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7.1 &nbsp; 二叉树
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7.2 &nbsp; 二叉树遍历
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7.3 &nbsp; 二叉树数组表示
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7.4 &nbsp; 二叉搜索树
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7.5 &nbsp; AVL 树 *
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第 8 章 &nbsp;
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第 9 章 &nbsp;
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9.2 &nbsp; 图基础操作
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9.3 &nbsp; 图的遍历
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第 10 章 &nbsp; 搜索
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第 10 章 &nbsp; 搜索
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10.1 &nbsp; 二分查找
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10.2 &nbsp; 二分查找插入点
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10.3 &nbsp; 二分查找边界
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10.4 &nbsp; 哈希优化策略
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10.5 &nbsp; 重识搜索算法
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第 11 章 &nbsp; 排序
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11.1 &nbsp; 排序算法
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11.2 &nbsp; 选择排序
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11.3 &nbsp; 冒泡排序
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11.4 &nbsp; 插入排序
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11.6 &nbsp; 归并排序
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11.7 &nbsp; 堆排序
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11.8 &nbsp; 桶排序
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11.9 &nbsp; 计数排序
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11.10 &nbsp; 基数排序
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第 12 章 &nbsp; 分治
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12.1 &nbsp; 分治算法
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第 13 章 &nbsp; 回溯
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13.4 &nbsp; N 皇后问题
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第 14 章 &nbsp; 动态规划
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14.1 &nbsp; 初探动态规划
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14.2 &nbsp; DP 问题特性
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14.3 &nbsp; DP 解题思路
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14.4 &nbsp; 0-1 背包问题
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<h1 id="02">0.2 &nbsp; 如何使用本书<a class="headerlink" href="#02" title="Permanent link">&para;</a></h1>
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<div class="highlight"><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a><span class="sd">&quot;&quot;&quot;标题注释,用于标注函数、类、测试样例等&quot;&quot;&quot;</span>
<a id="__codelineno-0-2" name="__codelineno-0-2" href="#__codelineno-0-2"></a>
<a id="__codelineno-0-3" name="__codelineno-0-3" href="#__codelineno-0-3"></a><span class="c1"># 内容注释,用于详解代码</span>
<a id="__codelineno-0-4" name="__codelineno-0-4" href="#__codelineno-0-4"></a>
<a id="__codelineno-0-5" name="__codelineno-0-5" href="#__codelineno-0-5"></a><span class="sd">&quot;&quot;&quot;</span>
