="__codelineno-0-18" name="__codelineno-0-18" href="#__codelineno-0-18"></a>                <span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="nb">min</span><span class="p">(</span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span> <span class="o">-</span> <span class="mi">1</span><span class="p">],</span> <span class="n">dp</span><span class="p">[</span><span class="n">i</span> <span class="o">-</span> <span class="mi">1</span><span class="p">][</span><span class="n">j</span><span class="p">],</span> <span class="n">dp</span><span class="p">[</span><span class="n">i</span> <span class="o">-</span> <span class="mi">1</span><span class="p">][</span><span class="n">j</span> <span class="o">-</span> <span class="mi">1</span><span class="p">])</span> <span class="o">+</span> <span class="mi">1</span>
<a id="__codelineno-0-19" name="__codelineno-0-19" href="#__codelineno-0-19"></a>    <span class="k">return</span> <span class="n">dp</span><span class="p">[</span><span class="n">n</span><span class="p">][</span><span class="n">m</span><span class="p">]</span>
</code></pre></div>
</div>
<div class="tabbed-block">
<div class="highlight"><span class="filename">edit_distance.cpp</span><pre><span></span><code><a id="__codelineno-1-1" name="__codelineno-1-1" href="#__codelineno-1-1"></a><span class="cm">/* 编辑距离：动态规划 */</span>
<a id="__codelineno-1-2" name="__codelineno-1-2" href="#__codelineno-1-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">editDistanceDP</span><span class="p">(</span><span class="n">string</span><span class="w"> </span><span class="n">s</span><span class="p">,</span><span class="w"> </span><span class="n">string</span><span class="w"> </span><span class="n">t</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-1-3" name="__codelineno-1-3" href="#__codelineno-1-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">s</span><span class="p">.</span><span class="n">length</span><span class="p">(),</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">t</span><span class="p">.</span><span class="n">length</span><span class="p">();</span>
<a id="__codelineno-1-4" name="__codelineno-1-4" href="#__codelineno-1-4"></a><span class="w">    </span><span class="n">vector</span><span class="o">&lt;</span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;&gt;</span><span class="w"> </span><span class="n">dp</span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="p">(</span><span class="n">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="mi">0</span><span class="p">));</span>
<a id="__codelineno-1-5" name="__codelineno-1-5" href="#__codelineno-1-5"></a><span class="w">    </span><span class="c1">// 状态转移：首行首列</span>
<a id="__codelineno-1-6" name="__codelineno-1-6" href="#__codelineno-1-6"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-1-7" name="__codelineno-1-7" href="#__codelineno-1-7"></a><span class="w">        </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">i</span><span class="p">;</span>
<a id="__codelineno-1-8" name="__codelineno-1-8" href="#__codelineno-1-8"></a><span class="w">    </span><span class="p">}</span>
<a id="__codelineno-1-9" name="__codelineno-1-9" href="#__codelineno-1-9"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">m</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-1-10" name="__codelineno-1-10" href="#__codelineno-1-10"></a><span class="w">        </span><span class="n">dp</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">j</span><span class="p">;</span>
<a id="__codelineno-1-11" name="__codelineno-1-11" href="#__codelineno-1-11"></a><span class="w">    </span><span class="p">}</span>
<a id="__codelineno-1-12" name="__codelineno-1-12" href="#__codelineno-1-12"></a><span class="w">    </span><span class="c1">// 状态转移：其余行和列</span>
<a id="__codelineno-1-13" name="__codelineno-1-13" href="#__codelineno-1-13"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-1-14" name="__codelineno-1-14" href="#__codelineno-1-14"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">m</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-1-15" name="__codelineno-1-15" href="#__codelineno-1-15"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">s</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">t</span><span class="p">[</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">])</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-1-16" name="__codelineno-1-16" href="#__codelineno-1-16"></a><span class="w">                </span><span class="c1">// 若两字符相等，则直接跳过此两字符</span>
<a id="__codelineno-1-17" name="__codelineno-1-17" href="#__codelineno-1-17"></a><span class="w">                </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">][</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">];</span>
<a id="__codelineno-1-18" name="__codelineno-1-18" href="#__codelineno-1-18"></a><span class="w">            </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-1-19" name="__codelineno-1-19" href="#__codelineno-1-19"></a><span class="w">                </span><span class="c1">// 最少编辑步数 = 插入、删除、替换这三种操作的最少编辑步数 + 1</span>
<a id="__codelineno-1-20" name="__codelineno-1-20" href="#__codelineno-1-20"></a><span class="w">                </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">min</span><span class="p">(</span><span class="n">min</span><span class="p">(</span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">],</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">][</span><span class="n">j</span><span class="p">]),</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">][</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">])</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
<a id="__codelineno-1-21" name="__codelineno-1-21" href="#__codelineno-1-21"></a><span class="w">            </span><span class="p">}</span>
<a id="__codelineno-1-22" name="__codelineno-1-22" href="#__codelineno-1-22"></a><span class="w">        </span><span class="p">}</span>
<a id="__codelineno-1-23" name="__codelineno-1-23" href="#__codelineno-1-23"></a><span class="w">    </span><span class="p">}</span>
<a id="__codelineno-1-24" name="__codelineno-1-24" href="#__codelineno-1-24"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">n</span><span class="p">][</span><span class="n">m</span><span class="p">];</span>
<a id="__codelineno-1-25" name="__codelineno-1-25" href="#__codelineno-1-25"></a><span class="p">}</span>
</code></pre></div>
</div>
<div class="tabbed-block">
<div class="highlight"><span class="filename">edit_distance.java</span><pre><span></span><code><a id="__codelineno-2-1" name="__codelineno-2-1" href="#__codelineno-2-1"></a><span class="cm">/* 编辑距离：动态规划 */</span>
<a id="__codelineno-2-2" name="__codelineno-2-2" href="#__codelineno-2-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">editDistanceDP</span><span class="p">(</span><span class="n">String</span><span class="w"> </span><span class="n">s</span><span class="p">,</span><span class="w"> </span><span class="n">String</span><span class="w"> </span><span class="n">t</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-2-3" name="__codelineno-2-3" href="#__codelineno-2-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">s</span><span class="p">.</span><span class="na">length</span><span class="p">(),</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">t</span><span class="p">.</span><span class="na">length</span><span class="p">();</span>
<a id="__codelineno-2-4" name="__codelineno-2-4" href="#__codelineno-2-4"></a><span class="w">    </span><span class="kt">int</span><span class="o">[][]</span><span class="w"> </span><span class="n">dp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="kt">int</span><span class="o">[</span><span class="n">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="o">][</span><span class="n">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="o">]</span><span class="p">;</span>
<a id="__codelineno-2-5" name="__codelineno-2-5" href="#__codelineno-2-5"></a><span class="w">    </span><span class="c1">// 状态转移：首行首列</span>
<a id="__codelineno-2-6" name="__codelineno-2-6" href="#__codelineno-2-6"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-2-7" name="__codelineno-2-7" href="#__codelineno-2-7"></a><span class="w">        </span><span class="n">dp</span><span class="o">[</span><span class="n">i</span><span class="o">][</span><span class="mi">0</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">i</span><span class="p">;</span>
<a id="__codelineno-2-8" name="__codelineno-2-8" href="#__codelineno-2-8"></a><span class="w">    </span><span class="p">}</span>
<a id="__codelineno-2-9" name="__codelineno-2-9" href="#__codelineno-2-9"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">m</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-2-10" name="__codelineno-2-10" href="#__codelineno-2-10"></a><span class="w">        </span><span class="n">dp</span><span class="o">[</span><span class="mi">0</span><span class="o">][</span><span class="n">j</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">j</span><span class="p">;</span>
<a id="__codelineno-2-11" name="__codelineno-2-11" href="#__codelineno-2-11"></a><span class="w">    </span><span class="p">}</span>
<a id="__codelineno-2-12" name="__codelineno-2-12" href="#__codelineno-2-12"></a><span class="w">    </span><span class="c1">// 状态转移：其余行和列</span>
<a id="__codelineno-2-13" name="__codelineno-2-13" href="#__codelineno-2-13"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-2-14" name="__codelineno-2-14" href="#__codelineno-2-14"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">m</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-2-15" name="__codelineno-2-15" href="#__codelineno-2-15"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">s</span><span class="p">.</span><span class="na">charAt</span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">t</span><span class="p">.</span><span class="na">charAt</span><span class="p">(</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">))</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-2-16" name="__codelineno-2-16" href="#__codelineno-2-16"></a><span class="w">                </span><span class="c1">// 若两字符相等，则直接跳过此两字符</span>
<a id="__codelineno-2-17" name="__codelineno-2-17" href="#__codelineno-2-17"></a><span class="w">                </span><span class="n">dp</span><span class="o">[</span><span class="n">i</span><span class="o">][</span><span class="n">j</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="o">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="o">][</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="o">]</span><span class="p">;</span>
<a id="__codelineno-2-18" name="__codelineno-2-18" href="#__codelineno-2-18"></a><span class="w">            </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-2-19" name="__codelineno-2-19" href="#__codelineno-2-19"></a><span class="w">                </span><span class="c1">// 最少编辑步数 = 插入、删除、替换这三种操作的最少编辑步数 + 1</span>
<a id="__codelineno-2-20" name="__codelineno-2-20" href="#__codelineno-2-20"></a><span class="w">                </span><span class="n">dp</span><span class="o">[</span><span class="n">i</span><span class="o">][</span><span class="n">j</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">Math</span><span class="p">.</span><span class="na">min</span><span class="p">(</span><span class="n">Math</span><span class="p">.</span><span class="na">min</span><span class="p">(</span><span class="n">dp</span><span class="o">[</span><span class="n">i</span><span class="o">][</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="o">]</span><span class="p">,</span><span class="w"> </span><span class="n">dp</span><span class="o">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="o">][</span><span class="n">j</span><span class="o">]</span><span class="p">),</span><span class="w"> </span><span class="n">dp</span><span class="o">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="o">][</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="o">]</span><span class="p">)</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
<a id="__codelineno-2-21" name="__codelineno-2-21" href="#__codelineno-2-21"></a><span class="w">            </span><span class="p">}</span>
<a id="__codelineno-2-22" name="__codelineno-2-22" href="#__codelineno-2-22"></a><span class="w">        </span><span class="p">}</span>
<a id="__codelineno-2-23" name="__codelineno-2-23" href="#__codelineno-2-23"></a><span class="w">    </span><span class="p">}</span>
<a id="__codelineno-2-24" name="__codelineno-2-24" href="#__codelineno-2-24"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">dp</span><span class="o">[</span><span class="n">n</span><span class="o">][</span><span class="n">m</span><span class="o">]</span><span class="p">;</span>
<a id="__codelineno-2-25" name="__codelineno-2-25" href="#__codelineno-2-25"></a><span class="p">}</span>
</code></pre></div>
</div>
<div class="tabbed-block">
<div class="highlight"><span class="filename">edit_distance.cs</span><pre><span></span><code><a id="__codelineno-3-1" name="__codelineno-3-1" href="#__codelineno-3-1"></a><span class="cm">/* 编辑距离：动态规划 */</span>
<a id="__codelineno-3-2" name="__codelineno-3-2" href="#__codelineno-3-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">EditDistanceDP</span><span class="p">(</span><span class="kt">string</span><span class="w"> </span><span class="n">s</span><span class="p">,</span><span class="w"> </span><span class="kt">string</span><span class="w"> </span><span class="n">t</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-3-3" name="__codelineno-3-3" href="#__codelineno-3-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">s</span><span class="p">.</span><span class="n">Length</span><span class="p">,</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">t</span><span class="p">.</span><span class="n">Length</span><span class="p">;</span>
<a id="__codelineno-3-4" name="__codelineno-3-4" href="#__codelineno-3-4"></a><span class="w">    </span><span class="kt">int</span><span class="p">[,]</span><span class="w"> </span><span class="n">dp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="kt">int</span><span class="p">[</span><span class="n">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span><span class="p">,</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span><span class="p">];</span>
<a id="__codelineno-3-5" name="__codelineno-3-5" href="#__codelineno-3-5"></a><span class="w">    </span><span class="c1">// 状态转移：首行首列</span>
<a id="__codelineno-3-6" name="__codelineno-3-6" href="#__codelineno-3-6"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-3-7" name="__codelineno-3-7" href="#__codelineno-3-7"></a><span class="w">        </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="m">0</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">i</span><span class="p">;</span>
<a id="__codelineno-3-8" name="__codelineno-3-8" href="#__codelineno-3-8"></a><span class="w">    </span><span class="p">}</span>
<a id="__codelineno-3-9" name="__codelineno-3-9" href="#__codelineno-3-9"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">1</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">m</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-3-10" name="__codelineno-3-10" href="#__codelineno-3-10"></a><span class="w">        </span><span class="n">dp</span><span class="p">[</span><span class="m">0</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">j</span><span class="p">;</span>
<a id="__codelineno-3-11" name="__codelineno-3-11" href="#__codelineno-3-11"></a><span class="w">    </span><span class="p">}</span>
<a id="__codelineno-3-12" name="__codelineno-3-12" href="#__codelineno-3-12"></a><span class="w">    </span><span class="c1">// 状态转移：其余行和列</span>
<a id="__codelineno-3-13" name="__codelineno-3-13" href="#__codelineno-3-13"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-3-14" name="__codelineno-3-14" href="#__codelineno-3-14"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">1</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">m</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-3-15" name="__codelineno-3-15" href="#__codelineno-3-15"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">s</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">]</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">t</span><span class="p">[</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">])</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-3-16" name="__codelineno-3-16" href="#__codelineno-3-16"></a><span class="w">                </span><span class="c1">// 若两字符相等，则直接跳过此两字符</span>
<a id="__codelineno-3-17" name="__codelineno-3-17" href="#__codelineno-3-17"></a><span class="w">                </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">];</span>
<a id="__codelineno-3-18" name="__codelineno-3-18" href="#__codelineno-3-18"></a><span class="w">            </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-3-19" name="__codelineno-3-19" href="#__codelineno-3-19"></a><span class="w">                </span><span class="c1">// 最少编辑步数 = 插入、删除、替换这三种操作的最少编辑步数 + 1</span>
<a id="__codelineno-3-20" name="__codelineno-3-20" href="#__codelineno-3-20"></a><span class="w">                </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">Math</span><span class="p">.</span><span class="n">Min</span><span class="p">(</span><span class="n">Math</span><span class="p">.</span><span class="n">Min</span><span class="p">(</span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">],</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="p">]),</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">])</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span><span class="p">;</span>
<a id="__codelineno-3-21" name="__codelineno-3-21" href="#__codelineno-3-21"></a><span class="w">            </span><span class="p">}</span>
<a id="__codelineno-3-22" name="__codelineno-3-22" href="#__codelineno-3-22"></a><span class="w">        </span><span class="p">}</span>
<a id="__codelineno-3-23" name="__codelineno-3-23" href="#__codelineno-3-23"></a><span class="w">    </span><span class="p">}</span>
<a id="__codelineno-3-24" name="__codelineno-3-24" href="#__codelineno-3-24"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="n">m</span><span class="p">];</span>
<a id="__codelineno-3-25" name="__codelineno-3-25" href="#__codelineno-3-25"></a><span class="p">}</span>
</code></pre></div>
</div>
<div class="tabbed-block">
<div class="highlight"><span class="filename">edit_distance.go</span><pre><span></span><code><a id="__codelineno-4-1" name="__codelineno-4-1" href="#__codelineno-4-1"></a><span class="cm">/* 编辑距离：动态规划 */</span>
<a id="__codelineno-4-2" name="__codelineno-4-2" href="#__codelineno-4-2"></a><span class="kd">func</span><span class="w"> </span><span class="nx">editDistanceDP</span><span class="p">(</span><span class="nx">s</span><span class="w"> </span><span class="kt">string</span><span class="p">,</span><span class="w"> </span><span class="nx">t</span><span class="w"> </span><span class="kt">string</span><span class="p">)</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-4-3" name="__codelineno-4-3" href="#__codelineno-4-3"></a><span class="w">    </span><span class="nx">n</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nb">len</span><span class="p">(</span><span class="nx">s</span><span class="p">)</span>
<a id="__codelineno-4-4" name="__codelineno-4-4" href="#__codelineno-4-4"></a><span class="w">    </span><span class="nx">m</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nb">len</span><span class="p">(</span><span class="nx">t</span><span class="p">)</span>
<a id="__codelineno-4-5" name="__codelineno-4-5" href="#__codelineno-4-5"></a><span class="w">    </span><span class="nx">dp</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nb">make</span><span class="p">([][]</span><span class="kt">int</span><span class="p">,</span><span class="w"> </span><span class="nx">n</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span>
<a id="__codelineno-4-6" name="__codelineno-4-6" href="#__codelineno-4-6"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-4-7" name="__codelineno-4-7" href="#__codelineno-4-7"></a><span class="w">        </span><span class="nx">dp</span><span class="p">[</span><span class="nx">i</span><span class="p">]</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="nb">make</span><span class="p">([]</span><span class="kt">int</span><span class="p">,</span><span class="w"> </span><span class="nx">m</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span>
<a id="__codelineno-4-8" name="__codelineno-4-8" href="#__codelineno-4-8"></a><span class="w">    </span><span class="p">}</span>
<a id="__codelineno-4-9" name="__codelineno-4-9" href="#__codelineno-4-9"></a><span class="w">    </span><span class="c1">// 状态转移：首行首列</span>
<a id="__codelineno-4-10" name="__codelineno-4-10" href="#__codelineno-4-10"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-4-11" name="__codelineno-4-11" href="#__codelineno-4-11"></a><span class="w">        </span><span class="nx">dp</span><span class="p">[</span><span class="nx">i</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="nx">i</span>
<a id="__codelineno-4-12" name="__codelineno-4-12" href="#__codelineno-4-12"></a><span class="w">    </span><span class="p">}</span>
