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<h1 id="145-unbounded-knapsack-problem">14.5 &nbsp; Unbounded knapsack problem<a class="headerlink" href="#145-unbounded-knapsack-problem" title="Permanent link">&para;</a></h1>
<p>In this section, we first solve another common knapsack problem: the unbounded knapsack, and then explore a special case of it: the coin change problem.</p>
<h2 id="1451-unbounded-knapsack-problem">14.5.1 &nbsp; Unbounded knapsack problem<a class="headerlink" href="#1451-unbounded-knapsack-problem" title="Permanent link">&para;</a></h2>
<div class="admonition question">
<p class="admonition-title">Question</p>
<p>Given <span class="arithmatex">\(n\)</span> items, where the weight of the <span class="arithmatex">\(i^{th}\)</span> item is <span class="arithmatex">\(wgt[i-1]\)</span> and its value is <span class="arithmatex">\(val[i-1]\)</span>, and a backpack with a capacity of <span class="arithmatex">\(cap\)</span>. <strong>Each item can be selected multiple times</strong>. What is the maximum value of the items that can be put into the backpack without exceeding its capacity? See the example below.</p>
</div>
<p><a class="glightbox" href="../unbounded_knapsack_problem.assets/unbounded_knapsack_example.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Example data for the unbounded knapsack problem" class="animation-figure" src="../unbounded_knapsack_problem.assets/unbounded_knapsack_example.png" /></a></p>
<p align="center"> Figure 14-22 &nbsp; Example data for the unbounded knapsack problem </p>
<h3 id="1-dynamic-programming-approach">1. &nbsp; Dynamic programming approach<a class="headerlink" href="#1-dynamic-programming-approach" title="Permanent link">&para;</a></h3>
<p>The unbounded knapsack problem is very similar to the 0-1 knapsack problem, <strong>the only difference being that there is no limit on the number of times an item can be chosen</strong>.</p>
<ul>
<li>In the 0-1 knapsack problem, there is only one of each item, so after placing item <span class="arithmatex">\(i\)</span> into the backpack, you can only choose from the previous <span class="arithmatex">\(i-1\)</span> items.</li>
<li>In the unbounded knapsack problem, the quantity of each item is unlimited, so after placing item <span class="arithmatex">\(i\)</span> in the backpack, <strong>you can still choose from the previous <span class="arithmatex">\(i\)</span> items</strong>.</li>
</ul>
<p>Under the rules of the unbounded knapsack problem, the state <span class="arithmatex">\([i, c]\)</span> can change in two ways.</p>
<ul>
<li><strong>Not putting item <span class="arithmatex">\(i\)</span> in</strong>: As with the 0-1 knapsack problem, transition to <span class="arithmatex">\([i-1, c]\)</span>.</li>
<li><strong>Putting item <span class="arithmatex">\(i\)</span> in</strong>: Unlike the 0-1 knapsack problem, transition to <span class="arithmatex">\([i, c-wgt[i-1]]\)</span>.</li>
</ul>
<p>The state transition equation thus becomes:</p>
<div class="arithmatex">\[
dp[i, c] = \max(dp[i-1, c], dp[i, c - wgt[i-1]] + val[i-1])
\]</div>
<h3 id="2-code-implementation">2. &nbsp; Code implementation<a class="headerlink" href="#2-code-implementation" title="Permanent link">&para;</a></h3>
<p>Comparing the code for the two problems, the state transition changes from <span class="arithmatex">\(i-1\)</span> to <span class="arithmatex">\(i\)</span>, the rest is completely identical:</p>
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<div class="highlight"><span class="filename">unbounded_knapsack.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">unbounded_knapsack_dp</span><span class="p">(</span><span class="n">wgt</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">int</span><span class="p">],</span> <span class="n">val</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">int</span><span class="p">],</span> <span class="n">cap</span><span class="p">:</span> <span class="nb">int</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;Complete knapsack: Dynamic programming&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="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">wgt</span><span class="p">)</span>
<a id="__codelineno-0-4" name="__codelineno-0-4" href="#__codelineno-0-4"></a> <span class="c1"># Initialize dp table</span>
<a id="__codelineno-0-5" name="__codelineno-0-5" href="#__codelineno-0-5"></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">cap</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-0-6" name="__codelineno-0-6" href="#__codelineno-0-6"></a> <span class="c1"># State transition</span>
<a id="__codelineno-0-7" name="__codelineno-0-7" href="#__codelineno-0-7"></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-0-8" name="__codelineno-0-8" href="#__codelineno-0-8"></a> <span class="k">for</span> <span class="n">c</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">cap</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="k">if</span> <span class="n">wgt</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">&gt;</span> <span class="n">c</span><span class="p">:</span>