<a id="__codelineno-0-6" name="__codelineno-0-6" href="#__codelineno-0-6"></a><span class="sd">多行</span>
<a id="__codelineno-0-7" name="__codelineno-0-7" href="#__codelineno-0-7"></a><span class="sd">注释</span>
<a id="__codelineno-0-8" name="__codelineno-0-8" href="#__codelineno-0-8"></a><span class="sd">&quot;&quot;&quot;</span>
</code></pre></div>
</div>
<div class="tabbed-block">
<div class="highlight"><pre><span></span><code><a id="__codelineno-1-1" name="__codelineno-1-1" href="#__codelineno-1-1"></a><span class="cm">/* 标题注释,用于标注函数、类、测试样例等 */</span>
<a id="__codelineno-1-2" name="__codelineno-1-2" href="#__codelineno-1-2"></a>
<a id="__codelineno-1-3" name="__codelineno-1-3" href="#__codelineno-1-3"></a><span class="c1">// 内容注释,用于详解代码</span>
<a id="__codelineno-1-4" name="__codelineno-1-4" href="#__codelineno-1-4"></a>
<a id="__codelineno-1-5" name="__codelineno-1-5" href="#__codelineno-1-5"></a><span class="cm">/**</span>
<a id="__codelineno-1-6" name="__codelineno-1-6" href="#__codelineno-1-6"></a><span class="cm"> * 多行</span>
<a id="__codelineno-1-7" name="__codelineno-1-7" href="#__codelineno-1-7"></a><span class="cm"> * 注释</span>
<a id="__codelineno-1-8" name="__codelineno-1-8" href="#__codelineno-1-8"></a><span class="cm"> */</span>
</code></pre></div>
</div>
<div class="tabbed-block">
<div class="highlight"><pre><span></span><code><a id="__codelineno-2-1" name="__codelineno-2-1" href="#__codelineno-2-1"></a><span class="cm">/* 标题注释,用于标注函数、类、测试样例等 */</span>
<a id="__codelineno-2-2" name="__codelineno-2-2" href="#__codelineno-2-2"></a>
<a id="__codelineno-2-3" name="__codelineno-2-3" href="#__codelineno-2-3"></a><span class="c1">// 内容注释,用于详解代码</span>
<a id="__codelineno-2-4" name="__codelineno-2-4" href="#__codelineno-2-4"></a>
<a id="__codelineno-2-5" name="__codelineno-2-5" href="#__codelineno-2-5"></a><span class="cm">/**</span>
<a id="__codelineno-2-6" name="__codelineno-2-6" href="#__codelineno-2-6"></a><span class="cm"> * 多行</span>
<a id="__codelineno-2-7" name="__codelineno-2-7" href="#__codelineno-2-7"></a><span class="cm"> * 注释</span>
<a id="__codelineno-2-8" name="__codelineno-2-8" href="#__codelineno-2-8"></a><span class="cm"> */</span>
</code></pre></div>
</div>
<div class="tabbed-block">
<div class="highlight"><pre><span></span><code><a id="__codelineno-3-1" name="__codelineno-3-1" href="#__codelineno-3-1"></a><span class="cm">/* 标题注释,用于标注函数、类、测试样例等 */</span>
<a id="__codelineno-3-2" name="__codelineno-3-2" href="#__codelineno-3-2"></a>
<a id="__codelineno-3-3" name="__codelineno-3-3" href="#__codelineno-3-3"></a><span class="c1">// 内容注释,用于详解代码</span>
<a id="__codelineno-3-4" name="__codelineno-3-4" href="#__codelineno-3-4"></a>
<a id="__codelineno-3-5" name="__codelineno-3-5" href="#__codelineno-3-5"></a><span class="cm">/**</span>
<a id="__codelineno-3-6" name="__codelineno-3-6" href="#__codelineno-3-6"></a><span class="cm"> * 多行</span>
<a id="__codelineno-3-7" name="__codelineno-3-7" href="#__codelineno-3-7"></a><span class="cm"> * 注释</span>
<a id="__codelineno-3-8" name="__codelineno-3-8" href="#__codelineno-3-8"></a><span class="cm"> */</span>
</code></pre></div>
</div>
<div class="tabbed-block">