<a id="__codelineno-4-13" name="__codelineno-4-13" href="#__codelineno-4-13"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">m</span><span class="p">;</span><span class="w"> </span><span class="nx">j</span><span class="o">++</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-4-14" name="__codelineno-4-14" href="#__codelineno-4-14"></a><span class="w">        </span><span class="nx">dp</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="nx">j</span><span class="p">]</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="nx">j</span>
<a id="__codelineno-4-15" name="__codelineno-4-15" href="#__codelineno-4-15"></a><span class="w">    </span><span class="p">}</span>
<a id="__codelineno-4-16" name="__codelineno-4-16" href="#__codelineno-4-16"></a><span class="w">    </span><span class="c1">// 状态转移：其余行和列</span>
<a id="__codelineno-4-17" name="__codelineno-4-17" href="#__codelineno-4-17"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-4-18" name="__codelineno-4-18" href="#__codelineno-4-18"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">m</span><span class="p">;</span><span class="w"> </span><span class="nx">j</span><span class="o">++</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-4-19" name="__codelineno-4-19" href="#__codelineno-4-19"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="nx">s</span><span class="p">[</span><span class="nx">i</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="nx">t</span><span class="p">[</span><span class="nx">j</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-4-20" name="__codelineno-4-20" href="#__codelineno-4-20"></a><span class="w">                </span><span class="c1">// 若两字符相等，则直接跳过此两字符</span>
<a id="__codelineno-4-21" name="__codelineno-4-21" href="#__codelineno-4-21"></a><span class="w">                </span><span class="nx">dp</span><span class="p">[</span><span class="nx">i</span><span class="p">][</span><span class="nx">j</span><span class="p">]</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="nx">dp</span><span class="p">[</span><span class="nx">i</span><span class="o">-</span><span class="mi">1</span><span class="p">][</span><span class="nx">j</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
<a id="__codelineno-4-22" name="__codelineno-4-22" href="#__codelineno-4-22"></a><span class="w">            </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-4-23" name="__codelineno-4-23" href="#__codelineno-4-23"></a><span class="w">                </span><span class="c1">// 最少编辑步数 = 插入、删除、替换这三种操作的最少编辑步数 + 1</span>
<a id="__codelineno-4-24" name="__codelineno-4-24" href="#__codelineno-4-24"></a><span class="w">                </span><span class="nx">dp</span><span class="p">[</span><span class="nx">i</span><span class="p">][</span><span class="nx">j</span><span class="p">]</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="nx">MinInt</span><span class="p">(</span><span class="nx">MinInt</span><span class="p">(</span><span class="nx">dp</span><span class="p">[</span><span class="nx">i</span><span class="p">][</span><span class="nx">j</span><span class="o">-</span><span class="mi">1</span><span class="p">],</span><span class="w"> </span><span class="nx">dp</span><span class="p">[</span><span class="nx">i</span><span class="o">-</span><span class="mi">1</span><span class="p">][</span><span class="nx">j</span><span class="p">]),</span><span class="w"> </span><span class="nx">dp</span><span class="p">[</span><span class="nx">i</span><span class="o">-</span><span class="mi">1</span><span class="p">][</span><span class="nx">j</span><span class="o">-</span><span class="mi">1</span><span class="p">])</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span>
<a id="__codelineno-4-25" name="__codelineno-4-25" href="#__codelineno-4-25"></a><span class="w">            </span><span class="p">}</span>
<a id="__codelineno-4-26" name="__codelineno-4-26" href="#__codelineno-4-26"></a><span class="w">        </span><span class="p">}</span>
<a id="__codelineno-4-27" name="__codelineno-4-27" href="#__codelineno-4-27"></a><span class="w">    </span><span class="p">}</span>
<a id="__codelineno-4-28" name="__codelineno-4-28" href="#__codelineno-4-28"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">dp</span><span class="p">[</span><span class="nx">n</span><span class="p">][</span><span class="nx">m</span><span class="p">]</span>
<a id="__codelineno-4-29" name="__codelineno-4-29" href="#__codelineno-4-29"></a><span class="p">}</span>
</code></pre></div>
</div>
<div class="tabbed-block">
<div class="highlight"><span class="filename">edit_distance.swift</span><pre><span></span><code><a id="__codelineno-5-1" name="__codelineno-5-1" href="#__codelineno-5-1"></a><span class="cm">/* 编辑距离：动态规划 */</span>
<a id="__codelineno-5-2" name="__codelineno-5-2" href="#__codelineno-5-2"></a><span class="kd">func</span> <span class="nf">editDistanceDP</span><span class="p">(</span><span class="n">s</span><span class="p">:</span> <span class="nb">String</span><span class="p">,</span> <span class="n">t</span><span class="p">:</span> <span class="nb">String</span><span class="p">)</span> <span class="p">-&gt;</span> <span class="nb">Int</span> <span class="p">{</span>
<a id="__codelineno-5-3" name="__codelineno-5-3" href="#__codelineno-5-3"></a>    <span class="kd">let</span> <span class="nv">n</span> <span class="p">=</span> <span class="n">s</span><span class="p">.</span><span class="n">utf8CString</span><span class="p">.</span><span class="bp">count</span>
<a id="__codelineno-5-4" name="__codelineno-5-4" href="#__codelineno-5-4"></a>    <span class="kd">let</span> <span class="nv">m</span> <span class="p">=</span> <span class="n">t</span><span class="p">.</span><span class="n">utf8CString</span><span class="p">.</span><span class="bp">count</span>
<a id="__codelineno-5-5" name="__codelineno-5-5" href="#__codelineno-5-5"></a>    <span class="kd">var</span> <span class="nv">dp</span> <span class="p">=</span> <span class="nb">Array</span><span class="p">(</span><span class="n">repeating</span><span class="p">:</span> <span class="nb">Array</span><span class="p">(</span><span class="n">repeating</span><span class="p">:</span> <span class="mi">0</span><span class="p">,</span> <span class="bp">count</span><span class="p">:</span> <span class="n">m</span> <span class="o">+</span> <span class="mi">1</span><span class="p">),</span> <span class="bp">count</span><span class="p">:</span> <span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
<a id="__codelineno-5-6" name="__codelineno-5-6" href="#__codelineno-5-6"></a>    <span class="c1">// 状态转移：首行首列</span>
<a id="__codelineno-5-7" name="__codelineno-5-7" href="#__codelineno-5-7"></a>    <span class="k">for</span> <span class="n">i</span> <span class="k">in</span> <span class="bp">stride</span><span class="p">(</span><span class="n">from</span><span class="p">:</span> <span class="mi">1</span><span class="p">,</span> <span class="n">through</span><span class="p">:</span> <span class="n">n</span><span class="p">,</span> <span class="n">by</span><span class="p">:</span> <span class="mi">1</span><span class="p">)</span> <span class="p">{</span>
<a id="__codelineno-5-8" name="__codelineno-5-8" href="#__codelineno-5-8"></a>        <span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span> <span class="p">=</span> <span class="n">i</span>
<a id="__codelineno-5-9" name="__codelineno-5-9" href="#__codelineno-5-9"></a>    <span class="p">}</span>
<a id="__codelineno-5-10" name="__codelineno-5-10" href="#__codelineno-5-10"></a>    <span class="k">for</span> <span class="n">j</span> <span class="k">in</span> <span class="bp">stride</span><span class="p">(</span><span class="n">from</span><span class="p">:</span> <span class="mi">1</span><span class="p">,</span> <span class="n">through</span><span class="p">:</span> <span class="n">m</span><span class="p">,</span> <span class="n">by</span><span class="p">:</span> <span class="mi">1</span><span class="p">)</span> <span class="p">{</span>
<a id="__codelineno-5-11" name="__codelineno-5-11" href="#__codelineno-5-11"></a>        <span class="n">dp</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="n">j</span><span class="p">]</span> <span class="p">=</span> <span class="n">j</span>
<a id="__codelineno-5-12" name="__codelineno-5-12" href="#__codelineno-5-12"></a>    <span class="p">}</span>
<a id="__codelineno-5-13" name="__codelineno-5-13" href="#__codelineno-5-13"></a>    <span class="c1">// 状态转移：其余行和列</span>
<a id="__codelineno-5-14" name="__codelineno-5-14" href="#__codelineno-5-14"></a>    <span class="k">for</span> <span class="n">i</span> <span class="k">in</span> <span class="bp">stride</span><span class="p">(</span><span class="n">from</span><span class="p">:</span> <span class="mi">1</span><span class="p">,</span> <span class="n">through</span><span class="p">:</span> <span class="n">n</span><span class="p">,</span> <span class="n">by</span><span class="p">:</span> <span class="mi">1</span><span class="p">)</span> <span class="p">{</span>
<a id="__codelineno-5-15" name="__codelineno-5-15" href="#__codelineno-5-15"></a>        <span class="k">for</span> <span class="n">j</span> <span class="k">in</span> <span class="bp">stride</span><span class="p">(</span><span class="n">from</span><span class="p">:</span> <span class="mi">1</span><span class="p">,</span> <span class="n">through</span><span class="p">:</span> <span class="n">m</span><span class="p">,</span> <span class="n">by</span><span class="p">:</span> <span class="mi">1</span><span class="p">)</span> <span class="p">{</span>
<a id="__codelineno-5-16" name="__codelineno-5-16" href="#__codelineno-5-16"></a>            <span class="k">if</span> <span class="n">s</span><span class="p">.</span><span class="n">utf8CString</span><span class="p">[</span><span class="n">i</span> <span class="o">-</span> <span class="mi">1</span><span class="p">]</span> <span class="p">==</span> <span class="n">t</span><span class="p">.</span><span class="n">utf8CString</span><span class="p">[</span><span class="n">j</span> <span class="o">-</span> <span class="mi">1</span><span class="p">]</span> <span class="p">{</span>
<a id="__codelineno-5-17" name="__codelineno-5-17" href="#__codelineno-5-17"></a>                <span class="c1">// 若两字符相等，则直接跳过此两字符</span>
<a id="__codelineno-5-18" name="__codelineno-5-18" href="#__codelineno-5-18"></a>                <span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="p">]</span> <span class="p">=</span> <span class="n">dp</span><span class="p">[</span><span class="n">i</span> <span class="o">-</span> <span class="mi">1</span><span class="p">][</span><span class="n">j</span> <span class="o">-</span> <span class="mi">1</span><span class="p">]</span>
<a id="__codelineno-5-19" name="__codelineno-5-19" href="#__codelineno-5-19"></a>            <span class="p">}</span> <span class="k">else</span> <span class="p">{</span>
<a id="__codelineno-5-20" name="__codelineno-5-20" href="#__codelineno-5-20"></a>                <span class="c1">// 最少编辑步数 = 插入、删除、替换这三种操作的最少编辑步数 + 1</span>
<a id="__codelineno-5-21" name="__codelineno-5-21" href="#__codelineno-5-21"></a>                <span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="p">]</span> <span class="p">=</span> <span class="bp">min</span><span class="p">(</span><span class="bp">min</span><span class="p">(</span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span> <span class="o">-</span> <span class="mi">1</span><span class="p">],</span> <span class="n">dp</span><span class="p">[</span><span class="n">i</span> <span class="o">-</span> <span class="mi">1</span><span class="p">][</span><span class="n">j</span><span class="p">]),</span> <span class="n">dp</span><span class="p">[</span><span class="n">i</span> <span class="o">-</span> <span class="mi">1</span><span class="p">][</span><span class="n">j</span> <span class="o">-</span> <span class="mi">1</span><span class="p">])</span> <span class="o">+</span> <span class="mi">1</span>
<a id="__codelineno-5-22" name="__codelineno-5-22" href="#__codelineno-5-22"></a>            <span class="p">}</span>
<a id="__codelineno-5-23" name="__codelineno-5-23" href="#__codelineno-5-23"></a>        <span class="p">}</span>
<a id="__codelineno-5-24" name="__codelineno-5-24" href="#__codelineno-5-24"></a>    <span class="p">}</span>
<a id="__codelineno-5-25" name="__codelineno-5-25" href="#__codelineno-5-25"></a>    <span class="k">return</span> <span class="n">dp</span><span class="p">[</span><span class="n">n</span><span class="p">][</span><span class="n">m</span><span class="p">]</span>
<a id="__codelineno-5-26" name="__codelineno-5-26" href="#__codelineno-5-26"></a><span class="p">}</span>
</code></pre></div>
</div>
<div class="tabbed-block">
<div class="highlight"><span class="filename">edit_distance.js</span><pre><span></span><code><a id="__codelineno-6-1" name="__codelineno-6-1" href="#__codelineno-6-1"></a><span class="cm">/* 编辑距离：动态规划 */</span>
<a id="__codelineno-6-2" name="__codelineno-6-2" href="#__codelineno-6-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">editDistanceDP</span><span class="p">(</span><span class="nx">s</span><span class="p">,</span><span class="w"> </span><span class="nx">t</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-6-3" name="__codelineno-6-3" href="#__codelineno-6-3"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">s</span><span class="p">.</span><span class="nx">length</span><span class="p">,</span>
<a id="__codelineno-6-4" name="__codelineno-6-4" href="#__codelineno-6-4"></a><span class="w">        </span><span class="nx">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">t</span><span class="p">.</span><span class="nx">length</span><span class="p">;</span>
<a id="__codelineno-6-5" name="__codelineno-6-5" href="#__codelineno-6-5"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">dp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Array</span><span class="p">.</span><span class="kr">from</span><span class="p">({</span><span class="w"> </span><span class="nx">length</span><span class="o">:</span><span class="w"> </span><span class="nx">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="w"> </span><span class="p">},</span><span class="w"> </span><span class="p">()</span><span class="w"> </span><span class="p">=&gt;</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="nb">Array</span><span class="p">(</span><span class="nx">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="p">).</span><span class="nx">fill</span><span class="p">(</span><span class="mf">0</span><span class="p">));</span>
<a id="__codelineno-6-6" name="__codelineno-6-6" href="#__codelineno-6-6"></a><span class="w">    </span><span class="c1">// 状态转移：首行首列</span>
<a id="__codelineno-6-7" name="__codelineno-6-7" href="#__codelineno-6-7"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-6-8" name="__codelineno-6-8" href="#__codelineno-6-8"></a><span class="w">        </span><span class="nx">dp</span><span class="p">[</span><span class="nx">i</span><span class="p">][</span><span class="mf">0</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">i</span><span class="p">;</span>
<a id="__codelineno-6-9" name="__codelineno-6-9" href="#__codelineno-6-9"></a><span class="w">    </span><span class="p">}</span>
<a id="__codelineno-6-10" name="__codelineno-6-10" href="#__codelineno-6-10"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">m</span><span class="p">;</span><span class="w"> </span><span class="nx">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-6-11" name="__codelineno-6-11" href="#__codelineno-6-11"></a><span class="w">        </span><span class="nx">dp</span><span class="p">[</span><span class="mf">0</span><span class="p">][</span><span class="nx">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">j</span><span class="p">;</span>
<a id="__codelineno-6-12" name="__codelineno-6-12" href="#__codelineno-6-12"></a><span class="w">    </span><span class="p">}</span>
<a id="__codelineno-6-13" name="__codelineno-6-13" href="#__codelineno-6-13"></a><span class="w">    </span><span class="c1">// 状态转移：其余行和列</span>
<a id="__codelineno-6-14" name="__codelineno-6-14" href="#__codelineno-6-14"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-6-15" name="__codelineno-6-15" href="#__codelineno-6-15"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">m</span><span class="p">;</span><span class="w"> </span><span class="nx">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-6-16" name="__codelineno-6-16" href="#__codelineno-6-16"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">s</span><span class="p">.</span><span class="nx">charAt</span><span class="p">(</span><span class="nx">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">)</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="nx">t</span><span class="p">.</span><span class="nx">charAt</span><span class="p">(</span><span class="nx">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">))</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-6-17" name="__codelineno-6-17" href="#__codelineno-6-17"></a><span class="w">                </span><span class="c1">// 若两字符相等，则直接跳过此两字符</span>
<a id="__codelineno-6-18" name="__codelineno-6-18" href="#__codelineno-6-18"></a><span class="w">                </span><span class="nx">dp</span><span class="p">[</span><span class="nx">i</span><span class="p">][</span><span class="nx">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">dp</span><span class="p">[</span><span class="nx">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">][</span><span class="nx">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">];</span>
<a id="__codelineno-6-19" name="__codelineno-6-19" href="#__codelineno-6-19"></a><span class="w">            </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-6-20" name="__codelineno-6-20" href="#__codelineno-6-20"></a><span class="w">                </span><span class="c1">// 最少编辑步数 = 插入、删除、替换这三种操作的最少编辑步数 + 1</span>
<a id="__codelineno-6-21" name="__codelineno-6-21" href="#__codelineno-6-21"></a><span class="w">                </span><span class="nx">dp</span><span class="p">[</span><span class="nx">i</span><span class="p">][</span><span class="nx">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span>
<a id="__codelineno-6-22" name="__codelineno-6-22" href="#__codelineno-6-22"></a><span class="w">                    </span><span class="nb">Math</span><span class="p">.</span><span class="nx">min</span><span class="p">(</span><span class="nx">dp</span><span class="p">[</span><span class="nx">i</span><span class="p">][</span><span class="nx">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">],</span><span class="w"> </span><span class="nx">dp</span><span class="p">[</span><span class="nx">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">][</span><span class="nx">j</span><span class="p">],</span><span class="w"> </span><span class="nx">dp</span><span class="p">[</span><span class="nx">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">][</span><span class="nx">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">])</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span>
<a id="__codelineno-6-23" name="__codelineno-6-23" href="#__codelineno-6-23"></a><span class="w">            </span><span class="p">}</span>
<a id="__codelineno-6-24" name="__codelineno-6-24" href="#__codelineno-6-24"></a><span class="w">        </span><span class="p">}</span>
<a id="__codelineno-6-25" name="__codelineno-6-25" href="#__codelineno-6-25"></a><span class="w">    </span><span class="p">}</span>
<a id="__codelineno-6-26" name="__codelineno-6-26" href="#__codelineno-6-26"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">dp</span><span class="p">[</span><span class="nx">n</span><span class="p">][</span><span class="nx">m</span><span class="p">];</span>
<a id="__codelineno-6-27" name="__codelineno-6-27" href="#__codelineno-6-27"></a><span class="p">}</span>
</code></pre></div>
</div>
<div class="tabbed-block">
<div class="highlight"><span class="filename">edit_distance.ts</span><pre><span></span><code><a id="__codelineno-7-1" name="__codelineno-7-1" href="#__codelineno-7-1"></a><span class="cm">/* 编辑距离：动态规划 */</span>
<a id="__codelineno-7-2" name="__codelineno-7-2" href="#__codelineno-7-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">editDistanceDP</span><span class="p">(</span><span class="nx">s</span><span class="o">:</span><span class="w"> </span><span class="kt">string</span><span class="p">,</span><span class="w"> </span><span class="nx">t</span><span class="o">:</span><span class="w"> </span><span class="kt">string</span><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-7-3" name="__codelineno-7-3" href="#__codelineno-7-3"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">s</span><span class="p">.</span><span class="nx">length</span><span class="p">,</span>
<a id="__codelineno-7-4" name="__codelineno-7-4" href="#__codelineno-7-4"></a><span class="w">        </span><span class="nx">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">t</span><span class="p">.</span><span class="nx">length</span><span class="p">;</span>