<a id="__codelineno-0-10" name="__codelineno-0-10" href="#__codelineno-0-10"></a> <span class="c1"># If exceeding the knapsack capacity, do not choose item i</span>
<a id="__codelineno-0-11" name="__codelineno-0-11" href="#__codelineno-0-11"></a> <span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">c</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">c</span><span class="p">]</span>
<a id="__codelineno-0-12" name="__codelineno-0-12" href="#__codelineno-0-12"></a> <span class="k">else</span><span class="p">:</span>
<a id="__codelineno-0-13" name="__codelineno-0-13" href="#__codelineno-0-13"></a> <span class="c1"># The greater value between not choosing and choosing item i</span>
<a id="__codelineno-0-14" name="__codelineno-0-14" href="#__codelineno-0-14"></a> <span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">c</span><span class="p">]</span> <span class="o">=</span> <span class="nb">max</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">c</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">c</span> <span class="o">-</span> <span class="n">wgt</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">val</span><span class="p">[</span><span class="n">i</span> <span class="o">-</span> <span class="mi">1</span><span class="p">])</span>
<a id="__codelineno-0-15" name="__codelineno-0-15" href="#__codelineno-0-15"></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">cap</span><span class="p">]</span>
</code></pre></div>
</div>
<div class="tabbed-block">
<div class="highlight"><span class="filename">unbounded_knapsack.cpp</span><pre><span></span><code><a id="__codelineno-1-1" name="__codelineno-1-1" href="#__codelineno-1-1"></a><span class="cm">/* Complete knapsack: Dynamic programming */</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">unboundedKnapsackDP</span><span class="p">(</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="o">&amp;</span><span class="n">wgt</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="w"> </span><span class="o">&amp;</span><span class="n">val</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">cap</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">wgt</span><span class="p">.</span><span class="n">size</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="c1">// Initialize dp table</span>
<a id="__codelineno-1-5" name="__codelineno-1-5" href="#__codelineno-1-5"></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">cap</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-1-6" name="__codelineno-1-6" href="#__codelineno-1-6"></a><span class="w"> </span><span class="c1">// State transition</span>
<a id="__codelineno-1-7" name="__codelineno-1-7" href="#__codelineno-1-7"></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-1-8" name="__codelineno-1-8" href="#__codelineno-1-8"></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">c</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">c</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">cap</span><span class="p">;</span><span class="w"> </span><span class="n">c</span><span class="o">++</span><span class="p">)</span><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">if</span><span class="w"> </span><span class="p">(</span><span class="n">wgt</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">&gt;</span><span class="w"> </span><span class="n">c</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="c1">// If exceeding the knapsack capacity, do not choose item i</span>
<a id="__codelineno-1-11" name="__codelineno-1-11" href="#__codelineno-1-11"></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">c</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">c</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="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-1-13" name="__codelineno-1-13" href="#__codelineno-1-13"></a><span class="w"> </span><span class="c1">// The greater value between not choosing and choosing item i</span>
<a id="__codelineno-1-14" name="__codelineno-1-14" href="#__codelineno-1-14"></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">c</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">max</span><span class="p">(</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">c</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="p">][</span><span class="n">c</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">wgt</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">val</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>
<a id="__codelineno-1-15" name="__codelineno-1-15" href="#__codelineno-1-15"></a><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="p">}</span>
<a id="__codelineno-1-17" name="__codelineno-1-17" href="#__codelineno-1-17"></a><span class="w"> </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="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">cap</span><span class="p">];</span>
<a id="__codelineno-1-19" name="__codelineno-1-19" href="#__codelineno-1-19"></a><span class="p">}</span>
</code></pre></div>
</div>
<div class="tabbed-block">