<div class="highlight"><pre><span></span><code><a id="__codelineno-4-1" name="__codelineno-4-1" href="#__codelineno-4-1"></a><span class="cm">/* 标题注释,用于标注函数、类、测试样例等 */</span>
<a id="__codelineno-4-2" name="__codelineno-4-2" href="#__codelineno-4-2"></a>
<a id="__codelineno-4-3" name="__codelineno-4-3" href="#__codelineno-4-3"></a><span class="c1">// 内容注释,用于详解代码</span>
<a id="__codelineno-4-4" name="__codelineno-4-4" href="#__codelineno-4-4"></a>
<a id="__codelineno-4-5" name="__codelineno-4-5" href="#__codelineno-4-5"></a><span class="cm">/**</span>
<a id="__codelineno-4-6" name="__codelineno-4-6" href="#__codelineno-4-6"></a><span class="cm"> * 多行</span>
<a id="__codelineno-4-7" name="__codelineno-4-7" href="#__codelineno-4-7"></a><span class="cm"> * 注释</span>
<a id="__codelineno-4-8" name="__codelineno-4-8" href="#__codelineno-4-8"></a><span class="cm"> */</span>
</code></pre></div>
</div>
<div class="tabbed-block">
<div class="highlight"><pre><span></span><code><a id="__codelineno-5-1" name="__codelineno-5-1" href="#__codelineno-5-1"></a><span class="cm">/* 标题注释,用于标注函数、类、测试样例等 */</span>
<a id="__codelineno-5-2" name="__codelineno-5-2" href="#__codelineno-5-2"></a>
<a id="__codelineno-5-3" name="__codelineno-5-3" href="#__codelineno-5-3"></a><span class="c1">// 内容注释,用于详解代码</span>
<a id="__codelineno-5-4" name="__codelineno-5-4" href="#__codelineno-5-4"></a>
<a id="__codelineno-5-5" name="__codelineno-5-5" href="#__codelineno-5-5"></a><span class="cm">/**</span>
<a id="__codelineno-5-6" name="__codelineno-5-6" href="#__codelineno-5-6"></a><span class="cm"> * 多行</span>
<a id="__codelineno-5-7" name="__codelineno-5-7" href="#__codelineno-5-7"></a><span class="cm"> * 注释</span>
<a id="__codelineno-5-8" name="__codelineno-5-8" href="#__codelineno-5-8"></a><span class="cm"> */</span>
</code></pre></div>
</div>
<div class="tabbed-block">
<div class="highlight"><pre><span></span><code><a id="__codelineno-6-1" name="__codelineno-6-1" href="#__codelineno-6-1"></a><span class="cm">/* 标题注释,用于标注函数、类、测试样例等 */</span>
<a id="__codelineno-6-2" name="__codelineno-6-2" href="#__codelineno-6-2"></a>
<a id="__codelineno-6-3" name="__codelineno-6-3" href="#__codelineno-6-3"></a><span class="c1">// 内容注释,用于详解代码</span>
<a id="__codelineno-6-4" name="__codelineno-6-4" href="#__codelineno-6-4"></a>
<a id="__codelineno-6-5" name="__codelineno-6-5" href="#__codelineno-6-5"></a><span class="cm">/**</span>
<a id="__codelineno-6-6" name="__codelineno-6-6" href="#__codelineno-6-6"></a><span class="cm"> * 多行</span>
<a id="__codelineno-6-7" name="__codelineno-6-7" href="#__codelineno-6-7"></a><span class="cm"> * 注释</span>
<a id="__codelineno-6-8" name="__codelineno-6-8" href="#__codelineno-6-8"></a><span class="cm"> */</span>
</code></pre></div>
</div>
<div class="tabbed-block">
<div class="highlight"><pre><span></span><code><a id="__codelineno-7-1" name="__codelineno-7-1" href="#__codelineno-7-1"></a><span class="cm">/* 标题注释,用于标注函数、类、测试样例等 */</span>
<a id="__codelineno-7-2" name="__codelineno-7-2" href="#__codelineno-7-2"></a>
<a id="__codelineno-7-3" name="__codelineno-7-3" href="#__codelineno-7-3"></a><span class="c1">// 内容注释,用于详解代码</span>
<a id="__codelineno-7-4" name="__codelineno-7-4" href="#__codelineno-7-4"></a>