<a id="__codelineno-7-5" name="__codelineno-7-5" href="#__codelineno-7-5"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">dp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Array</span><span class="p">.</span><span class="kr">from</span><span class="p">({</span><span class="w"> </span><span class="nx">length</span><span class="o">:</span><span class="w"> </span><span class="kt">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="w"> </span><span class="p">},</span><span class="w"> </span><span class="p">()</span><span class="w"> </span><span class="p">=&gt;</span>
<a id="__codelineno-7-6" name="__codelineno-7-6" href="#__codelineno-7-6"></a><span class="w">        </span><span class="nb">Array</span><span class="p">.</span><span class="kr">from</span><span class="p">({</span><span class="w"> </span><span class="nx">length</span><span class="o">:</span><span class="w"> </span><span class="kt">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="w"> </span><span class="p">},</span><span class="w"> </span><span class="p">()</span><span class="w"> </span><span class="p">=&gt;</span><span class="w"> </span><span class="mf">0</span><span class="p">)</span>
<a id="__codelineno-7-7" name="__codelineno-7-7" href="#__codelineno-7-7"></a><span class="w">    </span><span class="p">);</span>
<a id="__codelineno-7-8" name="__codelineno-7-8" href="#__codelineno-7-8"></a><span class="w">    </span><span class="c1">// 状态转移：首行首列</span>
<a id="__codelineno-7-9" name="__codelineno-7-9" href="#__codelineno-7-9"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-7-10" name="__codelineno-7-10" href="#__codelineno-7-10"></a><span class="w">        </span><span class="nx">dp</span><span class="p">[</span><span class="nx">i</span><span class="p">][</span><span class="mf">0</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">i</span><span class="p">;</span>
<a id="__codelineno-7-11" name="__codelineno-7-11" href="#__codelineno-7-11"></a><span class="w">    </span><span class="p">}</span>
<a id="__codelineno-7-12" name="__codelineno-7-12" href="#__codelineno-7-12"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">m</span><span class="p">;</span><span class="w"> </span><span class="nx">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-7-13" name="__codelineno-7-13" href="#__codelineno-7-13"></a><span class="w">        </span><span class="nx">dp</span><span class="p">[</span><span class="mf">0</span><span class="p">][</span><span class="nx">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">j</span><span class="p">;</span>
<a id="__codelineno-7-14" name="__codelineno-7-14" href="#__codelineno-7-14"></a><span class="w">    </span><span class="p">}</span>
<a id="__codelineno-7-15" name="__codelineno-7-15" href="#__codelineno-7-15"></a><span class="w">    </span><span class="c1">// 状态转移：其余行和列</span>
<a id="__codelineno-7-16" name="__codelineno-7-16" href="#__codelineno-7-16"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-7-17" name="__codelineno-7-17" href="#__codelineno-7-17"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">m</span><span class="p">;</span><span class="w"> </span><span class="nx">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-7-18" name="__codelineno-7-18" href="#__codelineno-7-18"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">s</span><span class="p">.</span><span class="nx">charAt</span><span class="p">(</span><span class="nx">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">)</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="nx">t</span><span class="p">.</span><span class="nx">charAt</span><span class="p">(</span><span class="nx">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">))</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-7-19" name="__codelineno-7-19" href="#__codelineno-7-19"></a><span class="w">                </span><span class="c1">// 若两字符相等，则直接跳过此两字符</span>
<a id="__codelineno-7-20" name="__codelineno-7-20" href="#__codelineno-7-20"></a><span class="w">                </span><span class="nx">dp</span><span class="p">[</span><span class="nx">i</span><span class="p">][</span><span class="nx">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">dp</span><span class="p">[</span><span class="nx">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">][</span><span class="nx">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">];</span>
<a id="__codelineno-7-21" name="__codelineno-7-21" href="#__codelineno-7-21"></a><span class="w">            </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-7-22" name="__codelineno-7-22" href="#__codelineno-7-22"></a><span class="w">                </span><span class="c1">// 最少编辑步数 = 插入、删除、替换这三种操作的最少编辑步数 + 1</span>
<a id="__codelineno-7-23" name="__codelineno-7-23" href="#__codelineno-7-23"></a><span class="w">                </span><span class="nx">dp</span><span class="p">[</span><span class="nx">i</span><span class="p">][</span><span class="nx">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span>
<a id="__codelineno-7-24" name="__codelineno-7-24" href="#__codelineno-7-24"></a><span class="w">                    </span><span class="nb">Math</span><span class="p">.</span><span class="nx">min</span><span class="p">(</span><span class="nx">dp</span><span class="p">[</span><span class="nx">i</span><span class="p">][</span><span class="nx">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">],</span><span class="w"> </span><span class="nx">dp</span><span class="p">[</span><span class="nx">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">][</span><span class="nx">j</span><span class="p">],</span><span class="w"> </span><span class="nx">dp</span><span class="p">[</span><span class="nx">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">][</span><span class="nx">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">])</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span>
<a id="__codelineno-7-25" name="__codelineno-7-25" href="#__codelineno-7-25"></a><span class="w">            </span><span class="p">}</span>
<a id="__codelineno-7-26" name="__codelineno-7-26" href="#__codelineno-7-26"></a><span class="w">        </span><span class="p">}</span>
<a id="__codelineno-7-27" name="__codelineno-7-27" href="#__codelineno-7-27"></a><span class="w">    </span><span class="p">}</span>
<a id="__codelineno-7-28" name="__codelineno-7-28" href="#__codelineno-7-28"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">dp</span><span class="p">[</span><span class="nx">n</span><span class="p">][</span><span class="nx">m</span><span class="p">];</span>
<a id="__codelineno-7-29" name="__codelineno-7-29" href="#__codelineno-7-29"></a><span class="p">}</span>
</code></pre></div>
</div>
<div class="tabbed-block">
<div class="highlight"><span class="filename">edit_distance.dart</span><pre><span></span><code><a id="__codelineno-8-1" name="__codelineno-8-1" href="#__codelineno-8-1"></a><span class="cm">/* 编辑距离：动态规划 */</span>
<a id="__codelineno-8-2" name="__codelineno-8-2" href="#__codelineno-8-2"></a><span class="kt">int</span><span class="w"> </span><span class="n">editDistanceDP</span><span class="p">(</span><span class="kt">String</span><span class="w"> </span><span class="n">s</span><span class="p">,</span><span class="w"> </span><span class="kt">String</span><span class="w"> </span><span class="n">t</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-8-3" name="__codelineno-8-3" href="#__codelineno-8-3"></a><span class="w">  </span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">s</span><span class="p">.</span><span class="n">length</span><span class="p">,</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">t</span><span class="p">.</span><span class="n">length</span><span class="p">;</span>
<a id="__codelineno-8-4" name="__codelineno-8-4" href="#__codelineno-8-4"></a><span class="w">  </span><span class="n">List</span><span class="o">&lt;</span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;&gt;</span><span class="w"> </span><span class="n">dp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">List</span><span class="p">.</span><span class="n">generate</span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span><span class="p">,</span><span class="w"> </span><span class="p">(</span><span class="n">_</span><span class="p">)</span><span class="w"> </span><span class="o">=&gt;</span><span class="w"> </span><span class="n">List</span><span class="p">.</span><span class="n">filled</span><span class="p">(</span><span class="n">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span><span class="p">,</span><span class="w"> </span><span class="m">0</span><span class="p">));</span>
<a id="__codelineno-8-5" name="__codelineno-8-5" href="#__codelineno-8-5"></a><span class="w">  </span><span class="c1">// 状态转移：首行首列</span>
<a id="__codelineno-8-6" name="__codelineno-8-6" href="#__codelineno-8-6"></a><span class="w">  </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-8-7" name="__codelineno-8-7" href="#__codelineno-8-7"></a><span class="w">    </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="m">0</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">i</span><span class="p">;</span>
<a id="__codelineno-8-8" name="__codelineno-8-8" href="#__codelineno-8-8"></a><span class="w">  </span><span class="p">}</span>
<a id="__codelineno-8-9" name="__codelineno-8-9" href="#__codelineno-8-9"></a><span class="w">  </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">1</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">m</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-8-10" name="__codelineno-8-10" href="#__codelineno-8-10"></a><span class="w">    </span><span class="n">dp</span><span class="p">[</span><span class="m">0</span><span class="p">][</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">j</span><span class="p">;</span>
<a id="__codelineno-8-11" name="__codelineno-8-11" href="#__codelineno-8-11"></a><span class="w">  </span><span class="p">}</span>
<a id="__codelineno-8-12" name="__codelineno-8-12" href="#__codelineno-8-12"></a><span class="w">  </span><span class="c1">// 状态转移：其余行和列</span>
<a id="__codelineno-8-13" name="__codelineno-8-13" href="#__codelineno-8-13"></a><span class="w">  </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-8-14" name="__codelineno-8-14" href="#__codelineno-8-14"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">1</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">m</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-8-15" name="__codelineno-8-15" href="#__codelineno-8-15"></a><span class="w">      </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">s</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">]</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">t</span><span class="p">[</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">])</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-8-16" name="__codelineno-8-16" href="#__codelineno-8-16"></a><span class="w">        </span><span class="c1">// 若两字符相等，则直接跳过此两字符</span>
<a id="__codelineno-8-17" name="__codelineno-8-17" href="#__codelineno-8-17"></a><span class="w">        </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">][</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">];</span>
<a id="__codelineno-8-18" name="__codelineno-8-18" href="#__codelineno-8-18"></a><span class="w">      </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-8-19" name="__codelineno-8-19" href="#__codelineno-8-19"></a><span class="w">        </span><span class="c1">// 最少编辑步数 = 插入、删除、替换这三种操作的最少编辑步数 + 1</span>
<a id="__codelineno-8-20" name="__codelineno-8-20" href="#__codelineno-8-20"></a><span class="w">        </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">min</span><span class="p">(</span><span class="n">min</span><span class="p">(</span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">],</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">][</span><span class="n">j</span><span class="p">]),</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">][</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">])</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span><span class="p">;</span>
<a id="__codelineno-8-21" name="__codelineno-8-21" href="#__codelineno-8-21"></a><span class="w">      </span><span class="p">}</span>
<a id="__codelineno-8-22" name="__codelineno-8-22" href="#__codelineno-8-22"></a><span class="w">    </span><span class="p">}</span>
<a id="__codelineno-8-23" name="__codelineno-8-23" href="#__codelineno-8-23"></a><span class="w">  </span><span class="p">}</span>
<a id="__codelineno-8-24" name="__codelineno-8-24" href="#__codelineno-8-24"></a><span class="w">  </span><span class="k">return</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">n</span><span class="p">][</span><span class="n">m</span><span class="p">];</span>
<a id="__codelineno-8-25" name="__codelineno-8-25" href="#__codelineno-8-25"></a><span class="p">}</span>
</code></pre></div>
</div>
<div class="tabbed-block">
<div class="highlight"><span class="filename">edit_distance.rs</span><pre><span></span><code><a id="__codelineno-9-1" name="__codelineno-9-1" href="#__codelineno-9-1"></a><span class="cm">/* 编辑距离：动态规划 */</span>
<a id="__codelineno-9-2" name="__codelineno-9-2" href="#__codelineno-9-2"></a><span class="k">fn</span> <span class="nf">edit_distance_dp</span><span class="p">(</span><span class="n">s</span>: <span class="kp">&amp;</span><span class="kt">str</span><span class="p">,</span><span class="w"> </span><span class="n">t</span>: <span class="kp">&amp;</span><span class="kt">str</span><span class="p">)</span><span class="w"> </span>-&gt; <span class="kt">i32</span> <span class="p">{</span>
<a id="__codelineno-9-3" name="__codelineno-9-3" href="#__codelineno-9-3"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="p">(</span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="n">m</span><span class="p">)</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">(</span><span class="n">s</span><span class="p">.</span><span class="n">len</span><span class="p">(),</span><span class="w"> </span><span class="n">t</span><span class="p">.</span><span class="n">len</span><span class="p">());</span>
<a id="__codelineno-9-4" name="__codelineno-9-4" href="#__codelineno-9-4"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">dp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="fm">vec!</span><span class="p">[</span><span class="fm">vec!</span><span class="p">[</span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">];</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">];</span>
<a id="__codelineno-9-5" name="__codelineno-9-5" href="#__codelineno-9-5"></a><span class="w">    </span><span class="c1">// 状态转移：首行首列</span>
<a id="__codelineno-9-6" name="__codelineno-9-6" href="#__codelineno-9-6"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">1</span><span class="o">..=</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-9-7" name="__codelineno-9-7" href="#__codelineno-9-7"></a><span class="w">        </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="kt">i32</span><span class="p">;</span>
<a id="__codelineno-9-8" name="__codelineno-9-8" href="#__codelineno-9-8"></a><span class="w">    </span><span class="p">}</span>
<a id="__codelineno-9-9" name="__codelineno-9-9" href="#__codelineno-9-9"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">1</span><span class="o">..</span><span class="n">m</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-9-10" name="__codelineno-9-10" href="#__codelineno-9-10"></a><span class="w">        </span><span class="n">dp</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="kt">i32</span><span class="p">;</span>
<a id="__codelineno-9-11" name="__codelineno-9-11" href="#__codelineno-9-11"></a><span class="w">    </span><span class="p">}</span>
<a id="__codelineno-9-12" name="__codelineno-9-12" href="#__codelineno-9-12"></a><span class="w">    </span><span class="c1">// 状态转移：其余行和列</span>
<a id="__codelineno-9-13" name="__codelineno-9-13" href="#__codelineno-9-13"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">1</span><span class="o">..=</span><span class="n">n</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-9-14" name="__codelineno-9-14" href="#__codelineno-9-14"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">1</span><span class="o">..=</span><span class="n">m</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-9-15" name="__codelineno-9-15" href="#__codelineno-9-15"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="n">s</span><span class="p">.</span><span class="n">chars</span><span class="p">().</span><span class="n">nth</span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">t</span><span class="p">.</span><span class="n">chars</span><span class="p">().</span><span class="n">nth</span><span class="p">(</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-9-16" name="__codelineno-9-16" href="#__codelineno-9-16"></a><span class="w">                </span><span class="c1">// 若两字符相等，则直接跳过此两字符</span>
<a id="__codelineno-9-17" name="__codelineno-9-17" href="#__codelineno-9-17"></a><span class="w">                </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">][</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">];</span>
<a id="__codelineno-9-18" name="__codelineno-9-18" href="#__codelineno-9-18"></a><span class="w">            </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-9-19" name="__codelineno-9-19" href="#__codelineno-9-19"></a><span class="w">                </span><span class="c1">// 最少编辑步数 = 插入、删除、替换这三种操作的最少编辑步数 + 1</span>
<a id="__codelineno-9-20" name="__codelineno-9-20" href="#__codelineno-9-20"></a><span class="w">                </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">std</span>::<span class="n">cmp</span>::<span class="n">min</span><span class="p">(</span><span class="n">std</span>::<span class="n">cmp</span>::<span class="n">min</span><span class="p">(</span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">],</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">][</span><span class="n">j</span><span class="p">]),</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">][</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">])</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
<a id="__codelineno-9-21" name="__codelineno-9-21" href="#__codelineno-9-21"></a><span class="w">            </span><span class="p">}</span>
<a id="__codelineno-9-22" name="__codelineno-9-22" href="#__codelineno-9-22"></a><span class="w">        </span><span class="p">}</span>
<a id="__codelineno-9-23" name="__codelineno-9-23" href="#__codelineno-9-23"></a><span class="w">    </span><span class="p">}</span>
<a id="__codelineno-9-24" name="__codelineno-9-24" href="#__codelineno-9-24"></a><span class="w">    </span><span class="n">dp</span><span class="p">[</span><span class="n">n</span><span class="p">][</span><span class="n">m</span><span class="p">]</span>
<a id="__codelineno-9-25" name="__codelineno-9-25" href="#__codelineno-9-25"></a><span class="p">}</span>
</code></pre></div>
</div>
<div class="tabbed-block">
<div class="highlight"><span class="filename">edit_distance.c</span><pre><span></span><code><a id="__codelineno-10-1" name="__codelineno-10-1" href="#__codelineno-10-1"></a><span class="cm">/* 编辑距离：动态规划 */</span>
<a id="__codelineno-10-2" name="__codelineno-10-2" href="#__codelineno-10-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">editDistanceDP</span><span class="p">(</span><span class="kt">char</span><span class="w"> </span><span class="o">*</span><span class="n">s</span><span class="p">,</span><span class="w"> </span><span class="kt">char</span><span class="w"> </span><span class="o">*</span><span class="n">t</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">m</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-10-3" name="__codelineno-10-3" href="#__codelineno-10-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="o">**</span><span class="n">dp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">malloc</span><span class="p">((</span><span class="n">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="k">sizeof</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="o">*</span><span class="p">));</span>
<a id="__codelineno-10-4" name="__codelineno-10-4" href="#__codelineno-10-4"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-10-5" name="__codelineno-10-5" href="#__codelineno-10-5"></a><span class="w">        </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">calloc</span><span class="p">(</span><span class="n">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="k">sizeof</span><span class="p">(</span><span class="kt">int</span><span class="p">));</span>
<a id="__codelineno-10-6" name="__codelineno-10-6" href="#__codelineno-10-6"></a><span class="w">    </span><span class="p">}</span>
<a id="__codelineno-10-7" name="__codelineno-10-7" href="#__codelineno-10-7"></a><span class="w">    </span><span class="c1">// 状态转移：首行首列</span>
<a id="__codelineno-10-8" name="__codelineno-10-8" href="#__codelineno-10-8"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-10-9" name="__codelineno-10-9" href="#__codelineno-10-9"></a><span class="w">        </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">i</span><span class="p">;</span>
<a id="__codelineno-10-10" name="__codelineno-10-10" href="#__codelineno-10-10"></a><span class="w">    </span><span class="p">}</span>