<div class="highlight"><span class="filename">unbounded_knapsack.java</span><pre><span></span><code><a id="__codelineno-2-1" name="__codelineno-2-1" href="#__codelineno-2-1"></a><span class="cm">/* Complete knapsack: Dynamic programming */</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">unboundedKnapsackDP</span><span class="p">(</span><span class="kt">int</span><span class="o">[]</span><span class="w"> </span><span class="n">wgt</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="o">[]</span><span class="w"> </span><span class="n">val</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">cap</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">wgt</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="c1">// Initialize dp table</span>
<a id="__codelineno-2-5" name="__codelineno-2-5" href="#__codelineno-2-5"></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">cap</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-2-6" name="__codelineno-2-6" href="#__codelineno-2-6"></a><span class="w"> </span><span class="c1">// State transition</span>
<a id="__codelineno-2-7" name="__codelineno-2-7" href="#__codelineno-2-7"></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-2-8" name="__codelineno-2-8" href="#__codelineno-2-8"></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">c</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">c</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">cap</span><span class="p">;</span><span class="w"> </span><span class="n">c</span><span class="o">++</span><span class="p">)</span><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">if</span><span class="w"> </span><span class="p">(</span><span class="n">wgt</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="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">c</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="c1">// If exceeding the knapsack capacity, do not choose item i</span>
<a id="__codelineno-2-11" name="__codelineno-2-11" href="#__codelineno-2-11"></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">c</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">c</span><span class="o">]</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="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-2-13" name="__codelineno-2-13" href="#__codelineno-2-13"></a><span class="w"> </span><span class="c1">// The greater value between not choosing and choosing item i</span>
<a id="__codelineno-2-14" name="__codelineno-2-14" href="#__codelineno-2-14"></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">c</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">max</span><span class="p">(</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">c</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="o">][</span><span class="n">c</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">wgt</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="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">val</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="p">);</span>
<a id="__codelineno-2-15" name="__codelineno-2-15" href="#__codelineno-2-15"></a><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="p">}</span>
<a id="__codelineno-2-17" name="__codelineno-2-17" href="#__codelineno-2-17"></a><span class="w"> </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="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">cap</span><span class="o">]</span><span class="p">;</span>
<a id="__codelineno-2-19" name="__codelineno-2-19" href="#__codelineno-2-19"></a><span class="p">}</span>
</code></pre></div>
</div>
<div class="tabbed-block">
<div class="highlight"><span class="filename">unbounded_knapsack.cs</span><pre><span></span><code><a id="__codelineno-3-1" name="__codelineno-3-1" href="#__codelineno-3-1"></a><span class="na">[class]</span><span class="p">{</span><span class="n">unbounded_knapsack</span><span class="p">}</span><span class="o">-</span><span class="p">[</span><span class="n">func</span><span class="p">]{</span><span class="n">UnboundedKnapsackDP</span><span class="p">}</span>
</code></pre></div>
</div>
<div class="tabbed-block">
<div class="highlight"><span class="filename">unbounded_knapsack.go</span><pre><span></span><code><a id="__codelineno-4-1" name="__codelineno-4-1" href="#__codelineno-4-1"></a><span class="p">[</span><span class="nx">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="kd">func</span><span class="p">]{</span><span class="nx">unboundedKnapsackDP</span><span class="p">}</span>
</code></pre></div>
</div>
<div class="tabbed-block">
<div class="highlight"><span class="filename">unbounded_knapsack.swift</span><pre><span></span><code><a id="__codelineno-5-1" name="__codelineno-5-1" href="#__codelineno-5-1"></a><span class="p">[</span><span class="kd">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="kd">func</span><span class="p">]{</span><span class="n">unboundedKnapsackDP</span><span class="p">}</span>
</code></pre></div>
</div>
<div class="tabbed-block">
<div class="highlight"><span class="filename">unbounded_knapsack.js</span><pre><span></span><code><a id="__codelineno-6-1" name="__codelineno-6-1" href="#__codelineno-6-1"></a><span class="p">[</span><span class="kd">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="nx">func</span><span class="p">]{</span><span class="nx">unboundedKnapsackDP</span><span class="p">}</span>