<a id="__codelineno-7-5" name="__codelineno-7-5" href="#__codelineno-7-5"></a><span class="cm">/**</span>
<a id="__codelineno-7-6" name="__codelineno-7-6" href="#__codelineno-7-6"></a><span class="cm"> * 多行</span>
<a id="__codelineno-7-7" name="__codelineno-7-7" href="#__codelineno-7-7"></a><span class="cm"> * 注释</span>
<a id="__codelineno-7-8" name="__codelineno-7-8" href="#__codelineno-7-8"></a><span class="cm"> */</span>
</code></pre></div>
</div>
<div class="tabbed-block">
<div class="highlight"><pre><span></span><code><a id="__codelineno-8-1" name="__codelineno-8-1" href="#__codelineno-8-1"></a><span class="cm">/* 标题注释,用于标注函数、类、测试样例等 */</span>
<a id="__codelineno-8-2" name="__codelineno-8-2" href="#__codelineno-8-2"></a>
<a id="__codelineno-8-3" name="__codelineno-8-3" href="#__codelineno-8-3"></a><span class="c1">// 内容注释,用于详解代码</span>
<a id="__codelineno-8-4" name="__codelineno-8-4" href="#__codelineno-8-4"></a>
<a id="__codelineno-8-5" name="__codelineno-8-5" href="#__codelineno-8-5"></a><span class="cm">/**</span>
<a id="__codelineno-8-6" name="__codelineno-8-6" href="#__codelineno-8-6"></a><span class="cm"> * 多行</span>
<a id="__codelineno-8-7" name="__codelineno-8-7" href="#__codelineno-8-7"></a><span class="cm"> * 注释</span>
<a id="__codelineno-8-8" name="__codelineno-8-8" href="#__codelineno-8-8"></a><span class="cm"> */</span>
</code></pre></div>
</div>
<div class="tabbed-block">
<div class="highlight"><pre><span></span><code><a id="__codelineno-9-1" name="__codelineno-9-1" href="#__codelineno-9-1"></a><span class="cm">/* 标题注释,用于标注函数、类、测试样例等 */</span>
<a id="__codelineno-9-2" name="__codelineno-9-2" href="#__codelineno-9-2"></a>
<a id="__codelineno-9-3" name="__codelineno-9-3" href="#__codelineno-9-3"></a><span class="c1">// 内容注释,用于详解代码</span>
<a id="__codelineno-9-4" name="__codelineno-9-4" href="#__codelineno-9-4"></a>
<a id="__codelineno-9-5" name="__codelineno-9-5" href="#__codelineno-9-5"></a><span class="sd">/**</span>
<a id="__codelineno-9-6" name="__codelineno-9-6" href="#__codelineno-9-6"></a><span class="sd"> * 多行</span>
<a id="__codelineno-9-7" name="__codelineno-9-7" href="#__codelineno-9-7"></a><span class="sd"> * 注释</span>
<a id="__codelineno-9-8" name="__codelineno-9-8" href="#__codelineno-9-8"></a><span class="sd"> */</span>
</code></pre></div>
</div>
<div class="tabbed-block">
<div class="highlight"><pre><span></span><code><a id="__codelineno-10-1" name="__codelineno-10-1" href="#__codelineno-10-1"></a><span class="cm">/* 标题注释,用于标注函数、类、测试样例等 */</span>
<a id="__codelineno-10-2" name="__codelineno-10-2" href="#__codelineno-10-2"></a>
<a id="__codelineno-10-3" name="__codelineno-10-3" href="#__codelineno-10-3"></a><span class="c1">// 内容注释,用于详解代码</span>
<a id="__codelineno-10-4" name="__codelineno-10-4" href="#__codelineno-10-4"></a>
<a id="__codelineno-10-5" name="__codelineno-10-5" href="#__codelineno-10-5"></a><span class="cm">/**</span>
<a id="__codelineno-10-6" name="__codelineno-10-6" href="#__codelineno-10-6"></a><span class="cm"> * 多行</span>
<a id="__codelineno-10-7" name="__codelineno-10-7" href="#__codelineno-10-7"></a><span class="cm"> * 注释</span>