<a id="__codelineno-10-11" name="__codelineno-10-11" href="#__codelineno-10-11"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">m</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-10-12" name="__codelineno-10-12" href="#__codelineno-10-12"></a><span class="w">        </span><span class="n">dp</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">j</span><span class="p">;</span>
<a id="__codelineno-10-13" name="__codelineno-10-13" href="#__codelineno-10-13"></a><span class="w">    </span><span class="p">}</span>
<a id="__codelineno-10-14" name="__codelineno-10-14" href="#__codelineno-10-14"></a><span class="w">    </span><span class="c1">// 状态转移：其余行和列</span>
<a id="__codelineno-10-15" name="__codelineno-10-15" href="#__codelineno-10-15"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-10-16" name="__codelineno-10-16" href="#__codelineno-10-16"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">m</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-10-17" name="__codelineno-10-17" href="#__codelineno-10-17"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">s</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">t</span><span class="p">[</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">])</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-10-18" name="__codelineno-10-18" href="#__codelineno-10-18"></a><span class="w">                </span><span class="c1">// 若两字符相等，则直接跳过此两字符</span>
<a id="__codelineno-10-19" name="__codelineno-10-19" href="#__codelineno-10-19"></a><span class="w">                </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">][</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">];</span>
<a id="__codelineno-10-20" name="__codelineno-10-20" href="#__codelineno-10-20"></a><span class="w">            </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-10-21" name="__codelineno-10-21" href="#__codelineno-10-21"></a><span class="w">                </span><span class="c1">// 最少编辑步数 = 插入、删除、替换这三种操作的最少编辑步数 + 1</span>
<a id="__codelineno-10-22" name="__codelineno-10-22" href="#__codelineno-10-22"></a><span class="w">                </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">myMin</span><span class="p">(</span><span class="n">myMin</span><span class="p">(</span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">],</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">][</span><span class="n">j</span><span class="p">]),</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">][</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">])</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
<a id="__codelineno-10-23" name="__codelineno-10-23" href="#__codelineno-10-23"></a><span class="w">            </span><span class="p">}</span>
<a id="__codelineno-10-24" name="__codelineno-10-24" href="#__codelineno-10-24"></a><span class="w">        </span><span class="p">}</span>
<a id="__codelineno-10-25" name="__codelineno-10-25" href="#__codelineno-10-25"></a><span class="w">    </span><span class="p">}</span>
<a id="__codelineno-10-26" name="__codelineno-10-26" href="#__codelineno-10-26"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">n</span><span class="p">][</span><span class="n">m</span><span class="p">];</span>
<a id="__codelineno-10-27" name="__codelineno-10-27" href="#__codelineno-10-27"></a><span class="w">    </span><span class="c1">// 释放内存</span>
<a id="__codelineno-10-28" name="__codelineno-10-28" href="#__codelineno-10-28"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-10-29" name="__codelineno-10-29" href="#__codelineno-10-29"></a><span class="w">        </span><span class="n">free</span><span class="p">(</span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">]);</span>
<a id="__codelineno-10-30" name="__codelineno-10-30" href="#__codelineno-10-30"></a><span class="w">    </span><span class="p">}</span>
<a id="__codelineno-10-31" name="__codelineno-10-31" href="#__codelineno-10-31"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">res</span><span class="p">;</span>
<a id="__codelineno-10-32" name="__codelineno-10-32" href="#__codelineno-10-32"></a><span class="p">}</span>
</code></pre></div>
</div>
<div class="tabbed-block">
<div class="highlight"><span class="filename">edit_distance.zig</span><pre><span></span><code><a id="__codelineno-11-1" name="__codelineno-11-1" href="#__codelineno-11-1"></a><span class="c1">// 编辑距离：动态规划</span>
<a id="__codelineno-11-2" name="__codelineno-11-2" href="#__codelineno-11-2"></a><span class="k">fn</span><span class="w"> </span><span class="n">editDistanceDP</span><span class="p">(</span><span class="kr">comptime</span><span class="w"> </span><span class="n">s</span><span class="o">:</span><span class="w"> </span><span class="p">[]</span><span class="kr">const</span><span class="w"> </span><span class="kt">u8</span><span class="p">,</span><span class="w"> </span><span class="kr">comptime</span><span class="w"> </span><span class="n">t</span><span class="o">:</span><span class="w"> </span><span class="p">[]</span><span class="kr">const</span><span class="w"> </span><span class="kt">u8</span><span class="p">)</span><span class="w"> </span><span class="kt">i32</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-11-3" name="__codelineno-11-3" href="#__codelineno-11-3"></a><span class="w">    </span><span class="kr">comptime</span><span class="w"> </span><span class="kr">var</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">s</span><span class="p">.</span><span class="n">len</span><span class="p">;</span>
<a id="__codelineno-11-4" name="__codelineno-11-4" href="#__codelineno-11-4"></a><span class="w">    </span><span class="kr">comptime</span><span class="w"> </span><span class="kr">var</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">t</span><span class="p">.</span><span class="n">len</span><span class="p">;</span>
<a id="__codelineno-11-5" name="__codelineno-11-5" href="#__codelineno-11-5"></a><span class="w">    </span><span class="kr">var</span><span class="w"> </span><span class="n">dp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[</span><span class="n">_</span><span class="p">][</span><span class="n">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">]</span><span class="kt">i32</span><span class="p">{[</span><span class="n">_</span><span class="p">]</span><span class="kt">i32</span><span class="p">{</span><span class="mi">0</span><span class="p">}</span><span class="w"> </span><span class="o">**</span><span class="w"> </span><span class="p">(</span><span class="n">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">)}</span><span class="w"> </span><span class="o">**</span><span class="w"> </span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">);</span>
<a id="__codelineno-11-6" name="__codelineno-11-6" href="#__codelineno-11-6"></a><span class="w">    </span><span class="c1">// 状态转移：首行首列</span>
<a id="__codelineno-11-7" name="__codelineno-11-7" href="#__codelineno-11-7"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="mi">1</span><span class="p">..</span><span class="n">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="o">|</span><span class="n">i</span><span class="o">|</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-11-8" name="__codelineno-11-8" href="#__codelineno-11-8"></a><span class="w">        </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">@intCast</span><span class="p">(</span><span class="n">i</span><span class="p">);</span>
<a id="__codelineno-11-9" name="__codelineno-11-9" href="#__codelineno-11-9"></a><span class="w">    </span><span class="p">}</span>
<a id="__codelineno-11-10" name="__codelineno-11-10" href="#__codelineno-11-10"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="mi">1</span><span class="p">..</span><span class="n">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="o">|</span><span class="n">j</span><span class="o">|</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-11-11" name="__codelineno-11-11" href="#__codelineno-11-11"></a><span class="w">        </span><span class="n">dp</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">@intCast</span><span class="p">(</span><span class="n">j</span><span class="p">);</span>
<a id="__codelineno-11-12" name="__codelineno-11-12" href="#__codelineno-11-12"></a><span class="w">    </span><span class="p">}</span>
<a id="__codelineno-11-13" name="__codelineno-11-13" href="#__codelineno-11-13"></a><span class="w">    </span><span class="c1">// 状态转移：其余行和列</span>
<a id="__codelineno-11-14" name="__codelineno-11-14" href="#__codelineno-11-14"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="mi">1</span><span class="p">..</span><span class="n">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="o">|</span><span class="n">i</span><span class="o">|</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-11-15" name="__codelineno-11-15" href="#__codelineno-11-15"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="mi">1</span><span class="p">..</span><span class="n">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="o">|</span><span class="n">j</span><span class="o">|</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-11-16" name="__codelineno-11-16" href="#__codelineno-11-16"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">s</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">t</span><span class="p">[</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">])</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-11-17" name="__codelineno-11-17" href="#__codelineno-11-17"></a><span class="w">                </span><span class="c1">// 若两字符相等，则直接跳过此两字符</span>
<a id="__codelineno-11-18" name="__codelineno-11-18" href="#__codelineno-11-18"></a><span class="w">                </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">][</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">];</span>
<a id="__codelineno-11-19" name="__codelineno-11-19" href="#__codelineno-11-19"></a><span class="w">            </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-11-20" name="__codelineno-11-20" href="#__codelineno-11-20"></a><span class="w">                </span><span class="c1">// 最少编辑步数 = 插入、删除、替换这三种操作的最少编辑步数 + 1</span>
<a id="__codelineno-11-21" name="__codelineno-11-21" href="#__codelineno-11-21"></a><span class="w">                </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">@min</span><span class="p">(</span><span class="nb">@min</span><span class="p">(</span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">],</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">][</span><span class="n">j</span><span class="p">]),</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">][</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">])</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
<a id="__codelineno-11-22" name="__codelineno-11-22" href="#__codelineno-11-22"></a><span class="w">            </span><span class="p">}</span>
<a id="__codelineno-11-23" name="__codelineno-11-23" href="#__codelineno-11-23"></a><span class="w">        </span><span class="p">}</span>
<a id="__codelineno-11-24" name="__codelineno-11-24" href="#__codelineno-11-24"></a><span class="w">    </span><span class="p">}</span>
<a id="__codelineno-11-25" name="__codelineno-11-25" href="#__codelineno-11-25"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">n</span><span class="p">][</span><span class="n">m</span><span class="p">];</span>
<a id="__codelineno-11-26" name="__codelineno-11-26" href="#__codelineno-11-26"></a><span class="p">}</span>
</code></pre></div>
</div>
</div>
</div>
<details class="pythontutor">
<summary>可视化运行</summary>
<p><div style="height: 549px; width: 100%;"><iframe class="pythontutor-iframe" src="https://pythontutor.com/iframe-embed.html#code=def%20edit_distance_dp%28s%3A%20str,%20t%3A%20str%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22%E7%BC%96%E8%BE%91%E8%B7%9D%E7%A6%BB%EF%BC%9A%E5%8A%A8%E6%80%81%E8%A7%84%E5%88%92%22%22%22%0A%20%20%20%20n,%20m%20%3D%20len%28s%29,%20len%28t%29%0A%20%20%20%20dp%20%3D%20%5B%5B0%5D%20*%20%28m%20%2B%201%29%20for%20_%20in%20range%28n%20%2B%201%29%5D%0A%20%20%20%20%23%20%E7%8A%B6%E6%80%81%E8%BD%AC%E7%A7%BB%EF%BC%9A%E9%A6%96%E8%A1%8C%E9%A6%96%E5%88%97%0A%20%20%20%20for%20i%20in%20range%281,%20n%20%2B%201%29%3A%0A%20%20%20%20%20%20%20%20dp%5Bi%5D%5B0%5D%20%3D%20i%0A%20%20%20%20for%20j%20in%20range%281,%20m%20%2B%201%29%3A%0A%20%20%20%20%20%20%20%20dp%5B0%5D%5Bj%5D%20%3D%20j%0A%20%20%20%20%23%20%E7%8A%B6%E6%80%81%E8%BD%AC%E7%A7%BB%EF%BC%9A%E5%85%B6%E4%BD%99%E8%A1%8C%E5%92%8C%E5%88%97%0A%20%20%20%20for%20i%20in%20range%281,%20n%20%2B%201%29%3A%0A%20%20%20%20%20%20%20%20for%20j%20in%20range%281,%20m%20%2B%201%29%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20if%20s%5Bi%20-%201%5D%20%3D%3D%20t%5Bj%20-%201%5D%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%23%20%E8%8B%A5%E4%B8%A4%E5%AD%97%E7%AC%A6%E7%9B%B8%E7%AD%89%EF%BC%8C%E5%88%99%E7%9B%B4%E6%8E%A5%E8%B7%B3%E8%BF%87%E6%AD%A4%E4%B8%A4%E5%AD%97%E7%AC%A6%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20dp%5Bi%5D%5Bj%5D%20%3D%20dp%5Bi%20-%201%5D%5Bj%20-%201%5D%0A%20%20%20%20%20%20%20%20%20%20%20%20else%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%23%20%E6%9C%80%E5%B0%91%E7%BC%96%E8%BE%91%E6%AD%A5%E6%95%B0%20%3D%20%E6%8F%92%E5%85%A5%E3%80%81%E5%88%A0%E9%99%A4%E3%80%81%E6%9B%BF%E6%8D%A2%E8%BF%99%E4%B8%89%E7%A7%8D%E6%93%8D%E4%BD%9C%E7%9A%84%E6%9C%80%E5%B0%91%E7%BC%96%E8%BE%91%E6%AD%A5%E6%95%B0%20%2B%201%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20dp%5Bi%5D%5Bj%5D%20%3D%20min%28dp%5Bi%5D%5Bj%20-%201%5D,%20dp%5Bi%20-%201%5D%5Bj%5D,%20dp%5Bi%20-%201%5D%5Bj%20-%201%5D%29%20%2B%201%0A%20%20%20%20return%20dp%5Bn%5D%5Bm%5D%0A%0A%0A%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20s%20%3D%20%22bag%22%0A%20%20%20%20t%20%3D%20%22pack%22%0A%20%20%20%20n,%20m%20%3D%20len%28s%29,%20len%28t%29%0A%0A%20%20%20%20%23%20%E5%8A%A8%E6%80%81%E8%A7%84%E5%88%92%0A%20%20%20%20res%20%3D%20edit_distance_dp%28s,%20t%29%0A%20%20%20%20print%28f%22%E5%B0%86%20%7Bs%7D%20%E6%9B%B4%E6%94%B9%E4%B8%BA%20%7Bt%7D%20%E6%9C%80%E5%B0%91%E9%9C%80%E8%A6%81%E7%BC%96%E8%BE%91%20%7Bres%7D%20%E6%AD%A5%22%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=6&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=def%20edit_distance_dp%28s%3A%20str,%20t%3A%20str%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22%E7%BC%96%E8%BE%91%E8%B7%9D%E7%A6%BB%EF%BC%9A%E5%8A%A8%E6%80%81%E8%A7%84%E5%88%92%22%22%22%0A%20%20%20%20n,%20m%20%3D%20len%28s%29,%20len%28t%29%0A%20%20%20%20dp%20%3D%20%5B%5B0%5D%20*%20%28m%20%2B%201%29%20for%20_%20in%20range%28n%20%2B%201%29%5D%0A%20%20%20%20%23%20%E7%8A%B6%E6%80%81%E8%BD%AC%E7%A7%BB%EF%BC%9A%E9%A6%96%E8%A1%8C%E9%A6%96%E5%88%97%0A%20%20%20%20for%20i%20in%20range%281,%20n%20%2B%201%29%3A%0A%20%20%20%20%20%20%20%20dp%5Bi%5D%5B0%5D%20%3D%20i%0A%20%20%20%20for%20j%20in%20range%281,%20m%20%2B%201%29%3A%0A%20%20%20%20%20%20%20%20dp%5B0%5D%5Bj%5D%20%3D%20j%0A%20%20%20%20%23%20%E7%8A%B6%E6%80%81%E8%BD%AC%E7%A7%BB%EF%BC%9A%E5%85%B6%E4%BD%99%E8%A1%8C%E5%92%8C%E5%88%97%0A%20%20%20%20for%20i%20in%20range%281,%20n%20%2B%201%29%3A%0A%20%20%20%20%20%20%20%20for%20j%20in%20range%281,%20m%20%2B%201%29%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20if%20s%5Bi%20-%201%5D%20%3D%3D%20t%5Bj%20-%201%5D%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%23%20%E8%8B%A5%E4%B8%A4%E5%AD%97%E7%AC%A6%E7%9B%B8%E7%AD%89%EF%BC%8C%E5%88%99%E7%9B%B4%E6%8E%A5%E8%B7%B3%E8%BF%87%E6%AD%A4%E4%B8%A4%E5%AD%97%E7%AC%A6%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20dp%5Bi%5D%5Bj%5D%20%3D%20dp%5Bi%20-%201%5D%5Bj%20-%201%5D%0A%20%20%20%20%20%20%20%20%20%20%20%20else%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%23%20%E6%9C%80%E5%B0%91%E7%BC%96%E8%BE%91%E6%AD%A5%E6%95%B0%20%3D%20%E6%8F%92%E5%85%A5%E3%80%81%E5%88%A0%E9%99%A4%E3%80%81%E6%9B%BF%E6%8D%A2%E8%BF%99%E4%B8%89%E7%A7%8D%E6%93%8D%E4%BD%9C%E7%9A%84%E6%9C%80%E5%B0%91%E7%BC%96%E8%BE%91%E6%AD%A5%E6%95%B0%20%2B%201%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20dp%5Bi%5D%5Bj%5D%20%3D%20min%28dp%5Bi%5D%5Bj%20-%201%5D,%20dp%5Bi%20-%201%5D%5Bj%5D,%20dp%5Bi%20-%201%5D%5Bj%20-%201%5D%29%20%2B%201%0A%20%20%20%20return%20dp%5Bn%5D%5Bm%5D%0A%0A%0A%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20s%20%3D%20%22bag%22%0A%20%20%20%20t%20%3D%20%22pack%22%0A%20%20%20%20n,%20m%20%3D%20len%28s%29,%20len%28t%29%0A%0A%20%20%20%20%23%20%E5%8A%A8%E6%80%81%E8%A7%84%E5%88%92%0A%20%20%20%20res%20%3D%20edit_distance_dp%28s,%20t%29%0A%20%20%20%20print%28f%22%E5%B0%86%20%7Bs%7D%20%E6%9B%B4%E6%94%B9%E4%B8%BA%20%7Bt%7D%20%E6%9C%80%E5%B0%91%E9%9C%80%E8%A6%81%E7%BC%96%E8%BE%91%20%7Bres%7D%20%E6%AD%A5%22%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=6&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">全屏观看 &gt;</a></div></p>
</details>
<p>如图 14-30 所示，编辑距离问题的状态转移过程与背包问题非常类似，都可以看作填写一个二维网格的过程。</p>
<div class="tabbed-set tabbed-alternate" data-tabs="2:15"><input checked="checked" id="__tabbed_2_1" name="__tabbed_2" type="radio" /><input id="__tabbed_2_2" name="__tabbed_2" type="radio" /><input id="__tabbed_2_3" name="__tabbed_2" type="radio" /><input id="__tabbed_2_4" name="__tabbed_2" type="radio" /><input id="__tabbed_2_5" name="__tabbed_2" type="radio" /><input id="__tabbed_2_6" name="__tabbed_2" type="radio" /><input id="__tabbed_2_7" name="__tabbed_2" type="radio" /><input id="__tabbed_2_8" name="__tabbed_2" type="radio" /><input id="__tabbed_2_9" name="__tabbed_2" type="radio" /><input id="__tabbed_2_10" name="__tabbed_2" type="radio" /><input id="__tabbed_2_11" name="__tabbed_2" type="radio" /><input id="__tabbed_2_12" name="__tabbed_2" type="radio" /><input id="__tabbed_2_13" name="__tabbed_2" type="radio" /><input id="__tabbed_2_14" name="__tabbed_2" type="radio" /><input id="__tabbed_2_15" name="__tabbed_2" type="radio" /><div class="tabbed-labels"><label for="__tabbed_2_1">&lt;1&gt;</label><label for="__tabbed_2_2">&lt;2&gt;</label><label for="__tabbed_2_3">&lt;3&gt;</label><label for="__tabbed_2_4">&lt;4&gt;</label><label for="__tabbed_2_5">&lt;5&gt;</label><label for="__tabbed_2_6">&lt;6&gt;</label><label for="__tabbed_2_7">&lt;7&gt;</label><label for="__tabbed_2_8">&lt;8&gt;</label><label for="__tabbed_2_9">&lt;9&gt;</label><label for="__tabbed_2_10">&lt;10&gt;</label><label for="__tabbed_2_11">&lt;11&gt;</label><label for="__tabbed_2_12">&lt;12&gt;</label><label for="__tabbed_2_13">&lt;13&gt;</label><label for="__tabbed_2_14">&lt;14&gt;</label><label for="__tabbed_2_15">&lt;15&gt;</label></div>
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<p><a class="glightbox" href="../edit_distance_problem.assets/edit_distance_dp_step1.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="编辑距离的动态规划过程" class="animation-figure" src="../edit_distance_problem.assets/edit_distance_dp_step1.png" /></a></p>
</div>
<div class="tabbed-block">
<p><a class="glightbox" href="../edit_distance_problem.assets/edit_distance_dp_step2.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="edit_distance_dp_step2" class="animation-figure" src="../edit_distance_problem.assets/edit_distance_dp_step2.png" /></a></p>
</div>
<div class="tabbed-block">
<p><a class="glightbox" href="../edit_distance_problem.assets/edit_distance_dp_step3.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="edit_distance_dp_step3" class="animation-figure" src="../edit_distance_problem.assets/edit_distance_dp_step3.png" /></a></p>
</div>
<div class="tabbed-block">
<p><a class="glightbox" href="../edit_distance_problem.assets/edit_distance_dp_step4.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="edit_distance_dp_step4" class="animation-figure" src="../edit_distance_problem.assets/edit_distance_dp_step4.png" /></a></p>
</div>
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<p><a class="glightbox" href="../edit_distance_problem.assets/edit_distance_dp_step5.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="edit_distance_dp_step5" class="animation-figure" src="../edit_distance_problem.assets/edit_distance_dp_step5.png" /></a></p>
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<p><a class="glightbox" href="../edit_distance_problem.assets/edit_distance_dp_step6.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="edit_distance_dp_step6" class="animation-figure" src="../edit_distance_problem.assets/edit_distance_dp_step6.png" /></a></p>
</div>
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<p><a class="glightbox" href="../edit_distance_problem.assets/edit_distance_dp_step7.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="edit_distance_dp_step7" class="animation-figure" src="../edit_distance_problem.assets/edit_distance_dp_step7.png" /></a></p>
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<p><a class="glightbox" href="../edit_distance_problem.assets/edit_distance_dp_step8.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="edit_distance_dp_step8" class="animation-figure" src="../edit_distance_problem.assets/edit_distance_dp_step8.png" /></a></p>
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<p><a class="glightbox" href="../edit_distance_problem.assets/edit_distance_dp_step9.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="edit_distance_dp_step9" class="animation-figure" src="../edit_distance_problem.assets/edit_distance_dp_step9.png" /></a></p>
</div>