</code></pre></div>
</div>
<div class="tabbed-block">
<div class="highlight"><span class="filename">unbounded_knapsack.ts</span><pre><span></span><code><a id="__codelineno-7-1" name="__codelineno-7-1" href="#__codelineno-7-1"></a><span class="p">[</span><span class="kd">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="nx">func</span><span class="p">]{</span><span class="nx">unboundedKnapsackDP</span><span class="p">}</span>
</code></pre></div>
</div>
<div class="tabbed-block">
<div class="highlight"><span class="filename">unbounded_knapsack.dart</span><pre><span></span><code><a id="__codelineno-8-1" name="__codelineno-8-1" href="#__codelineno-8-1"></a><span class="p">[</span><span class="n">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="n">func</span><span class="p">]{</span><span class="n">unboundedKnapsackDP</span><span class="p">}</span>
</code></pre></div>
</div>
<div class="tabbed-block">
<div class="highlight"><span class="filename">unbounded_knapsack.rs</span><pre><span></span><code><a id="__codelineno-9-1" name="__codelineno-9-1" href="#__codelineno-9-1"></a><span class="p">[</span><span class="n">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="n">func</span><span class="p">]{</span><span class="n">unbounded_knapsack_dp</span><span class="p">}</span>
</code></pre></div>
</div>
<div class="tabbed-block">
<div class="highlight"><span class="filename">unbounded_knapsack.c</span><pre><span></span><code><a id="__codelineno-10-1" name="__codelineno-10-1" href="#__codelineno-10-1"></a><span class="p">[</span><span class="n">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="n">func</span><span class="p">]{</span><span class="n">unboundedKnapsackDP</span><span class="p">}</span>
</code></pre></div>
</div>
<div class="tabbed-block">
<div class="highlight"><span class="filename">unbounded_knapsack.kt</span><pre><span></span><code><a id="__codelineno-11-1" name="__codelineno-11-1" href="#__codelineno-11-1"></a><span class="o">[</span><span class="n">class</span><span class="o">]</span><span class="p">{}</span><span class="o">-[</span><span class="n">func</span><span class="o">]</span><span class="p">{</span><span class="n">unboundedKnapsackDP</span><span class="p">}</span>
</code></pre></div>
</div>
<div class="tabbed-block">
<div class="highlight"><span class="filename">unbounded_knapsack.rb</span><pre><span></span><code><a id="__codelineno-12-1" name="__codelineno-12-1" href="#__codelineno-12-1"></a><span class="o">[</span><span class="n">class</span><span class="o">]</span><span class="p">{}</span><span class="o">-[</span><span class="n">func</span><span class="o">]</span><span class="p">{</span><span class="n">unbounded_knapsack_dp</span><span class="p">}</span>
</code></pre></div>
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<div class="highlight"><span class="filename">unbounded_knapsack.zig</span><pre><span></span><code><a id="__codelineno-13-1" name="__codelineno-13-1" href="#__codelineno-13-1"></a><span class="p">[</span><span class="n">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="n">func</span><span class="p">]{</span><span class="n">unboundedKnapsackDP</span><span class="p">}</span>
</code></pre></div>
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<h3 id="3-space-optimization">3. &nbsp; Space optimization<a class="headerlink" href="#3-space-optimization" title="Permanent link">&para;</a></h3>
<p>Since the current state comes from the state to the left and above, <strong>the space-optimized solution should perform a forward traversal for each row in the <span class="arithmatex">\(dp\)</span> table</strong>.</p>
<p>This traversal order is the opposite of that for the 0-1 knapsack. Please refer to Figure 14-23 to understand the difference.</p>
<div class="tabbed-set tabbed-alternate" data-tabs="2:6"><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" /><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></div>
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<p><a class="glightbox" href="../unbounded_knapsack_problem.assets/unbounded_knapsack_dp_comp_step1.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Dynamic programming process for the unbounded knapsack problem after space optimization" class="animation-figure" src="../unbounded_knapsack_problem.assets/unbounded_knapsack_dp_comp_step1.png" /></a></p>
</div>
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<p><a class="glightbox" href="../unbounded_knapsack_problem.assets/unbounded_knapsack_dp_comp_step2.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="unbounded_knapsack_dp_comp_step2" class="animation-figure" src="../unbounded_knapsack_problem.assets/unbounded_knapsack_dp_comp_step2.png" /></a></p>
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<p><a class="glightbox" href="../unbounded_knapsack_problem.assets/unbounded_knapsack_dp_comp_step3.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="unbounded_knapsack_dp_comp_step3" class="animation-figure" src="../unbounded_knapsack_problem.assets/unbounded_knapsack_dp_comp_step3.png" /></a></p>