<a id="__codelineno-10-8" name="__codelineno-10-8" href="#__codelineno-10-8"></a><span class="cm"> */</span>
</code></pre></div>
</div>
<div class="tabbed-block">
<div class="highlight"><pre><span></span><code><a id="__codelineno-11-1" name="__codelineno-11-1" href="#__codelineno-11-1"></a><span class="cm">/* 标题注释,用于标注函数、类、测试样例等 */</span>
<a id="__codelineno-11-2" name="__codelineno-11-2" href="#__codelineno-11-2"></a>
<a id="__codelineno-11-3" name="__codelineno-11-3" href="#__codelineno-11-3"></a><span class="c1">// 内容注释,用于详解代码</span>
<a id="__codelineno-11-4" name="__codelineno-11-4" href="#__codelineno-11-4"></a>
<a id="__codelineno-11-5" name="__codelineno-11-5" href="#__codelineno-11-5"></a><span class="cm">/**</span>
<a id="__codelineno-11-6" name="__codelineno-11-6" href="#__codelineno-11-6"></a><span class="cm"> * 多行</span>
<a id="__codelineno-11-7" name="__codelineno-11-7" href="#__codelineno-11-7"></a><span class="cm"> * 注释</span>
<a id="__codelineno-11-8" name="__codelineno-11-8" href="#__codelineno-11-8"></a><span class="cm"> */</span>
</code></pre></div>
</div>
<div class="tabbed-block">
<div class="highlight"><pre><span></span><code><a id="__codelineno-12-1" name="__codelineno-12-1" href="#__codelineno-12-1"></a>
</code></pre></div>
</div>
<div class="tabbed-block">
<div class="highlight"><pre><span></span><code><a id="__codelineno-13-1" name="__codelineno-13-1" href="#__codelineno-13-1"></a><span class="c1">// 标题注释,用于标注函数、类、测试样例等</span>
<a id="__codelineno-13-2" name="__codelineno-13-2" href="#__codelineno-13-2"></a>
<a id="__codelineno-13-3" name="__codelineno-13-3" href="#__codelineno-13-3"></a><span class="c1">// 内容注释,用于详解代码</span>
<a id="__codelineno-13-4" name="__codelineno-13-4" href="#__codelineno-13-4"></a>
<a id="__codelineno-13-5" name="__codelineno-13-5" href="#__codelineno-13-5"></a><span class="c1">// 多行</span>
<a id="__codelineno-13-6" name="__codelineno-13-6" href="#__codelineno-13-6"></a><span class="c1">// 注释</span>
</code></pre></div>
</div>
</div>
</div>
<h2 id="022">0.2.2 &nbsp; 在动画图解中高效学习<a class="headerlink" href="#022" title="Permanent link">&para;</a></h2>
<p>相较于文字,视频和图片具有更高的信息密度和结构化程度,更易于理解。在本书中,<strong>重点和难点知识将主要通过动画以图解形式展示</strong>,而文字则作为解释与补充。</p>
<p>如果你在阅读本书时,发现某段内容提供了如图 0-2 所示的动画图解,<strong>请以图为主、以文字为辅</strong>,综合两者来理解内容。</p>
<p><a class="glightbox" href="../../index.assets/animation.gif" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="动画图解示例" class="animation-figure" src="../../index.assets/animation.gif" /></a></p>
<p align="center"> 图 0-2 &nbsp; 动画图解示例 </p>
<h2 id="023">0.2.3 &nbsp; 在代码实践中加深理解<a class="headerlink" href="#023" title="Permanent link">&para;</a></h2>
<p>本书的配套代码托管在 <a href="https://github.com/krahets/hello-algo">GitHub 仓库</a>。如图 0-3 所示,<strong>源代码附有测试样例,可一键运行</strong></p>
<p>如果时间允许,<strong>建议你参照代码自行敲一遍</strong>。如果学习时间有限,请至少通读并运行所有代码。</p>
<p>与阅读代码相比,编写代码的过程往往能带来更多收获。<strong>动手学,才是真的学</strong></p>
<p><a class="glightbox" href="../../index.assets/running_code.gif" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="运行代码示例" class="animation-figure" src="../../index.assets/running_code.gif" /></a></p>
<p align="center"> 图 0-3 &nbsp; 运行代码示例 </p>
<p>运行代码的前置工作主要分为三步。</p>