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<p><a class="glightbox" href="../edit_distance_problem.assets/edit_distance_dp_step10.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="edit_distance_dp_step10" class="animation-figure" src="../edit_distance_problem.assets/edit_distance_dp_step10.png" /></a></p>
</div>
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<p><a class="glightbox" href="../edit_distance_problem.assets/edit_distance_dp_step11.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="edit_distance_dp_step11" class="animation-figure" src="../edit_distance_problem.assets/edit_distance_dp_step11.png" /></a></p>
</div>
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<p><a class="glightbox" href="../edit_distance_problem.assets/edit_distance_dp_step12.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="edit_distance_dp_step12" class="animation-figure" src="../edit_distance_problem.assets/edit_distance_dp_step12.png" /></a></p>
</div>
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<p><a class="glightbox" href="../edit_distance_problem.assets/edit_distance_dp_step13.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="edit_distance_dp_step13" class="animation-figure" src="../edit_distance_problem.assets/edit_distance_dp_step13.png" /></a></p>
</div>
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<p><a class="glightbox" href="../edit_distance_problem.assets/edit_distance_dp_step14.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="edit_distance_dp_step14" class="animation-figure" src="../edit_distance_problem.assets/edit_distance_dp_step14.png" /></a></p>
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<p><a class="glightbox" href="../edit_distance_problem.assets/edit_distance_dp_step15.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="edit_distance_dp_step15" class="animation-figure" src="../edit_distance_problem.assets/edit_distance_dp_step15.png" /></a></p>
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</div>
<p align="center"> 图 14-30 &nbsp; 编辑距离的动态规划过程 </p>

<h3 id="3">3. &nbsp; 空间优化<a class="headerlink" href="#3" title="Permanent link">&para;</a></h3>
<p>由于 <span class="arithmatex">\(dp[i,j]\)</span> 是由上方 <span class="arithmatex">\(dp[i-1, j]\)</span>、左方 <span class="arithmatex">\(dp[i, j-1]\)</span>、左上方 <span class="arithmatex">\(dp[i-1, j-1]\)</span> 转移而来的，而正序遍历会丢失左上方 <span class="arithmatex">\(dp[i-1, j-1]\)</span> ，倒序遍历无法提前构建 <span class="arithmatex">\(dp[i, j-1]\)</span> ，因此两种遍历顺序都不可取。</p>
<p>为此，我们可以使用一个变量 <code>leftup</code> 来暂存左上方的解 <span class="arithmatex">\(dp[i-1, j-1]\)</span> ，从而只需考虑左方和上方的解。此时的情况与完全背包问题相同，可使用正序遍历。代码如下所示：</p>
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<div class="highlight"><span class="filename">edit_distance.py</span><pre><span></span><code><a id="__codelineno-12-1" name="__codelineno-12-1" href="#__codelineno-12-1"></a><span class="k">def</span> <span class="nf">edit_distance_dp_comp</span><span class="p">(</span><span class="n">s</span><span class="p">:</span> <span class="nb">str</span><span class="p">,</span> <span class="n">t</span><span class="p">:</span> <span class="nb">str</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="nb">int</span><span class="p">:</span>
<a id="__codelineno-12-2" name="__codelineno-12-2" href="#__codelineno-12-2"></a><span class="w">    </span><span class="sd">&quot;&quot;&quot;编辑距离：空间优化后的动态规划&quot;&quot;&quot;</span>
<a id="__codelineno-12-3" name="__codelineno-12-3" href="#__codelineno-12-3"></a>    <span class="n">n</span><span class="p">,</span> <span class="n">m</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">s</span><span class="p">),</span> <span class="nb">len</span><span class="p">(</span><span class="n">t</span><span class="p">)</span>
<a id="__codelineno-12-4" name="__codelineno-12-4" href="#__codelineno-12-4"></a>    <span class="n">dp</span> <span class="o">=</span> <span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">*</span> <span class="p">(</span><span class="n">m</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
<a id="__codelineno-12-5" name="__codelineno-12-5" href="#__codelineno-12-5"></a>    <span class="c1"># 状态转移：首行</span>
<a id="__codelineno-12-6" name="__codelineno-12-6" href="#__codelineno-12-6"></a>    <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">m</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
<a id="__codelineno-12-7" name="__codelineno-12-7" href="#__codelineno-12-7"></a>        <span class="n">dp</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="n">j</span>
<a id="__codelineno-12-8" name="__codelineno-12-8" href="#__codelineno-12-8"></a>    <span class="c1"># 状态转移：其余行</span>
<a id="__codelineno-12-9" name="__codelineno-12-9" href="#__codelineno-12-9"></a>    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
<a id="__codelineno-12-10" name="__codelineno-12-10" href="#__codelineno-12-10"></a>        <span class="c1"># 状态转移：首列</span>
<a id="__codelineno-12-11" name="__codelineno-12-11" href="#__codelineno-12-11"></a>        <span class="n">leftup</span> <span class="o">=</span> <span class="n">dp</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>  <span class="c1"># 暂存 dp[i-1, j-1]</span>
<a id="__codelineno-12-12" name="__codelineno-12-12" href="#__codelineno-12-12"></a>        <span class="n">dp</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">+=</span> <span class="mi">1</span>
<a id="__codelineno-12-13" name="__codelineno-12-13" href="#__codelineno-12-13"></a>        <span class="c1"># 状态转移：其余列</span>
<a id="__codelineno-12-14" name="__codelineno-12-14" href="#__codelineno-12-14"></a>        <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">m</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
<a id="__codelineno-12-15" name="__codelineno-12-15" href="#__codelineno-12-15"></a>            <span class="n">temp</span> <span class="o">=</span> <span class="n">dp</span><span class="p">[</span><span class="n">j</span><span class="p">]</span>
<a id="__codelineno-12-16" name="__codelineno-12-16" href="#__codelineno-12-16"></a>            <span class="k">if</span> <span class="n">s</span><span class="p">[</span><span class="n">i</span> <span class="o">-</span> <span class="mi">1</span><span class="p">]</span> <span class="o">==</span> <span class="n">t</span><span class="p">[</span><span class="n">j</span> <span class="o">-</span> <span class="mi">1</span><span class="p">]:</span>
<a id="__codelineno-12-17" name="__codelineno-12-17" href="#__codelineno-12-17"></a>                <span class="c1"># 若两字符相等，则直接跳过此两字符</span>
<a id="__codelineno-12-18" name="__codelineno-12-18" href="#__codelineno-12-18"></a>                <span class="n">dp</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="n">leftup</span>
<a id="__codelineno-12-19" name="__codelineno-12-19" href="#__codelineno-12-19"></a>            <span class="k">else</span><span class="p">:</span>
<a id="__codelineno-12-20" name="__codelineno-12-20" href="#__codelineno-12-20"></a>                <span class="c1"># 最少编辑步数 = 插入、删除、替换这三种操作的最少编辑步数 + 1</span>
<a id="__codelineno-12-21" name="__codelineno-12-21" href="#__codelineno-12-21"></a>                <span class="n">dp</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="nb">min</span><span class="p">(</span><span class="n">dp</span><span class="p">[</span><span class="n">j</span> <span class="o">-</span> <span class="mi">1</span><span class="p">],</span> <span class="n">dp</span><span class="p">[</span><span class="n">j</span><span class="p">],</span> <span class="n">leftup</span><span class="p">)</span> <span class="o">+</span> <span class="mi">1</span>
<a id="__codelineno-12-22" name="__codelineno-12-22" href="#__codelineno-12-22"></a>            <span class="n">leftup</span> <span class="o">=</span> <span class="n">temp</span>  <span class="c1"># 更新为下一轮的 dp[i-1, j-1]</span>
<a id="__codelineno-12-23" name="__codelineno-12-23" href="#__codelineno-12-23"></a>    <span class="k">return</span> <span class="n">dp</span><span class="p">[</span><span class="n">m</span><span class="p">]</span>
</code></pre></div>
</div>
<div class="tabbed-block">
<div class="highlight"><span class="filename">edit_distance.cpp</span><pre><span></span><code><a id="__codelineno-13-1" name="__codelineno-13-1" href="#__codelineno-13-1"></a><span class="cm">/* 编辑距离：空间优化后的动态规划 */</span>
<a id="__codelineno-13-2" name="__codelineno-13-2" href="#__codelineno-13-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">editDistanceDPComp</span><span class="p">(</span><span class="n">string</span><span class="w"> </span><span class="n">s</span><span class="p">,</span><span class="w"> </span><span class="n">string</span><span class="w"> </span><span class="n">t</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-13-3" name="__codelineno-13-3" href="#__codelineno-13-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">s</span><span class="p">.</span><span class="n">length</span><span class="p">(),</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">t</span><span class="p">.</span><span class="n">length</span><span class="p">();</span>
<a id="__codelineno-13-4" name="__codelineno-13-4" href="#__codelineno-13-4"></a><span class="w">    </span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">dp</span><span class="p">(</span><span class="n">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="mi">0</span><span class="p">);</span>
<a id="__codelineno-13-5" name="__codelineno-13-5" href="#__codelineno-13-5"></a><span class="w">    </span><span class="c1">// 状态转移：首行</span>
<a id="__codelineno-13-6" name="__codelineno-13-6" href="#__codelineno-13-6"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">m</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-13-7" name="__codelineno-13-7" href="#__codelineno-13-7"></a><span class="w">        </span><span class="n">dp</span><span class="p">[</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">j</span><span class="p">;</span>
<a id="__codelineno-13-8" name="__codelineno-13-8" href="#__codelineno-13-8"></a><span class="w">    </span><span class="p">}</span>
<a id="__codelineno-13-9" name="__codelineno-13-9" href="#__codelineno-13-9"></a><span class="w">    </span><span class="c1">// 状态转移：其余行</span>
<a id="__codelineno-13-10" name="__codelineno-13-10" href="#__codelineno-13-10"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-13-11" name="__codelineno-13-11" href="#__codelineno-13-11"></a><span class="w">        </span><span class="c1">// 状态转移：首列</span>
<a id="__codelineno-13-12" name="__codelineno-13-12" href="#__codelineno-13-12"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">leftup</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="mi">0</span><span class="p">];</span><span class="w"> </span><span class="c1">// 暂存 dp[i-1, j-1]</span>
<a id="__codelineno-13-13" name="__codelineno-13-13" href="#__codelineno-13-13"></a><span class="w">        </span><span class="n">dp</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">i</span><span class="p">;</span>
<a id="__codelineno-13-14" name="__codelineno-13-14" href="#__codelineno-13-14"></a><span class="w">        </span><span class="c1">// 状态转移：其余列</span>
<a id="__codelineno-13-15" name="__codelineno-13-15" href="#__codelineno-13-15"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">m</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-13-16" name="__codelineno-13-16" href="#__codelineno-13-16"></a><span class="w">            </span><span class="kt">int</span><span class="w"> </span><span class="n">temp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">j</span><span class="p">];</span>
<a id="__codelineno-13-17" name="__codelineno-13-17" href="#__codelineno-13-17"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">s</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">t</span><span class="p">[</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">])</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-13-18" name="__codelineno-13-18" href="#__codelineno-13-18"></a><span class="w">                </span><span class="c1">// 若两字符相等，则直接跳过此两字符</span>
<a id="__codelineno-13-19" name="__codelineno-13-19" href="#__codelineno-13-19"></a><span class="w">                </span><span class="n">dp</span><span class="p">[</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">leftup</span><span class="p">;</span>
<a id="__codelineno-13-20" name="__codelineno-13-20" href="#__codelineno-13-20"></a><span class="w">            </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-13-21" name="__codelineno-13-21" href="#__codelineno-13-21"></a><span class="w">                </span><span class="c1">// 最少编辑步数 = 插入、删除、替换这三种操作的最少编辑步数 + 1</span>
<a id="__codelineno-13-22" name="__codelineno-13-22" href="#__codelineno-13-22"></a><span class="w">                </span><span class="n">dp</span><span class="p">[</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">min</span><span class="p">(</span><span class="n">min</span><span class="p">(</span><span class="n">dp</span><span class="p">[</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">],</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">j</span><span class="p">]),</span><span class="w"> </span><span class="n">leftup</span><span class="p">)</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
<a id="__codelineno-13-23" name="__codelineno-13-23" href="#__codelineno-13-23"></a><span class="w">            </span><span class="p">}</span>
<a id="__codelineno-13-24" name="__codelineno-13-24" href="#__codelineno-13-24"></a><span class="w">            </span><span class="n">leftup</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">temp</span><span class="p">;</span><span class="w"> </span><span class="c1">// 更新为下一轮的 dp[i-1, j-1]</span>
<a id="__codelineno-13-25" name="__codelineno-13-25" href="#__codelineno-13-25"></a><span class="w">        </span><span class="p">}</span>
<a id="__codelineno-13-26" name="__codelineno-13-26" href="#__codelineno-13-26"></a><span class="w">    </span><span class="p">}</span>
<a id="__codelineno-13-27" name="__codelineno-13-27" href="#__codelineno-13-27"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">m</span><span class="p">];</span>
<a id="__codelineno-13-28" name="__codelineno-13-28" href="#__codelineno-13-28"></a><span class="p">}</span>
</code></pre></div>
</div>
<div class="tabbed-block">
<div class="highlight"><span class="filename">edit_distance.java</span><pre><span></span><code><a id="__codelineno-14-1" name="__codelineno-14-1" href="#__codelineno-14-1"></a><span class="cm">/* 编辑距离：空间优化后的动态规划 */</span>
<a id="__codelineno-14-2" name="__codelineno-14-2" href="#__codelineno-14-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">editDistanceDPComp</span><span class="p">(</span><span class="n">String</span><span class="w"> </span><span class="n">s</span><span class="p">,</span><span class="w"> </span><span class="n">String</span><span class="w"> </span><span class="n">t</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-14-3" name="__codelineno-14-3" href="#__codelineno-14-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">s</span><span class="p">.</span><span class="na">length</span><span class="p">(),</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">t</span><span class="p">.</span><span class="na">length</span><span class="p">();</span>
<a id="__codelineno-14-4" name="__codelineno-14-4" href="#__codelineno-14-4"></a><span class="w">    </span><span class="kt">int</span><span class="o">[]</span><span class="w"> </span><span class="n">dp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="kt">int</span><span class="o">[</span><span class="n">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="o">]</span><span class="p">;</span>
<a id="__codelineno-14-5" name="__codelineno-14-5" href="#__codelineno-14-5"></a><span class="w">    </span><span class="c1">// 状态转移：首行</span>
<a id="__codelineno-14-6" name="__codelineno-14-6" href="#__codelineno-14-6"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">m</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-14-7" name="__codelineno-14-7" href="#__codelineno-14-7"></a><span class="w">        </span><span class="n">dp</span><span class="o">[</span><span class="n">j</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">j</span><span class="p">;</span>
<a id="__codelineno-14-8" name="__codelineno-14-8" href="#__codelineno-14-8"></a><span class="w">    </span><span class="p">}</span>
<a id="__codelineno-14-9" name="__codelineno-14-9" href="#__codelineno-14-9"></a><span class="w">    </span><span class="c1">// 状态转移：其余行</span>
<a id="__codelineno-14-10" name="__codelineno-14-10" href="#__codelineno-14-10"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-14-11" name="__codelineno-14-11" href="#__codelineno-14-11"></a><span class="w">        </span><span class="c1">// 状态转移：首列</span>
<a id="__codelineno-14-12" name="__codelineno-14-12" href="#__codelineno-14-12"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">leftup</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="o">[</span><span class="mi">0</span><span class="o">]</span><span class="p">;</span><span class="w"> </span><span class="c1">// 暂存 dp[i-1, j-1]</span>
<a id="__codelineno-14-13" name="__codelineno-14-13" href="#__codelineno-14-13"></a><span class="w">        </span><span class="n">dp</span><span class="o">[</span><span class="mi">0</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">i</span><span class="p">;</span>
<a id="__codelineno-14-14" name="__codelineno-14-14" href="#__codelineno-14-14"></a><span class="w">        </span><span class="c1">// 状态转移：其余列</span>
<a id="__codelineno-14-15" name="__codelineno-14-15" href="#__codelineno-14-15"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">m</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-14-16" name="__codelineno-14-16" href="#__codelineno-14-16"></a><span class="w">            </span><span class="kt">int</span><span class="w"> </span><span class="n">temp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="o">[</span><span class="n">j</span><span class="o">]</span><span class="p">;</span>
<a id="__codelineno-14-17" name="__codelineno-14-17" href="#__codelineno-14-17"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">s</span><span class="p">.</span><span class="na">charAt</span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">t</span><span class="p">.</span><span class="na">charAt</span><span class="p">(</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">))</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-14-18" name="__codelineno-14-18" href="#__codelineno-14-18"></a><span class="w">                </span><span class="c1">// 若两字符相等，则直接跳过此两字符</span>
<a id="__codelineno-14-19" name="__codelineno-14-19" href="#__codelineno-14-19"></a><span class="w">                </span><span class="n">dp</span><span class="o">[</span><span class="n">j</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">leftup</span><span class="p">;</span>
<a id="__codelineno-14-20" name="__codelineno-14-20" href="#__codelineno-14-20"></a><span class="w">            </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-14-21" name="__codelineno-14-21" href="#__codelineno-14-21"></a><span class="w">                </span><span class="c1">// 最少编辑步数 = 插入、删除、替换这三种操作的最少编辑步数 + 1</span>
<a id="__codelineno-14-22" name="__codelineno-14-22" href="#__codelineno-14-22"></a><span class="w">                </span><span class="n">dp</span><span class="o">[</span><span class="n">j</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">Math</span><span class="p">.</span><span class="na">min</span><span class="p">(</span><span class="n">Math</span><span class="p">.</span><span class="na">min</span><span class="p">(</span><span class="n">dp</span><span class="o">[</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="o">]</span><span class="p">,</span><span class="w"> </span><span class="n">dp</span><span class="o">[</span><span class="n">j</span><span class="o">]</span><span class="p">),</span><span class="w"> </span><span class="n">leftup</span><span class="p">)</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
<a id="__codelineno-14-23" name="__codelineno-14-23" href="#__codelineno-14-23"></a><span class="w">            </span><span class="p">}</span>
<a id="__codelineno-14-24" name="__codelineno-14-24" href="#__codelineno-14-24"></a><span class="w">            </span><span class="n">leftup</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">temp</span><span class="p">;</span><span class="w"> </span><span class="c1">// 更新为下一轮的 dp[i-1, j-1]</span>
<a id="__codelineno-14-25" name="__codelineno-14-25" href="#__codelineno-14-25"></a><span class="w">        </span><span class="p">}</span>
<a id="__codelineno-14-26" name="__codelineno-14-26" href="#__codelineno-14-26"></a><span class="w">    </span><span class="p">}</span>
<a id="__codelineno-14-27" name="__codelineno-14-27" href="#__codelineno-14-27"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">dp</span><span class="o">[</span><span class="n">m</span><span class="o">]</span><span class="p">;</span>
<a id="__codelineno-14-28" name="__codelineno-14-28" href="#__codelineno-14-28"></a><span class="p">}</span>
</code></pre></div>
</div>
<div class="tabbed-block">
<div class="highlight"><span class="filename">edit_distance.cs</span><pre><span></span><code><a id="__codelineno-15-1" name="__codelineno-15-1" href="#__codelineno-15-1"></a><span class="cm">/* 编辑距离：空间优化后的动态规划 */</span>