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<p><a class="glightbox" href="../unbounded_knapsack_problem.assets/unbounded_knapsack_dp_comp_step4.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="unbounded_knapsack_dp_comp_step4" class="animation-figure" src="../unbounded_knapsack_problem.assets/unbounded_knapsack_dp_comp_step4.png" /></a></p>
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<p><a class="glightbox" href="../unbounded_knapsack_problem.assets/unbounded_knapsack_dp_comp_step5.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="unbounded_knapsack_dp_comp_step5" class="animation-figure" src="../unbounded_knapsack_problem.assets/unbounded_knapsack_dp_comp_step5.png" /></a></p>
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<p><a class="glightbox" href="../unbounded_knapsack_problem.assets/unbounded_knapsack_dp_comp_step6.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="unbounded_knapsack_dp_comp_step6" class="animation-figure" src="../unbounded_knapsack_problem.assets/unbounded_knapsack_dp_comp_step6.png" /></a></p>
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<p align="center"> Figure 14-23 &nbsp; Dynamic programming process for the unbounded knapsack problem after space optimization </p>
<p>The code implementation is quite simple, just remove the first dimension of the array <code>dp</code>:</p>
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<div class="highlight"><span class="filename">unbounded_knapsack.py</span><pre><span></span><code><a id="__codelineno-14-1" name="__codelineno-14-1" href="#__codelineno-14-1"></a><span class="k">def</span> <span class="nf">unbounded_knapsack_dp_comp</span><span class="p">(</span><span class="n">wgt</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">int</span><span class="p">],</span> <span class="n">val</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">int</span><span class="p">],</span> <span class="n">cap</span><span class="p">:</span> <span class="nb">int</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="nb">int</span><span class="p">:</span>
<a id="__codelineno-14-2" name="__codelineno-14-2" href="#__codelineno-14-2"></a><span class="w"> </span><span class="sd">&quot;&quot;&quot;Complete knapsack: Space-optimized dynamic programming&quot;&quot;&quot;</span>
<a id="__codelineno-14-3" name="__codelineno-14-3" href="#__codelineno-14-3"></a> <span class="n">n</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">wgt</span><span class="p">)</span>
<a id="__codelineno-14-4" name="__codelineno-14-4" href="#__codelineno-14-4"></a> <span class="c1"># Initialize dp table</span>
<a id="__codelineno-14-5" name="__codelineno-14-5" href="#__codelineno-14-5"></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">cap</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
<a id="__codelineno-14-6" name="__codelineno-14-6" href="#__codelineno-14-6"></a> <span class="c1"># State transition</span>
<a id="__codelineno-14-7" name="__codelineno-14-7" href="#__codelineno-14-7"></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-14-8" name="__codelineno-14-8" href="#__codelineno-14-8"></a> <span class="c1"># Traverse in order</span>
<a id="__codelineno-14-9" name="__codelineno-14-9" href="#__codelineno-14-9"></a> <span class="k">for</span> <span class="n">c</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">cap</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
<a id="__codelineno-14-10" name="__codelineno-14-10" href="#__codelineno-14-10"></a> <span class="k">if</span> <span class="n">wgt</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">&gt;</span> <span class="n">c</span><span class="p">:</span>
<a id="__codelineno-14-11" name="__codelineno-14-11" href="#__codelineno-14-11"></a> <span class="c1"># If exceeding the knapsack capacity, do not choose item i</span>
<a id="__codelineno-14-12" name="__codelineno-14-12" href="#__codelineno-14-12"></a> <span class="n">dp</span><span class="p">[</span><span class="n">c</span><span class="p">]</span> <span class="o">=</span> <span class="n">dp</span><span class="p">[</span><span class="n">c</span><span class="p">]</span>
<a id="__codelineno-14-13" name="__codelineno-14-13" href="#__codelineno-14-13"></a> <span class="k">else</span><span class="p">:</span>
<a id="__codelineno-14-14" name="__codelineno-14-14" href="#__codelineno-14-14"></a> <span class="c1"># The greater value between not choosing and choosing item i</span>
<a id="__codelineno-14-15" name="__codelineno-14-15" href="#__codelineno-14-15"></a> <span class="n">dp</span><span class="p">[</span><span class="n">c</span><span class="p">]</span> <span class="o">=</span> <span class="nb">max</span><span class="p">(</span><span class="n">dp</span><span class="p">[</span><span class="n">c</span><span class="p">],</span> <span class="n">dp</span><span class="p">[</span><span class="n">c</span> <span class="o">-</span> <span class="n">wgt</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">val</span><span class="p">[</span><span class="n">i</span> <span class="o">-</span> <span class="mi">1</span><span class="p">])</span>
<a id="__codelineno-14-16" name="__codelineno-14-16" href="#__codelineno-14-16"></a> <span class="k">return</span> <span class="n">dp</span><span class="p">[</span><span class="n">cap</span><span class="p">]</span>