<p><strong>第一步:安装本地编程环境</strong>。请参照附录所示的<a href="https://www.hello-algo.com/chapter_appendix/installation/">教程</a>进行安装,如果已安装,则可跳过此步骤。</p>
<p><strong>第二步:克隆或下载代码仓库</strong>。前往 <a href="https://github.com/krahets/hello-algo">GitHub 仓库</a>。如果已经安装 <a href="https://git-scm.com/downloads">Git</a> ,可以通过以下命令克隆本仓库:</p>
<div class="highlight"><pre><span></span><code><a id="__codelineno-14-1" name="__codelineno-14-1" href="#__codelineno-14-1"></a>git<span class="w"> </span>clone<span class="w"> </span>https://github.com/krahets/hello-algo.git
</code></pre></div>
<p>当然,你也可以在图 0-4 所示的位置点击“Download ZIP”按钮直接下载代码压缩包然后在本地解压即可。</p>
<p><a class="glightbox" href="../suggestions.assets/download_code.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="克隆仓库与下载代码" class="animation-figure" src="../suggestions.assets/download_code.png" /></a></p>
<p align="center"> 图 0-4 &nbsp; 克隆仓库与下载代码 </p>
<p><strong>第三步:运行源代码</strong>。如图 0-5 所示,对于顶部标有文件名称的代码块,我们可以在仓库的 <code>codes</code> 文件夹内找到对应的源代码文件。源代码文件可一键运行,将帮助你节省不必要的调试时间,让你能够专注于学习内容。</p>
<p><a class="glightbox" href="../suggestions.assets/code_md_to_repo.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="代码块与对应的源代码文件" class="animation-figure" src="../suggestions.assets/code_md_to_repo.png" /></a></p>
<p align="center"> 图 0-5 &nbsp; 代码块与对应的源代码文件 </p>
<p>除了本地运行代码,<strong>网页版还支持 Python 代码的可视化运行</strong>(基于 <a href="https://pythontutor.com/">pythontutor</a> 实现)。如图 0-6 所示,你可以点击代码块下方的“可视化运行”来展开视图,观察算法代码的执行过程;也可以点击“全屏观看”,以获得更好的阅览体验。</p>
<p><a class="glightbox" href="../suggestions.assets/pythontutor_example.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Python 代码的可视化运行" class="animation-figure" src="../suggestions.assets/pythontutor_example.png" /></a></p>
<p align="center"> 图 0-6 &nbsp; Python 代码的可视化运行 </p>
<h2 id="024">0.2.4 &nbsp; 在提问讨论中共同成长<a class="headerlink" href="#024" title="Permanent link">&para;</a></h2>
<p>在阅读本书时,请不要轻易跳过那些没学明白的知识点。<strong>欢迎在评论区提出你的问题</strong>,我和小伙伴们将竭诚为你解答,一般情况下可在两天内回复。</p>
<p>如图 0-7 所示,网页版每个章节的底部都配有评论区。希望你能多关注评论区的内容。一方面,你可以了解大家遇到的问题,从而查漏补缺,激发更深入的思考。另一方面,期待你能慷慨地回答其他小伙伴的问题,分享你的见解,帮助他人进步。</p>
<p><a class="glightbox" href="../../index.assets/comment.gif" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="评论区示例" class="animation-figure" src="../../index.assets/comment.gif" /></a></p>
<p align="center"> 图 0-7 &nbsp; 评论区示例 </p>
<h2 id="025">0.2.5 &nbsp; 算法学习路线<a class="headerlink" href="#025" title="Permanent link">&para;</a></h2>
<p>从总体上看,我们可以将学习数据结构与算法的过程划分为三个阶段。</p>
<ol>
<li><strong>阶段一:算法入门</strong>。我们需要熟悉各种数据结构的特点和用法,学习不同算法的原理、流程、用途和效率等方面的内容。</li>
<li><strong>阶段二:刷算法题</strong>。建议从热门题目开刷,先积累至少 100 道题目,熟悉主流的算法问题。初次刷题时,“知识遗忘”可能是一个挑战,但请放心,这是很正常的。我们可以按照“艾宾浩斯遗忘曲线”来复习题目,通常在进行 35 轮的重复后,就能将其牢记在心。推荐的题单和刷题计划请见此 <a href="https://github.com/krahets/LeetCode-Book">GitHub 仓库</a></li>
<li><strong>阶段三:搭建知识体系</strong>。在学习方面,我们可以阅读算法专栏文章、解题框架和算法教材,以不断丰富知识体系。在刷题方面,可以尝试采用进阶刷题策略,如按专题分类、一题多解、一解多题等,相关的刷题心得可以在各个社区找到。</li>
</ol>
<p>如图 0-8 所示,本书内容主要涵盖“阶段一”,旨在帮助你更高效地展开阶段二和阶段三的学习。</p>
<p><a class="glightbox" href="../suggestions.assets/learning_route.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="算法学习路线" class="animation-figure" src="../suggestions.assets/learning_route.png" /></a></p>
<p align="center"> 图 0-8 &nbsp; 算法学习路线 </p>
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