<a id="__codelineno-15-2" name="__codelineno-15-2" href="#__codelineno-15-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">EditDistanceDPComp</span><span class="p">(</span><span class="kt">string</span><span class="w"> </span><span class="n">s</span><span class="p">,</span><span class="w"> </span><span class="kt">string</span><span class="w"> </span><span class="n">t</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-15-3" name="__codelineno-15-3" href="#__codelineno-15-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">s</span><span class="p">.</span><span class="n">Length</span><span class="p">,</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">t</span><span class="p">.</span><span class="n">Length</span><span class="p">;</span>
<a id="__codelineno-15-4" name="__codelineno-15-4" href="#__codelineno-15-4"></a><span class="w">    </span><span class="kt">int</span><span class="p">[]</span><span class="w"> </span><span class="n">dp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="kt">int</span><span class="p">[</span><span class="n">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span><span class="p">];</span>
<a id="__codelineno-15-5" name="__codelineno-15-5" href="#__codelineno-15-5"></a><span class="w">    </span><span class="c1">// 状态转移：首行</span>
<a id="__codelineno-15-6" name="__codelineno-15-6" href="#__codelineno-15-6"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">1</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">m</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-15-7" name="__codelineno-15-7" href="#__codelineno-15-7"></a><span class="w">        </span><span class="n">dp</span><span class="p">[</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">j</span><span class="p">;</span>
<a id="__codelineno-15-8" name="__codelineno-15-8" href="#__codelineno-15-8"></a><span class="w">    </span><span class="p">}</span>
<a id="__codelineno-15-9" name="__codelineno-15-9" href="#__codelineno-15-9"></a><span class="w">    </span><span class="c1">// 状态转移：其余行</span>
<a id="__codelineno-15-10" name="__codelineno-15-10" href="#__codelineno-15-10"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-15-11" name="__codelineno-15-11" href="#__codelineno-15-11"></a><span class="w">        </span><span class="c1">// 状态转移：首列</span>
<a id="__codelineno-15-12" name="__codelineno-15-12" href="#__codelineno-15-12"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">leftup</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="m">0</span><span class="p">];</span><span class="w"> </span><span class="c1">// 暂存 dp[i-1, j-1]</span>
<a id="__codelineno-15-13" name="__codelineno-15-13" href="#__codelineno-15-13"></a><span class="w">        </span><span class="n">dp</span><span class="p">[</span><span class="m">0</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">i</span><span class="p">;</span>
<a id="__codelineno-15-14" name="__codelineno-15-14" href="#__codelineno-15-14"></a><span class="w">        </span><span class="c1">// 状态转移：其余列</span>
<a id="__codelineno-15-15" name="__codelineno-15-15" href="#__codelineno-15-15"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">1</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">m</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-15-16" name="__codelineno-15-16" href="#__codelineno-15-16"></a><span class="w">            </span><span class="kt">int</span><span class="w"> </span><span class="n">temp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">j</span><span class="p">];</span>
<a id="__codelineno-15-17" name="__codelineno-15-17" href="#__codelineno-15-17"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">s</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">]</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">t</span><span class="p">[</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">])</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-15-18" name="__codelineno-15-18" href="#__codelineno-15-18"></a><span class="w">                </span><span class="c1">// 若两字符相等，则直接跳过此两字符</span>
<a id="__codelineno-15-19" name="__codelineno-15-19" href="#__codelineno-15-19"></a><span class="w">                </span><span class="n">dp</span><span class="p">[</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">leftup</span><span class="p">;</span>
<a id="__codelineno-15-20" name="__codelineno-15-20" href="#__codelineno-15-20"></a><span class="w">            </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-15-21" name="__codelineno-15-21" href="#__codelineno-15-21"></a><span class="w">                </span><span class="c1">// 最少编辑步数 = 插入、删除、替换这三种操作的最少编辑步数 + 1</span>
<a id="__codelineno-15-22" name="__codelineno-15-22" href="#__codelineno-15-22"></a><span class="w">                </span><span class="n">dp</span><span class="p">[</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">Math</span><span class="p">.</span><span class="n">Min</span><span class="p">(</span><span class="n">Math</span><span class="p">.</span><span class="n">Min</span><span class="p">(</span><span class="n">dp</span><span class="p">[</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">],</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">j</span><span class="p">]),</span><span class="w"> </span><span class="n">leftup</span><span class="p">)</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span><span class="p">;</span>
<a id="__codelineno-15-23" name="__codelineno-15-23" href="#__codelineno-15-23"></a><span class="w">            </span><span class="p">}</span>
<a id="__codelineno-15-24" name="__codelineno-15-24" href="#__codelineno-15-24"></a><span class="w">            </span><span class="n">leftup</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">temp</span><span class="p">;</span><span class="w"> </span><span class="c1">// 更新为下一轮的 dp[i-1, j-1]</span>
<a id="__codelineno-15-25" name="__codelineno-15-25" href="#__codelineno-15-25"></a><span class="w">        </span><span class="p">}</span>
<a id="__codelineno-15-26" name="__codelineno-15-26" href="#__codelineno-15-26"></a><span class="w">    </span><span class="p">}</span>
<a id="__codelineno-15-27" name="__codelineno-15-27" href="#__codelineno-15-27"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">m</span><span class="p">];</span>
<a id="__codelineno-15-28" name="__codelineno-15-28" href="#__codelineno-15-28"></a><span class="p">}</span>
</code></pre></div>
</div>
<div class="tabbed-block">
<div class="highlight"><span class="filename">edit_distance.go</span><pre><span></span><code><a id="__codelineno-16-1" name="__codelineno-16-1" href="#__codelineno-16-1"></a><span class="cm">/* 编辑距离：空间优化后的动态规划 */</span>
<a id="__codelineno-16-2" name="__codelineno-16-2" href="#__codelineno-16-2"></a><span class="kd">func</span><span class="w"> </span><span class="nx">editDistanceDPComp</span><span class="p">(</span><span class="nx">s</span><span class="w"> </span><span class="kt">string</span><span class="p">,</span><span class="w"> </span><span class="nx">t</span><span class="w"> </span><span class="kt">string</span><span class="p">)</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-16-3" name="__codelineno-16-3" href="#__codelineno-16-3"></a><span class="w">    </span><span class="nx">n</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nb">len</span><span class="p">(</span><span class="nx">s</span><span class="p">)</span>
<a id="__codelineno-16-4" name="__codelineno-16-4" href="#__codelineno-16-4"></a><span class="w">    </span><span class="nx">m</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nb">len</span><span class="p">(</span><span class="nx">t</span><span class="p">)</span>
<a id="__codelineno-16-5" name="__codelineno-16-5" href="#__codelineno-16-5"></a><span class="w">    </span><span class="nx">dp</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nb">make</span><span class="p">([]</span><span class="kt">int</span><span class="p">,</span><span class="w"> </span><span class="nx">m</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span>
<a id="__codelineno-16-6" name="__codelineno-16-6" href="#__codelineno-16-6"></a><span class="w">    </span><span class="c1">// 状态转移：首行</span>
<a id="__codelineno-16-7" name="__codelineno-16-7" href="#__codelineno-16-7"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">m</span><span class="p">;</span><span class="w"> </span><span class="nx">j</span><span class="o">++</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-16-8" name="__codelineno-16-8" href="#__codelineno-16-8"></a><span class="w">        </span><span class="nx">dp</span><span class="p">[</span><span class="nx">j</span><span class="p">]</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="nx">j</span>
<a id="__codelineno-16-9" name="__codelineno-16-9" href="#__codelineno-16-9"></a><span class="w">    </span><span class="p">}</span>
<a id="__codelineno-16-10" name="__codelineno-16-10" href="#__codelineno-16-10"></a><span class="w">    </span><span class="c1">// 状态转移：其余行</span>
<a id="__codelineno-16-11" name="__codelineno-16-11" href="#__codelineno-16-11"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-16-12" name="__codelineno-16-12" href="#__codelineno-16-12"></a><span class="w">        </span><span class="c1">// 状态转移：首列</span>
<a id="__codelineno-16-13" name="__codelineno-16-13" href="#__codelineno-16-13"></a><span class="w">        </span><span class="nx">leftUp</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nx">dp</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="w"> </span><span class="c1">// 暂存 dp[i-1, j-1]</span>
<a id="__codelineno-16-14" name="__codelineno-16-14" href="#__codelineno-16-14"></a><span class="w">        </span><span class="nx">dp</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="nx">i</span>
<a id="__codelineno-16-15" name="__codelineno-16-15" href="#__codelineno-16-15"></a><span class="w">        </span><span class="c1">// 状态转移：其余列</span>
<a id="__codelineno-16-16" name="__codelineno-16-16" href="#__codelineno-16-16"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">m</span><span class="p">;</span><span class="w"> </span><span class="nx">j</span><span class="o">++</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-16-17" name="__codelineno-16-17" href="#__codelineno-16-17"></a><span class="w">            </span><span class="nx">temp</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nx">dp</span><span class="p">[</span><span class="nx">j</span><span class="p">]</span>
<a id="__codelineno-16-18" name="__codelineno-16-18" href="#__codelineno-16-18"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="nx">s</span><span class="p">[</span><span class="nx">i</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="nx">t</span><span class="p">[</span><span class="nx">j</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-16-19" name="__codelineno-16-19" href="#__codelineno-16-19"></a><span class="w">                </span><span class="c1">// 若两字符相等，则直接跳过此两字符</span>
<a id="__codelineno-16-20" name="__codelineno-16-20" href="#__codelineno-16-20"></a><span class="w">                </span><span class="nx">dp</span><span class="p">[</span><span class="nx">j</span><span class="p">]</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="nx">leftUp</span>
<a id="__codelineno-16-21" name="__codelineno-16-21" href="#__codelineno-16-21"></a><span class="w">            </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-16-22" name="__codelineno-16-22" href="#__codelineno-16-22"></a><span class="w">                </span><span class="c1">// 最少编辑步数 = 插入、删除、替换这三种操作的最少编辑步数 + 1</span>
<a id="__codelineno-16-23" name="__codelineno-16-23" href="#__codelineno-16-23"></a><span class="w">                </span><span class="nx">dp</span><span class="p">[</span><span class="nx">j</span><span class="p">]</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="nx">MinInt</span><span class="p">(</span><span class="nx">MinInt</span><span class="p">(</span><span class="nx">dp</span><span class="p">[</span><span class="nx">j</span><span class="o">-</span><span class="mi">1</span><span class="p">],</span><span class="w"> </span><span class="nx">dp</span><span class="p">[</span><span class="nx">j</span><span class="p">]),</span><span class="w"> </span><span class="nx">leftUp</span><span class="p">)</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span>
<a id="__codelineno-16-24" name="__codelineno-16-24" href="#__codelineno-16-24"></a><span class="w">            </span><span class="p">}</span>
<a id="__codelineno-16-25" name="__codelineno-16-25" href="#__codelineno-16-25"></a><span class="w">            </span><span class="nx">leftUp</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="nx">temp</span><span class="w"> </span><span class="c1">// 更新为下一轮的 dp[i-1, j-1]</span>
<a id="__codelineno-16-26" name="__codelineno-16-26" href="#__codelineno-16-26"></a><span class="w">        </span><span class="p">}</span>
<a id="__codelineno-16-27" name="__codelineno-16-27" href="#__codelineno-16-27"></a><span class="w">    </span><span class="p">}</span>
<a id="__codelineno-16-28" name="__codelineno-16-28" href="#__codelineno-16-28"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">dp</span><span class="p">[</span><span class="nx">m</span><span class="p">]</span>
<a id="__codelineno-16-29" name="__codelineno-16-29" href="#__codelineno-16-29"></a><span class="p">}</span>
</code></pre></div>
</div>
<div class="tabbed-block">
<div class="highlight"><span class="filename">edit_distance.swift</span><pre><span></span><code><a id="__codelineno-17-1" name="__codelineno-17-1" href="#__codelineno-17-1"></a><span class="cm">/* 编辑距离：空间优化后的动态规划 */</span>
<a id="__codelineno-17-2" name="__codelineno-17-2" href="#__codelineno-17-2"></a><span class="kd">func</span> <span class="nf">editDistanceDPComp</span><span class="p">(</span><span class="n">s</span><span class="p">:</span> <span class="nb">String</span><span class="p">,</span> <span class="n">t</span><span class="p">:</span> <span class="nb">String</span><span class="p">)</span> <span class="p">-&gt;</span> <span class="nb">Int</span> <span class="p">{</span>
<a id="__codelineno-17-3" name="__codelineno-17-3" href="#__codelineno-17-3"></a>    <span class="kd">let</span> <span class="nv">n</span> <span class="p">=</span> <span class="n">s</span><span class="p">.</span><span class="n">utf8CString</span><span class="p">.</span><span class="bp">count</span>
<a id="__codelineno-17-4" name="__codelineno-17-4" href="#__codelineno-17-4"></a>    <span class="kd">let</span> <span class="nv">m</span> <span class="p">=</span> <span class="n">t</span><span class="p">.</span><span class="n">utf8CString</span><span class="p">.</span><span class="bp">count</span>
<a id="__codelineno-17-5" name="__codelineno-17-5" href="#__codelineno-17-5"></a>    <span class="kd">var</span> <span class="nv">dp</span> <span class="p">=</span> <span class="nb">Array</span><span class="p">(</span><span class="n">repeating</span><span class="p">:</span> <span class="mi">0</span><span class="p">,</span> <span class="bp">count</span><span class="p">:</span> <span class="n">m</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
<a id="__codelineno-17-6" name="__codelineno-17-6" href="#__codelineno-17-6"></a>    <span class="c1">// 状态转移：首行</span>
<a id="__codelineno-17-7" name="__codelineno-17-7" href="#__codelineno-17-7"></a>    <span class="k">for</span> <span class="n">j</span> <span class="k">in</span> <span class="bp">stride</span><span class="p">(</span><span class="n">from</span><span class="p">:</span> <span class="mi">1</span><span class="p">,</span> <span class="n">through</span><span class="p">:</span> <span class="n">m</span><span class="p">,</span> <span class="n">by</span><span class="p">:</span> <span class="mi">1</span><span class="p">)</span> <span class="p">{</span>
<a id="__codelineno-17-8" name="__codelineno-17-8" href="#__codelineno-17-8"></a>        <span class="n">dp</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="p">=</span> <span class="n">j</span>
<a id="__codelineno-17-9" name="__codelineno-17-9" href="#__codelineno-17-9"></a>    <span class="p">}</span>
<a id="__codelineno-17-10" name="__codelineno-17-10" href="#__codelineno-17-10"></a>    <span class="c1">// 状态转移：其余行</span>
<a id="__codelineno-17-11" name="__codelineno-17-11" href="#__codelineno-17-11"></a>    <span class="k">for</span> <span class="n">i</span> <span class="k">in</span> <span class="bp">stride</span><span class="p">(</span><span class="n">from</span><span class="p">:</span> <span class="mi">1</span><span class="p">,</span> <span class="n">through</span><span class="p">:</span> <span class="n">n</span><span class="p">,</span> <span class="n">by</span><span class="p">:</span> <span class="mi">1</span><span class="p">)</span> <span class="p">{</span>
<a id="__codelineno-17-12" name="__codelineno-17-12" href="#__codelineno-17-12"></a>        <span class="c1">// 状态转移：首列</span>
<a id="__codelineno-17-13" name="__codelineno-17-13" href="#__codelineno-17-13"></a>        <span class="kd">var</span> <span class="nv">leftup</span> <span class="p">=</span> <span class="n">dp</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="c1">// 暂存 dp[i-1, j-1]</span>
<a id="__codelineno-17-14" name="__codelineno-17-14" href="#__codelineno-17-14"></a>        <span class="n">dp</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="p">=</span> <span class="n">i</span>
<a id="__codelineno-17-15" name="__codelineno-17-15" href="#__codelineno-17-15"></a>        <span class="c1">// 状态转移：其余列</span>
<a id="__codelineno-17-16" name="__codelineno-17-16" href="#__codelineno-17-16"></a>        <span class="k">for</span> <span class="n">j</span> <span class="k">in</span> <span class="bp">stride</span><span class="p">(</span><span class="n">from</span><span class="p">:</span> <span class="mi">1</span><span class="p">,</span> <span class="n">through</span><span class="p">:</span> <span class="n">m</span><span class="p">,</span> <span class="n">by</span><span class="p">:</span> <span class="mi">1</span><span class="p">)</span> <span class="p">{</span>
<a id="__codelineno-17-17" name="__codelineno-17-17" href="#__codelineno-17-17"></a>            <span class="kd">let</span> <span class="nv">temp</span> <span class="p">=</span> <span class="n">dp</span><span class="p">[</span><span class="n">j</span><span class="p">]</span>
<a id="__codelineno-17-18" name="__codelineno-17-18" href="#__codelineno-17-18"></a>            <span class="k">if</span> <span class="n">s</span><span class="p">.</span><span class="n">utf8CString</span><span class="p">[</span><span class="n">i</span> <span class="o">-</span> <span class="mi">1</span><span class="p">]</span> <span class="p">==</span> <span class="n">t</span><span class="p">.</span><span class="n">utf8CString</span><span class="p">[</span><span class="n">j</span> <span class="o">-</span> <span class="mi">1</span><span class="p">]</span> <span class="p">{</span>
<a id="__codelineno-17-19" name="__codelineno-17-19" href="#__codelineno-17-19"></a>                <span class="c1">// 若两字符相等，则直接跳过此两字符</span>
<a id="__codelineno-17-20" name="__codelineno-17-20" href="#__codelineno-17-20"></a>                <span class="n">dp</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="p">=</span> <span class="n">leftup</span>
<a id="__codelineno-17-21" name="__codelineno-17-21" href="#__codelineno-17-21"></a>            <span class="p">}</span> <span class="k">else</span> <span class="p">{</span>
<a id="__codelineno-17-22" name="__codelineno-17-22" href="#__codelineno-17-22"></a>                <span class="c1">// 最少编辑步数 = 插入、删除、替换这三种操作的最少编辑步数 + 1</span>
<a id="__codelineno-17-23" name="__codelineno-17-23" href="#__codelineno-17-23"></a>                <span class="n">dp</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="p">=</span> <span class="bp">min</span><span class="p">(</span><span class="bp">min</span><span class="p">(</span><span class="n">dp</span><span class="p">[</span><span class="n">j</span> <span class="o">-</span> <span class="mi">1</span><span class="p">],</span> <span class="n">dp</span><span class="p">[</span><span class="n">j</span><span class="p">]),</span> <span class="n">leftup</span><span class="p">)</span> <span class="o">+</span> <span class="mi">1</span>
<a id="__codelineno-17-24" name="__codelineno-17-24" href="#__codelineno-17-24"></a>            <span class="p">}</span>
<a id="__codelineno-17-25" name="__codelineno-17-25" href="#__codelineno-17-25"></a>            <span class="n">leftup</span> <span class="p">=</span> <span class="n">temp</span> <span class="c1">// 更新为下一轮的 dp[i-1, j-1]</span>
<a id="__codelineno-17-26" name="__codelineno-17-26" href="#__codelineno-17-26"></a>        <span class="p">}</span>
<a id="__codelineno-17-27" name="__codelineno-17-27" href="#__codelineno-17-27"></a>    <span class="p">}</span>
<a id="__codelineno-17-28" name="__codelineno-17-28" href="#__codelineno-17-28"></a>    <span class="k">return</span> <span class="n">dp</span><span class="p">[</span><span class="n">m</span><span class="p">]</span>
<a id="__codelineno-17-29" name="__codelineno-17-29" href="#__codelineno-17-29"></a><span class="p">}</span>
</code></pre></div>
</div>
<div class="tabbed-block">