</code></pre></div>
</div>
<div class="tabbed-block">
<div class="highlight"><span class="filename">unbounded_knapsack.cpp</span><pre><span></span><code><a id="__codelineno-15-1" name="__codelineno-15-1" href="#__codelineno-15-1"></a><span class="cm">/* Complete knapsack: Space-optimized dynamic programming */</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">unboundedKnapsackDPComp</span><span class="p">(</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="o">&amp;</span><span class="n">wgt</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="w"> </span><span class="o">&amp;</span><span class="n">val</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">cap</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">wgt</span><span class="p">.</span><span class="n">size</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="c1">// Initialize dp table</span>
<a id="__codelineno-15-5" name="__codelineno-15-5" href="#__codelineno-15-5"></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">cap</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-15-6" name="__codelineno-15-6" href="#__codelineno-15-6"></a><span class="w"> </span><span class="c1">// State transition</span>
<a id="__codelineno-15-7" name="__codelineno-15-7" href="#__codelineno-15-7"></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-15-8" name="__codelineno-15-8" href="#__codelineno-15-8"></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">c</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">c</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">cap</span><span class="p">;</span><span class="w"> </span><span class="n">c</span><span class="o">++</span><span class="p">)</span><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="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">wgt</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">&gt;</span><span class="w"> </span><span class="n">c</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-15-10" name="__codelineno-15-10" href="#__codelineno-15-10"></a><span class="w"> </span><span class="c1">// If exceeding the knapsack capacity, do not choose item i</span>
<a id="__codelineno-15-11" name="__codelineno-15-11" href="#__codelineno-15-11"></a><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">c</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">c</span><span class="p">];</span>
<a id="__codelineno-15-12" name="__codelineno-15-12" href="#__codelineno-15-12"></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-15-13" name="__codelineno-15-13" href="#__codelineno-15-13"></a><span class="w"> </span><span class="c1">// The greater value between not choosing and choosing item i</span>
<a id="__codelineno-15-14" name="__codelineno-15-14" href="#__codelineno-15-14"></a><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">c</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">max</span><span class="p">(</span><span class="n">dp</span><span class="p">[</span><span class="n">c</span><span class="p">],</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">c</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">wgt</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">val</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>
<a id="__codelineno-15-15" name="__codelineno-15-15" href="#__codelineno-15-15"></a><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="p">}</span>
<a id="__codelineno-15-17" name="__codelineno-15-17" href="#__codelineno-15-17"></a><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="k">return</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">cap</span><span class="p">];</span>
<a id="__codelineno-15-19" name="__codelineno-15-19" href="#__codelineno-15-19"></a><span class="p">}</span>
</code></pre></div>
</div>
<div class="tabbed-block">
<div class="highlight"><span class="filename">unbounded_knapsack.java</span><pre><span></span><code><a id="__codelineno-16-1" name="__codelineno-16-1" href="#__codelineno-16-1"></a><span class="cm">/* Complete knapsack: Space-optimized dynamic programming */</span>
<a id="__codelineno-16-2" name="__codelineno-16-2" href="#__codelineno-16-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">unboundedKnapsackDPComp</span><span class="p">(</span><span class="kt">int</span><span class="o">[]</span><span class="w"> </span><span class="n">wgt</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="o">[]</span><span class="w"> </span><span class="n">val</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">cap</span><span class="p">)</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="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">wgt</span><span class="p">.</span><span class="na">length</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="c1">// Initialize dp table</span>
<a id="__codelineno-16-5" name="__codelineno-16-5" href="#__codelineno-16-5"></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">cap</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-16-6" name="__codelineno-16-6" href="#__codelineno-16-6"></a><span class="w"> </span><span class="c1">// State transition</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="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-16-8" name="__codelineno-16-8" href="#__codelineno-16-8"></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">c</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">c</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">cap</span><span class="p">;</span><span class="w"> </span><span class="n">c</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-16-9" name="__codelineno-16-9" href="#__codelineno-16-9"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">wgt</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="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">c</span><span class="p">)</span><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">// If exceeding the knapsack capacity, do not choose item i</span>
<a id="__codelineno-16-11" name="__codelineno-16-11" href="#__codelineno-16-11"></a><span class="w"> </span><span class="n">dp</span><span class="o">[</span><span class="n">c</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">c</span><span class="o">]</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="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-16-13" name="__codelineno-16-13" href="#__codelineno-16-13"></a><span class="w"> </span><span class="c1">// The greater value between not choosing and choosing item i</span>
<a id="__codelineno-16-14" name="__codelineno-16-14" href="#__codelineno-16-14"></a><span class="w"> </span><span class="n">dp</span><span class="o">[</span><span class="n">c</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">max</span><span class="p">(</span><span class="n">dp</span><span class="o">[</span><span class="n">c</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">c</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">wgt</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="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">val</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="p">);</span>
<a id="__codelineno-16-15" name="__codelineno-16-15" href="#__codelineno-16-15"></a><span class="w"> </span><span class="p">}</span>
<a id="__codelineno-16-16" name="__codelineno-16-16" href="#__codelineno-16-16"></a><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="p">}</span>
<a id="__codelineno-16-18" name="__codelineno-16-18" href="#__codelineno-16-18"></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">cap</span><span class="o">]</span><span class="p">;</span>
<a id="__codelineno-16-19" name="__codelineno-16-19" href="#__codelineno-16-19"></a><span class="p">}</span>
</code></pre></div>
</div>
<div class="tabbed-block">
<div class="highlight"><span class="filename">unbounded_knapsack.cs</span><pre><span></span><code><a id="__codelineno-17-1" name="__codelineno-17-1" href="#__codelineno-17-1"></a><span class="na">[class]</span><span class="p">{</span><span class="n">unbounded_knapsack</span><span class="p">}</span><span class="o">-</span><span class="p">[</span><span class="n">func</span><span class="p">]{</span><span class="n">UnboundedKnapsackDPComp</span><span class="p">}</span>
</code></pre></div>
</div>
<div class="tabbed-block">
<div class="highlight"><span class="filename">unbounded_knapsack.go</span><pre><span></span><code><a id="__codelineno-18-1" name="__codelineno-18-1" href="#__codelineno-18-1"></a><span class="p">[</span><span class="nx">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="kd">func</span><span class="p">]{</span><span class="nx">unboundedKnapsackDPComp</span><span class="p">}</span>
</code></pre></div>
</div>
<div class="tabbed-block">
<div class="highlight"><span class="filename">unbounded_knapsack.swift</span><pre><span></span><code><a id="__codelineno-19-1" name="__codelineno-19-1" href="#__codelineno-19-1"></a><span class="p">[</span><span class="kd">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="kd">func</span><span class="p">]{</span><span class="n">unboundedKnapsackDPComp</span><span class="p">}</span>
</code></pre></div>
</div>
<div class="tabbed-block">
<div class="highlight"><span class="filename">unbounded_knapsack.js</span><pre><span></span><code><a id="__codelineno-20-1" name="__codelineno-20-1" href="#__codelineno-20-1"></a><span class="p">[</span><span class="kd">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="nx">func</span><span class="p">]{</span><span class="nx">unboundedKnapsackDPComp</span><span class="p">}</span>
</code></pre></div>
</div>
<div class="tabbed-block">
<div class="highlight"><span class="filename">unbounded_knapsack.ts</span><pre><span></span><code><a id="__codelineno-21-1" name="__codelineno-21-1" href="#__codelineno-21-1"></a><span class="p">[</span><span class="kd">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="nx">func</span><span class="p">]{</span><span class="nx">unboundedKnapsackDPComp</span><span class="p">}</span>
</code></pre></div>
</div>
<div class="tabbed-block">
<div class="highlight"><span class="filename">unbounded_knapsack.dart</span><pre><span></span><code><a id="__codelineno-22-1" name="__codelineno-22-1" href="#__codelineno-22-1"></a><span class="p">[</span><span class="n">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="n">func</span><span class="p">]{</span><span class="n">unboundedKnapsackDPComp</span><span class="p">}</span>