<div class="highlight"><span class="filename">edit_distance.js</span><pre><span></span><code><a id="__codelineno-18-1" name="__codelineno-18-1" href="#__codelineno-18-1"></a><span class="cm">/* 编辑距离：状态压缩后的动态规划 */</span>
<a id="__codelineno-18-2" name="__codelineno-18-2" href="#__codelineno-18-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">editDistanceDPComp</span><span class="p">(</span><span class="nx">s</span><span class="p">,</span><span class="w"> </span><span class="nx">t</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-18-3" name="__codelineno-18-3" href="#__codelineno-18-3"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">s</span><span class="p">.</span><span class="nx">length</span><span class="p">,</span>
<a id="__codelineno-18-4" name="__codelineno-18-4" href="#__codelineno-18-4"></a><span class="w">        </span><span class="nx">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">t</span><span class="p">.</span><span class="nx">length</span><span class="p">;</span>
<a id="__codelineno-18-5" name="__codelineno-18-5" href="#__codelineno-18-5"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">dp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="nb">Array</span><span class="p">(</span><span class="nx">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="p">).</span><span class="nx">fill</span><span class="p">(</span><span class="mf">0</span><span class="p">);</span>
<a id="__codelineno-18-6" name="__codelineno-18-6" href="#__codelineno-18-6"></a><span class="w">    </span><span class="c1">// 状态转移：首行</span>
<a id="__codelineno-18-7" name="__codelineno-18-7" href="#__codelineno-18-7"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">m</span><span class="p">;</span><span class="w"> </span><span class="nx">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-18-8" name="__codelineno-18-8" href="#__codelineno-18-8"></a><span class="w">        </span><span class="nx">dp</span><span class="p">[</span><span class="nx">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">j</span><span class="p">;</span>
<a id="__codelineno-18-9" name="__codelineno-18-9" href="#__codelineno-18-9"></a><span class="w">    </span><span class="p">}</span>
<a id="__codelineno-18-10" name="__codelineno-18-10" href="#__codelineno-18-10"></a><span class="w">    </span><span class="c1">// 状态转移：其余行</span>
<a id="__codelineno-18-11" name="__codelineno-18-11" href="#__codelineno-18-11"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-18-12" name="__codelineno-18-12" href="#__codelineno-18-12"></a><span class="w">        </span><span class="c1">// 状态转移：首列</span>
<a id="__codelineno-18-13" name="__codelineno-18-13" href="#__codelineno-18-13"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="nx">leftup</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">dp</span><span class="p">[</span><span class="mf">0</span><span class="p">];</span><span class="w"> </span><span class="c1">// 暂存 dp[i-1, j-1]</span>
<a id="__codelineno-18-14" name="__codelineno-18-14" href="#__codelineno-18-14"></a><span class="w">        </span><span class="nx">dp</span><span class="p">[</span><span class="mf">0</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">i</span><span class="p">;</span>
<a id="__codelineno-18-15" name="__codelineno-18-15" href="#__codelineno-18-15"></a><span class="w">        </span><span class="c1">// 状态转移：其余列</span>
<a id="__codelineno-18-16" name="__codelineno-18-16" href="#__codelineno-18-16"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">m</span><span class="p">;</span><span class="w"> </span><span class="nx">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-18-17" name="__codelineno-18-17" href="#__codelineno-18-17"></a><span class="w">            </span><span class="kd">const</span><span class="w"> </span><span class="nx">temp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">dp</span><span class="p">[</span><span class="nx">j</span><span class="p">];</span>
<a id="__codelineno-18-18" name="__codelineno-18-18" href="#__codelineno-18-18"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">s</span><span class="p">.</span><span class="nx">charAt</span><span class="p">(</span><span class="nx">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">)</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="nx">t</span><span class="p">.</span><span class="nx">charAt</span><span class="p">(</span><span class="nx">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">))</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-18-19" name="__codelineno-18-19" href="#__codelineno-18-19"></a><span class="w">                </span><span class="c1">// 若两字符相等，则直接跳过此两字符</span>
<a id="__codelineno-18-20" name="__codelineno-18-20" href="#__codelineno-18-20"></a><span class="w">                </span><span class="nx">dp</span><span class="p">[</span><span class="nx">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">leftup</span><span class="p">;</span>
<a id="__codelineno-18-21" name="__codelineno-18-21" href="#__codelineno-18-21"></a><span class="w">            </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-18-22" name="__codelineno-18-22" href="#__codelineno-18-22"></a><span class="w">                </span><span class="c1">// 最少编辑步数 = 插入、删除、替换这三种操作的最少编辑步数 + 1</span>
<a id="__codelineno-18-23" name="__codelineno-18-23" href="#__codelineno-18-23"></a><span class="w">                </span><span class="nx">dp</span><span class="p">[</span><span class="nx">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Math</span><span class="p">.</span><span class="nx">min</span><span class="p">(</span><span class="nx">dp</span><span class="p">[</span><span class="nx">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">],</span><span class="w"> </span><span class="nx">dp</span><span class="p">[</span><span class="nx">j</span><span class="p">],</span><span class="w"> </span><span class="nx">leftup</span><span class="p">)</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span>
<a id="__codelineno-18-24" name="__codelineno-18-24" href="#__codelineno-18-24"></a><span class="w">            </span><span class="p">}</span>
<a id="__codelineno-18-25" name="__codelineno-18-25" href="#__codelineno-18-25"></a><span class="w">            </span><span class="nx">leftup</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">temp</span><span class="p">;</span><span class="w"> </span><span class="c1">// 更新为下一轮的 dp[i-1, j-1]</span>
<a id="__codelineno-18-26" name="__codelineno-18-26" href="#__codelineno-18-26"></a><span class="w">        </span><span class="p">}</span>
<a id="__codelineno-18-27" name="__codelineno-18-27" href="#__codelineno-18-27"></a><span class="w">    </span><span class="p">}</span>
<a id="__codelineno-18-28" name="__codelineno-18-28" href="#__codelineno-18-28"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">dp</span><span class="p">[</span><span class="nx">m</span><span class="p">];</span>
<a id="__codelineno-18-29" name="__codelineno-18-29" href="#__codelineno-18-29"></a><span class="p">}</span>
</code></pre></div>
</div>
<div class="tabbed-block">
<div class="highlight"><span class="filename">edit_distance.ts</span><pre><span></span><code><a id="__codelineno-19-1" name="__codelineno-19-1" href="#__codelineno-19-1"></a><span class="cm">/* 编辑距离：状态压缩后的动态规划 */</span>
<a id="__codelineno-19-2" name="__codelineno-19-2" href="#__codelineno-19-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">editDistanceDPComp</span><span class="p">(</span><span class="nx">s</span><span class="o">:</span><span class="w"> </span><span class="kt">string</span><span class="p">,</span><span class="w"> </span><span class="nx">t</span><span class="o">:</span><span class="w"> </span><span class="kt">string</span><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-19-3" name="__codelineno-19-3" href="#__codelineno-19-3"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">s</span><span class="p">.</span><span class="nx">length</span><span class="p">,</span>
<a id="__codelineno-19-4" name="__codelineno-19-4" href="#__codelineno-19-4"></a><span class="w">        </span><span class="nx">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">t</span><span class="p">.</span><span class="nx">length</span><span class="p">;</span>
<a id="__codelineno-19-5" name="__codelineno-19-5" href="#__codelineno-19-5"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">dp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="nb">Array</span><span class="p">(</span><span class="nx">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="p">).</span><span class="nx">fill</span><span class="p">(</span><span class="mf">0</span><span class="p">);</span>
<a id="__codelineno-19-6" name="__codelineno-19-6" href="#__codelineno-19-6"></a><span class="w">    </span><span class="c1">// 状态转移：首行</span>
<a id="__codelineno-19-7" name="__codelineno-19-7" href="#__codelineno-19-7"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">m</span><span class="p">;</span><span class="w"> </span><span class="nx">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-19-8" name="__codelineno-19-8" href="#__codelineno-19-8"></a><span class="w">        </span><span class="nx">dp</span><span class="p">[</span><span class="nx">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">j</span><span class="p">;</span>
<a id="__codelineno-19-9" name="__codelineno-19-9" href="#__codelineno-19-9"></a><span class="w">    </span><span class="p">}</span>
<a id="__codelineno-19-10" name="__codelineno-19-10" href="#__codelineno-19-10"></a><span class="w">    </span><span class="c1">// 状态转移：其余行</span>
<a id="__codelineno-19-11" name="__codelineno-19-11" href="#__codelineno-19-11"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-19-12" name="__codelineno-19-12" href="#__codelineno-19-12"></a><span class="w">        </span><span class="c1">// 状态转移：首列</span>
<a id="__codelineno-19-13" name="__codelineno-19-13" href="#__codelineno-19-13"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="nx">leftup</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">dp</span><span class="p">[</span><span class="mf">0</span><span class="p">];</span><span class="w"> </span><span class="c1">// 暂存 dp[i-1, j-1]</span>
<a id="__codelineno-19-14" name="__codelineno-19-14" href="#__codelineno-19-14"></a><span class="w">        </span><span class="nx">dp</span><span class="p">[</span><span class="mf">0</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">i</span><span class="p">;</span>
<a id="__codelineno-19-15" name="__codelineno-19-15" href="#__codelineno-19-15"></a><span class="w">        </span><span class="c1">// 状态转移：其余列</span>
<a id="__codelineno-19-16" name="__codelineno-19-16" href="#__codelineno-19-16"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">m</span><span class="p">;</span><span class="w"> </span><span class="nx">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-19-17" name="__codelineno-19-17" href="#__codelineno-19-17"></a><span class="w">            </span><span class="kd">const</span><span class="w"> </span><span class="nx">temp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">dp</span><span class="p">[</span><span class="nx">j</span><span class="p">];</span>
<a id="__codelineno-19-18" name="__codelineno-19-18" href="#__codelineno-19-18"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">s</span><span class="p">.</span><span class="nx">charAt</span><span class="p">(</span><span class="nx">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">)</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="nx">t</span><span class="p">.</span><span class="nx">charAt</span><span class="p">(</span><span class="nx">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">))</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-19-19" name="__codelineno-19-19" href="#__codelineno-19-19"></a><span class="w">                </span><span class="c1">// 若两字符相等，则直接跳过此两字符</span>
<a id="__codelineno-19-20" name="__codelineno-19-20" href="#__codelineno-19-20"></a><span class="w">                </span><span class="nx">dp</span><span class="p">[</span><span class="nx">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">leftup</span><span class="p">;</span>
<a id="__codelineno-19-21" name="__codelineno-19-21" href="#__codelineno-19-21"></a><span class="w">            </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-19-22" name="__codelineno-19-22" href="#__codelineno-19-22"></a><span class="w">                </span><span class="c1">// 最少编辑步数 = 插入、删除、替换这三种操作的最少编辑步数 + 1</span>
<a id="__codelineno-19-23" name="__codelineno-19-23" href="#__codelineno-19-23"></a><span class="w">                </span><span class="nx">dp</span><span class="p">[</span><span class="nx">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Math</span><span class="p">.</span><span class="nx">min</span><span class="p">(</span><span class="nx">dp</span><span class="p">[</span><span class="nx">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">],</span><span class="w"> </span><span class="nx">dp</span><span class="p">[</span><span class="nx">j</span><span class="p">],</span><span class="w"> </span><span class="nx">leftup</span><span class="p">)</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span>
<a id="__codelineno-19-24" name="__codelineno-19-24" href="#__codelineno-19-24"></a><span class="w">            </span><span class="p">}</span>
<a id="__codelineno-19-25" name="__codelineno-19-25" href="#__codelineno-19-25"></a><span class="w">            </span><span class="nx">leftup</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">temp</span><span class="p">;</span><span class="w"> </span><span class="c1">// 更新为下一轮的 dp[i-1, j-1]</span>
<a id="__codelineno-19-26" name="__codelineno-19-26" href="#__codelineno-19-26"></a><span class="w">        </span><span class="p">}</span>
<a id="__codelineno-19-27" name="__codelineno-19-27" href="#__codelineno-19-27"></a><span class="w">    </span><span class="p">}</span>
<a id="__codelineno-19-28" name="__codelineno-19-28" href="#__codelineno-19-28"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">dp</span><span class="p">[</span><span class="nx">m</span><span class="p">];</span>
<a id="__codelineno-19-29" name="__codelineno-19-29" href="#__codelineno-19-29"></a><span class="p">}</span>
</code></pre></div>
</div>
<div class="tabbed-block">
<div class="highlight"><span class="filename">edit_distance.dart</span><pre><span></span><code><a id="__codelineno-20-1" name="__codelineno-20-1" href="#__codelineno-20-1"></a><span class="cm">/* 编辑距离：空间优化后的动态规划 */</span>
<a id="__codelineno-20-2" name="__codelineno-20-2" href="#__codelineno-20-2"></a><span class="kt">int</span><span class="w"> </span><span class="n">editDistanceDPComp</span><span class="p">(</span><span class="kt">String</span><span class="w"> </span><span class="n">s</span><span class="p">,</span><span class="w"> </span><span class="kt">String</span><span class="w"> </span><span class="n">t</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-20-3" name="__codelineno-20-3" href="#__codelineno-20-3"></a><span class="w">  </span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">s</span><span class="p">.</span><span class="n">length</span><span class="p">,</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">t</span><span class="p">.</span><span class="n">length</span><span class="p">;</span>
<a id="__codelineno-20-4" name="__codelineno-20-4" href="#__codelineno-20-4"></a><span class="w">  </span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">dp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">List</span><span class="p">.</span><span class="n">filled</span><span class="p">(</span><span class="n">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span><span class="p">,</span><span class="w"> </span><span class="m">0</span><span class="p">);</span>
<a id="__codelineno-20-5" name="__codelineno-20-5" href="#__codelineno-20-5"></a><span class="w">  </span><span class="c1">// 状态转移：首行</span>
<a id="__codelineno-20-6" name="__codelineno-20-6" href="#__codelineno-20-6"></a><span class="w">  </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">1</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">m</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-20-7" name="__codelineno-20-7" href="#__codelineno-20-7"></a><span class="w">    </span><span class="n">dp</span><span class="p">[</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">j</span><span class="p">;</span>
<a id="__codelineno-20-8" name="__codelineno-20-8" href="#__codelineno-20-8"></a><span class="w">  </span><span class="p">}</span>
<a id="__codelineno-20-9" name="__codelineno-20-9" href="#__codelineno-20-9"></a><span class="w">  </span><span class="c1">// 状态转移：其余行</span>
<a id="__codelineno-20-10" name="__codelineno-20-10" href="#__codelineno-20-10"></a><span class="w">  </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-20-11" name="__codelineno-20-11" href="#__codelineno-20-11"></a><span class="w">    </span><span class="c1">// 状态转移：首列</span>
<a id="__codelineno-20-12" name="__codelineno-20-12" href="#__codelineno-20-12"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">leftup</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="m">0</span><span class="p">];</span><span class="w"> </span><span class="c1">// 暂存 dp[i-1, j-1]</span>
<a id="__codelineno-20-13" name="__codelineno-20-13" href="#__codelineno-20-13"></a><span class="w">    </span><span class="n">dp</span><span class="p">[</span><span class="m">0</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">i</span><span class="p">;</span>
<a id="__codelineno-20-14" name="__codelineno-20-14" href="#__codelineno-20-14"></a><span class="w">    </span><span class="c1">// 状态转移：其余列</span>
<a id="__codelineno-20-15" name="__codelineno-20-15" href="#__codelineno-20-15"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">1</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">m</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-20-16" name="__codelineno-20-16" href="#__codelineno-20-16"></a><span class="w">      </span><span class="kt">int</span><span class="w"> </span><span class="n">temp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">j</span><span class="p">];</span>
<a id="__codelineno-20-17" name="__codelineno-20-17" href="#__codelineno-20-17"></a><span class="w">      </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">s</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">]</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">t</span><span class="p">[</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">])</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-20-18" name="__codelineno-20-18" href="#__codelineno-20-18"></a><span class="w">        </span><span class="c1">// 若两字符相等，则直接跳过此两字符</span>
<a id="__codelineno-20-19" name="__codelineno-20-19" href="#__codelineno-20-19"></a><span class="w">        </span><span class="n">dp</span><span class="p">[</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">leftup</span><span class="p">;</span>
<a id="__codelineno-20-20" name="__codelineno-20-20" href="#__codelineno-20-20"></a><span class="w">      </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-20-21" name="__codelineno-20-21" href="#__codelineno-20-21"></a><span class="w">        </span><span class="c1">// 最少编辑步数 = 插入、删除、替换这三种操作的最少编辑步数 + 1</span>
<a id="__codelineno-20-22" name="__codelineno-20-22" href="#__codelineno-20-22"></a><span class="w">        </span><span class="n">dp</span><span class="p">[</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">min</span><span class="p">(</span><span class="n">min</span><span class="p">(</span><span class="n">dp</span><span class="p">[</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">],</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">j</span><span class="p">]),</span><span class="w"> </span><span class="n">leftup</span><span class="p">)</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span><span class="p">;</span>
<a id="__codelineno-20-23" name="__codelineno-20-23" href="#__codelineno-20-23"></a><span class="w">      </span><span class="p">}</span>
<a id="__codelineno-20-24" name="__codelineno-20-24" href="#__codelineno-20-24"></a><span class="w">      </span><span class="n">leftup</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">temp</span><span class="p">;</span><span class="w"> </span><span class="c1">// 更新为下一轮的 dp[i-1, j-1]</span>
<a id="__codelineno-20-25" name="__codelineno-20-25" href="#__codelineno-20-25"></a><span class="w">    </span><span class="p">}</span>
<a id="__codelineno-20-26" name="__codelineno-20-26" href="#__codelineno-20-26"></a><span class="w">  </span><span class="p">}</span>
<a id="__codelineno-20-27" name="__codelineno-20-27" href="#__codelineno-20-27"></a><span class="w">  </span><span class="k">return</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">m</span><span class="p">];</span>
<a id="__codelineno-20-28" name="__codelineno-20-28" href="#__codelineno-20-28"></a><span class="p">}</span>
</code></pre></div>
</div>
<div class="tabbed-block">
<div class="highlight"><span class="filename">edit_distance.rs</span><pre><span></span><code><a id="__codelineno-21-1" name="__codelineno-21-1" href="#__codelineno-21-1"></a><span class="cm">/* 编辑距离：空间优化后的动态规划 */</span>
<a id="__codelineno-21-2" name="__codelineno-21-2" href="#__codelineno-21-2"></a><span class="k">fn</span> <span class="nf">edit_distance_dp_comp</span><span class="p">(</span><span class="n">s</span>: <span class="kp">&amp;</span><span class="kt">str</span><span class="p">,</span><span class="w"> </span><span class="n">t</span>: <span class="kp">&amp;</span><span class="kt">str</span><span class="p">)</span><span class="w"> </span>-&gt; <span class="kt">i32</span> <span class="p">{</span>
<a id="__codelineno-21-3" name="__codelineno-21-3" href="#__codelineno-21-3"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="p">(</span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="n">m</span><span class="p">)</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">(</span><span class="n">s</span><span class="p">.</span><span class="n">len</span><span class="p">(),</span><span class="w"> </span><span class="n">t</span><span class="p">.</span><span class="n">len</span><span class="p">());</span>
<a id="__codelineno-21-4" name="__codelineno-21-4" href="#__codelineno-21-4"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">dp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="fm">vec!</span><span class="p">[</span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">];</span>
<a id="__codelineno-21-5" name="__codelineno-21-5" href="#__codelineno-21-5"></a><span class="w">    </span><span class="c1">// 状态转移：首行</span>
<a id="__codelineno-21-6" name="__codelineno-21-6" href="#__codelineno-21-6"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">1</span><span class="o">..</span><span class="n">m</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-21-7" name="__codelineno-21-7" href="#__codelineno-21-7"></a><span class="w">        </span><span class="n">dp</span><span class="p">[</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="kt">i32</span><span class="p">;</span>
<a id="__codelineno-21-8" name="__codelineno-21-8" href="#__codelineno-21-8"></a><span class="w">    </span><span class="p">}</span>
<a id="__codelineno-21-9" name="__codelineno-21-9" href="#__codelineno-21-9"></a><span class="w">    </span><span class="c1">// 状态转移：其余行</span>
<a id="__codelineno-21-10" name="__codelineno-21-10" href="#__codelineno-21-10"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">1</span><span class="o">..=</span><span class="n">n</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-21-11" name="__codelineno-21-11" href="#__codelineno-21-11"></a><span class="w">        </span><span class="c1">// 状态转移：首列</span>
<a id="__codelineno-21-12" name="__codelineno-21-12" href="#__codelineno-21-12"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">leftup</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="mi">0</span><span class="p">];</span><span class="w"> </span><span class="c1">// 暂存 dp[i-1, j-1]</span>
<a id="__codelineno-21-13" name="__codelineno-21-13" href="#__codelineno-21-13"></a><span class="w">        </span><span class="n">dp</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="kt">i32</span><span class="p">;</span>
<a id="__codelineno-21-14" name="__codelineno-21-14" href="#__codelineno-21-14"></a><span class="w">        </span><span class="c1">// 状态转移：其余列</span>
<a id="__codelineno-21-15" name="__codelineno-21-15" href="#__codelineno-21-15"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">1</span><span class="o">..=</span><span class="n">m</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-21-16" name="__codelineno-21-16" href="#__codelineno-21-16"></a><span class="w">            </span><span class="kd">let</span><span class="w"> </span><span class="n">temp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">j</span><span class="p">];</span>
<a id="__codelineno-21-17" name="__codelineno-21-17" href="#__codelineno-21-17"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="n">s</span><span class="p">.</span><span class="n">chars</span><span class="p">().</span><span class="n">nth</span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">t</span><span class="p">.</span><span class="n">chars</span><span class="p">().</span><span class="n">nth</span><span class="p">(</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-21-18" name="__codelineno-21-18" href="#__codelineno-21-18"></a><span class="w">                </span><span class="c1">// 若两字符相等，则直接跳过此两字符</span>
<a id="__codelineno-21-19" name="__codelineno-21-19" href="#__codelineno-21-19"></a><span class="w">                </span><span class="n">dp</span><span class="p">[</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">leftup</span><span class="p">;</span>
<a id="__codelineno-21-20" name="__codelineno-21-20" href="#__codelineno-21-20"></a><span class="w">            </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-21-21" name="__codelineno-21-21" href="#__codelineno-21-21"></a><span class="w">                </span><span class="c1">// 最少编辑步数 = 插入、删除、替换这三种操作的最少编辑步数 + 1</span>
<a id="__codelineno-21-22" name="__codelineno-21-22" href="#__codelineno-21-22"></a><span class="w">                </span><span class="n">dp</span><span class="p">[</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">std</span>::<span class="n">cmp</span>::<span class="n">min</span><span class="p">(</span><span class="n">std</span>::<span class="n">cmp</span>::<span class="n">min</span><span class="p">(</span><span class="n">dp</span><span class="p">[</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">],</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">j</span><span class="p">]),</span><span class="w"> </span><span class="n">leftup</span><span class="p">)</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
<a id="__codelineno-21-23" name="__codelineno-21-23" href="#__codelineno-21-23"></a><span class="w">            </span><span class="p">}</span>
<a id="__codelineno-21-24" name="__codelineno-21-24" href="#__codelineno-21-24"></a><span class="w">            </span><span class="n">leftup</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">temp</span><span class="p">;</span><span class="w"> </span><span class="c1">// 更新为下一轮的 dp[i-1, j-1]</span>
<a id="__codelineno-21-25" name="__codelineno-21-25" href="#__codelineno-21-25"></a><span class="w">        </span><span class="p">}</span>
<a id="__codelineno-21-26" name="__codelineno-21-26" href="#__codelineno-21-26"></a><span class="w">    </span><span class="p">}</span>
<a id="__codelineno-21-27" name="__codelineno-21-27" href="#__codelineno-21-27"></a><span class="w">    </span><span class="n">dp</span><span class="p">[</span><span class="n">m</span><span class="p">]</span>
<a id="__codelineno-21-28" name="__codelineno-21-28" href="#__codelineno-21-28"></a><span class="p">}</span>
</code></pre></div>
</div>
<div class="tabbed-block">
<div class="highlight"><span class="filename">edit_distance.c</span><pre><span></span><code><a id="__codelineno-22-1" name="__codelineno-22-1" href="#__codelineno-22-1"></a><span class="cm">/* 编辑距离：空间优化后的动态规划 */</span>
<a id="__codelineno-22-2" name="__codelineno-22-2" href="#__codelineno-22-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">editDistanceDPComp</span><span class="p">(</span><span class="kt">char</span><span class="w"> </span><span class="o">*</span><span class="n">s</span><span class="p">,</span><span class="w"> </span><span class="kt">char</span><span class="w"> </span><span class="o">*</span><span class="n">t</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">m</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-22-3" name="__codelineno-22-3" href="#__codelineno-22-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="o">*</span><span class="n">dp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">calloc</span><span class="p">(</span><span class="n">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="k">sizeof</span><span class="p">(</span><span class="kt">int</span><span class="p">));</span>
<a id="__codelineno-22-4" name="__codelineno-22-4" href="#__codelineno-22-4"></a><span class="w">    </span><span class="c1">// 状态转移：首行</span>
<a id="__codelineno-22-5" name="__codelineno-22-5" href="#__codelineno-22-5"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">m</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-22-6" name="__codelineno-22-6" href="#__codelineno-22-6"></a><span class="w">        </span><span class="n">dp</span><span class="p">[</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">j</span><span class="p">;</span>
<a id="__codelineno-22-7" name="__codelineno-22-7" href="#__codelineno-22-7"></a><span class="w">    </span><span class="p">}</span>
<a id="__codelineno-22-8" name="__codelineno-22-8" href="#__codelineno-22-8"></a><span class="w">    </span><span class="c1">// 状态转移：其余行</span>
<a id="__codelineno-22-9" name="__codelineno-22-9" href="#__codelineno-22-9"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-22-10" name="__codelineno-22-10" href="#__codelineno-22-10"></a><span class="w">        </span><span class="c1">// 状态转移：首列</span>
<a id="__codelineno-22-11" name="__codelineno-22-11" href="#__codelineno-22-11"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">leftup</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="mi">0</span><span class="p">];</span><span class="w"> </span><span class="c1">// 暂存 dp[i-1, j-1]</span>
<a id="__codelineno-22-12" name="__codelineno-22-12" href="#__codelineno-22-12"></a><span class="w">        </span><span class="n">dp</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">i</span><span class="p">;</span>
<a id="__codelineno-22-13" name="__codelineno-22-13" href="#__codelineno-22-13"></a><span class="w">        </span><span class="c1">// 状态转移：其余列</span>
<a id="__codelineno-22-14" name="__codelineno-22-14" href="#__codelineno-22-14"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">m</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-22-15" name="__codelineno-22-15" href="#__codelineno-22-15"></a><span class="w">            </span><span class="kt">int</span><span class="w"> </span><span class="n">temp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">j</span><span class="p">];</span>
<a id="__codelineno-22-16" name="__codelineno-22-16" href="#__codelineno-22-16"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">s</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">t</span><span class="p">[</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">])</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-22-17" name="__codelineno-22-17" href="#__codelineno-22-17"></a><span class="w">                </span><span class="c1">// 若两字符相等，则直接跳过此两字符</span>
<a id="__codelineno-22-18" name="__codelineno-22-18" href="#__codelineno-22-18"></a><span class="w">                </span><span class="n">dp</span><span class="p">[</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">leftup</span><span class="p">;</span>
<a id="__codelineno-22-19" name="__codelineno-22-19" href="#__codelineno-22-19"></a><span class="w">            </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-22-20" name="__codelineno-22-20" href="#__codelineno-22-20"></a><span class="w">                </span><span class="c1">// 最少编辑步数 = 插入、删除、替换这三种操作的最少编辑步数 + 1</span>
<a id="__codelineno-22-21" name="__codelineno-22-21" href="#__codelineno-22-21"></a><span class="w">                </span><span class="n">dp</span><span class="p">[</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">myMin</span><span class="p">(</span><span class="n">myMin</span><span class="p">(</span><span class="n">dp</span><span class="p">[</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">],</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">j</span><span class="p">]),</span><span class="w"> </span><span class="n">leftup</span><span class="p">)</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
<a id="__codelineno-22-22" name="__codelineno-22-22" href="#__codelineno-22-22"></a><span class="w">            </span><span class="p">}</span>
<a id="__codelineno-22-23" name="__codelineno-22-23" href="#__codelineno-22-23"></a><span class="w">            </span><span class="n">leftup</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">temp</span><span class="p">;</span><span class="w"> </span><span class="c1">// 更新为下一轮的 dp[i-1, j-1]</span>
<a id="__codelineno-22-24" name="__codelineno-22-24" href="#__codelineno-22-24"></a><span class="w">        </span><span class="p">}</span>
<a id="__codelineno-22-25" name="__codelineno-22-25" href="#__codelineno-22-25"></a><span class="w">    </span><span class="p">}</span>
<a id="__codelineno-22-26" name="__codelineno-22-26" href="#__codelineno-22-26"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">m</span><span class="p">];</span>
<a id="__codelineno-22-27" name="__codelineno-22-27" href="#__codelineno-22-27"></a><span class="w">    </span><span class="c1">// 释放内存</span>
<a id="__codelineno-22-28" name="__codelineno-22-28" href="#__codelineno-22-28"></a><span class="w">    </span><span class="n">free</span><span class="p">(</span><span class="n">dp</span><span class="p">);</span>
<a id="__codelineno-22-29" name="__codelineno-22-29" href="#__codelineno-22-29"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">res</span><span class="p">;</span>
<a id="__codelineno-22-30" name="__codelineno-22-30" href="#__codelineno-22-30"></a><span class="p">}</span>
</code></pre></div>
</div>
<div class="tabbed-block">
<div class="highlight"><span class="filename">edit_distance.zig</span><pre><span></span><code><a id="__codelineno-23-1" name="__codelineno-23-1" href="#__codelineno-23-1"></a><span class="c1">// 编辑距离：空间优化后的动态规划</span>
<a id="__codelineno-23-2" name="__codelineno-23-2" href="#__codelineno-23-2"></a><span class="k">fn</span><span class="w"> </span><span class="n">editDistanceDPComp</span><span class="p">(</span><span class="kr">comptime</span><span class="w"> </span><span class="n">s</span><span class="o">:</span><span class="w"> </span><span class="p">[]</span><span class="kr">const</span><span class="w"> </span><span class="kt">u8</span><span class="p">,</span><span class="w"> </span><span class="kr">comptime</span><span class="w"> </span><span class="n">t</span><span class="o">:</span><span class="w"> </span><span class="p">[]</span><span class="kr">const</span><span class="w"> </span><span class="kt">u8</span><span class="p">)</span><span class="w"> </span><span class="kt">i32</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-23-3" name="__codelineno-23-3" href="#__codelineno-23-3"></a><span class="w">    </span><span class="kr">comptime</span><span class="w"> </span><span class="kr">var</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">s</span><span class="p">.</span><span class="n">len</span><span class="p">;</span>
<a id="__codelineno-23-4" name="__codelineno-23-4" href="#__codelineno-23-4"></a><span class="w">    </span><span class="kr">comptime</span><span class="w"> </span><span class="kr">var</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">t</span><span class="p">.</span><span class="n">len</span><span class="p">;</span>
<a id="__codelineno-23-5" name="__codelineno-23-5" href="#__codelineno-23-5"></a><span class="w">    </span><span class="kr">var</span><span class="w"> </span><span class="n">dp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[</span><span class="n">_</span><span class="p">]</span><span class="kt">i32</span><span class="p">{</span><span class="mi">0</span><span class="p">}</span><span class="w"> </span><span class="o">**</span><span class="w"> </span><span class="p">(</span><span class="n">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">);</span>
<a id="__codelineno-23-6" name="__codelineno-23-6" href="#__codelineno-23-6"></a><span class="w">    </span><span class="c1">// 状态转移：首行</span>
<a id="__codelineno-23-7" name="__codelineno-23-7" href="#__codelineno-23-7"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="mi">1</span><span class="p">..</span><span class="n">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="o">|</span><span class="n">j</span><span class="o">|</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-23-8" name="__codelineno-23-8" href="#__codelineno-23-8"></a><span class="w">        </span><span class="n">dp</span><span class="p">[</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">@intCast</span><span class="p">(</span><span class="n">j</span><span class="p">);</span>
<a id="__codelineno-23-9" name="__codelineno-23-9" href="#__codelineno-23-9"></a><span class="w">    </span><span class="p">}</span>
<a id="__codelineno-23-10" name="__codelineno-23-10" href="#__codelineno-23-10"></a><span class="w">    </span><span class="c1">// 状态转移：其余行</span>
<a id="__codelineno-23-11" name="__codelineno-23-11" href="#__codelineno-23-11"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="mi">1</span><span class="p">..</span><span class="n">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="o">|</span><span class="n">i</span><span class="o">|</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-23-12" name="__codelineno-23-12" href="#__codelineno-23-12"></a><span class="w">        </span><span class="c1">// 状态转移：首列</span>
<a id="__codelineno-23-13" name="__codelineno-23-13" href="#__codelineno-23-13"></a><span class="w">        </span><span class="kr">var</span><span class="w"> </span><span class="n">leftup</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="mi">0</span><span class="p">];</span><span class="w"> </span><span class="c1">// 暂存 dp[i-1, j-1]</span>
<a id="__codelineno-23-14" name="__codelineno-23-14" href="#__codelineno-23-14"></a><span class="w">        </span><span class="n">dp</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">@intCast</span><span class="p">(</span><span class="n">i</span><span class="p">);</span>
<a id="__codelineno-23-15" name="__codelineno-23-15" href="#__codelineno-23-15"></a><span class="w">        </span><span class="c1">// 状态转移：其余列</span>
<a id="__codelineno-23-16" name="__codelineno-23-16" href="#__codelineno-23-16"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="mi">1</span><span class="p">..</span><span class="n">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="o">|</span><span class="n">j</span><span class="o">|</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-23-17" name="__codelineno-23-17" href="#__codelineno-23-17"></a><span class="w">            </span><span class="kr">var</span><span class="w"> </span><span class="n">temp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">j</span><span class="p">];</span>
<a id="__codelineno-23-18" name="__codelineno-23-18" href="#__codelineno-23-18"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">s</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">t</span><span class="p">[</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">])</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-23-19" name="__codelineno-23-19" href="#__codelineno-23-19"></a><span class="w">                </span><span class="c1">// 若两字符相等，则直接跳过此两字符</span>
<a id="__codelineno-23-20" name="__codelineno-23-20" href="#__codelineno-23-20"></a><span class="w">                </span><span class="n">dp</span><span class="p">[</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">leftup</span><span class="p">;</span>
<a id="__codelineno-23-21" name="__codelineno-23-21" href="#__codelineno-23-21"></a><span class="w">            </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-23-22" name="__codelineno-23-22" href="#__codelineno-23-22"></a><span class="w">                </span><span class="c1">// 最少编辑步数 = 插入、删除、替换这三种操作的最少编辑步数 + 1</span>
<a id="__codelineno-23-23" name="__codelineno-23-23" href="#__codelineno-23-23"></a><span class="w">                </span><span class="n">dp</span><span class="p">[</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">@min</span><span class="p">(</span><span class="nb">@min</span><span class="p">(</span><span class="n">dp</span><span class="p">[</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">],</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">j</span><span class="p">]),</span><span class="w"> </span><span class="n">leftup</span><span class="p">)</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
<a id="__codelineno-23-24" name="__codelineno-23-24" href="#__codelineno-23-24"></a><span class="w">            </span><span class="p">}</span>
<a id="__codelineno-23-25" name="__codelineno-23-25" href="#__codelineno-23-25"></a><span class="w">            </span><span class="n">leftup</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">temp</span><span class="p">;</span><span class="w"> </span><span class="c1">// 更新为下一轮的 dp[i-1, j-1]</span>
<a id="__codelineno-23-26" name="__codelineno-23-26" href="#__codelineno-23-26"></a><span class="w">        </span><span class="p">}</span>
<a id="__codelineno-23-27" name="__codelineno-23-27" href="#__codelineno-23-27"></a><span class="w">    </span><span class="p">}</span>
<a id="__codelineno-23-28" name="__codelineno-23-28" href="#__codelineno-23-28"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">m</span><span class="p">];</span>
<a id="__codelineno-23-29" name="__codelineno-23-29" href="#__codelineno-23-29"></a><span class="p">}</span>
</code></pre></div>
</div>
</div>
</div>
<details class="pythontutor">
<summary>可视化运行</summary>
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