</code></pre></div>
</div>
<div class="tabbed-block">
<div class="highlight"><span class="filename">unbounded_knapsack.rs</span><pre><span></span><code><a id="__codelineno-23-1" name="__codelineno-23-1" href="#__codelineno-23-1"></a><span class="p">[</span><span class="n">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="n">func</span><span class="p">]{</span><span class="n">unbounded_knapsack_dp_comp</span><span class="p">}</span>
</code></pre></div>
</div>
<div class="tabbed-block">
<div class="highlight"><span class="filename">unbounded_knapsack.c</span><pre><span></span><code><a id="__codelineno-24-1" name="__codelineno-24-1" href="#__codelineno-24-1"></a><span class="p">[</span><span class="n">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="n">func</span><span class="p">]{</span><span class="n">unboundedKnapsackDPComp</span><span class="p">}</span>
</code></pre></div>
</div>
<div class="tabbed-block">
<div class="highlight"><span class="filename">unbounded_knapsack.kt</span><pre><span></span><code><a id="__codelineno-25-1" name="__codelineno-25-1" href="#__codelineno-25-1"></a><span class="o">[</span><span class="n">class</span><span class="o">]</span><span class="p">{}</span><span class="o">-[</span><span class="n">func</span><span class="o">]</span><span class="p">{</span><span class="n">unboundedKnapsackDPComp</span><span class="p">}</span>
</code></pre></div>
</div>
<div class="tabbed-block">
<div class="highlight"><span class="filename">unbounded_knapsack.rb</span><pre><span></span><code><a id="__codelineno-26-1" name="__codelineno-26-1" href="#__codelineno-26-1"></a><span class="o">[</span><span class="n">class</span><span class="o">]</span><span class="p">{}</span><span class="o">-[</span><span class="n">func</span><span class="o">]</span><span class="p">{</span><span class="n">unbounded_knapsack_dp_comp</span><span class="p">}</span>
</code></pre></div>
</div>
<div class="tabbed-block">
<div class="highlight"><span class="filename">unbounded_knapsack.zig</span><pre><span></span><code><a id="__codelineno-27-1" name="__codelineno-27-1" href="#__codelineno-27-1"></a><span class="p">[</span><span class="n">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="n">func</span><span class="p">]{</span><span class="n">unboundedKnapsackDPComp</span><span class="p">}</span>
</code></pre></div>
</div>
</div>
</div>
<h2 id="1452-coin-change-problem">14.5.2 &nbsp; Coin change problem<a class="headerlink" href="#1452-coin-change-problem" title="Permanent link">&para;</a></h2>
<p>The knapsack problem is a representative of a large class of dynamic programming problems and has many variants, such as the coin change problem.</p>
<div class="admonition question">
<p class="admonition-title">Question</p>
<p>Given <span class="arithmatex">\(n\)</span> types of coins, the denomination of the <span class="arithmatex">\(i^{th}\)</span> type of coin is <span class="arithmatex">\(coins[i - 1]\)</span>, and the target amount is <span class="arithmatex">\(amt\)</span>. <strong>Each type of coin can be selected multiple times</strong>. What is the minimum number of coins needed to make up the target amount? If it is impossible to make up the target amount, return <span class="arithmatex">\(-1\)</span>. See the example below.</p>
</div>
<p><a class="glightbox" href="../unbounded_knapsack_problem.assets/coin_change_example.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Example data for the coin change problem" class="animation-figure" src="../unbounded_knapsack_problem.assets/coin_change_example.png" /></a></p>
<p align="center"> Figure 14-24 &nbsp; Example data for the coin change problem </p>
<h3 id="1-dynamic-programming-approach_1">1. &nbsp; Dynamic programming approach<a class="headerlink" href="#1-dynamic-programming-approach_1" title="Permanent link">&para;</a></h3>
<p><strong>The coin change can be seen as a special case of the unbounded knapsack problem</strong>, sharing the following similarities and differences.</p>
<ul>
<li>The two problems can be converted into each other: "item" corresponds to "coin", "item weight" corresponds to "coin denomination", and "backpack capacity" corresponds to "target amount".</li>
<li>The optimization goals are opposite: the unbounded knapsack problem aims to maximize the value of items, while the coin change problem aims to minimize the number of coins.</li>
<li>The unbounded knapsack problem seeks solutions "not exceeding" the backpack capacity, while the coin change seeks solutions that "exactly" make up the target amount.</li>
</ul>
<p><strong>First step: Think through each round's decision-making, define the state, and thus derive the <span class="arithmatex">\(dp\)</span> table</strong></p>
<p>The state <span class="arithmatex">\([i, a]\)</span> corresponds to the sub-problem: <strong>the minimum number of coins that can make up the amount <span class="arithmatex">\(a\)</span> using the first <span class="arithmatex">\(i\)</span> types of coins</strong>, denoted as <span class="arithmatex">\(dp[i, a]\)</span>.</p>
<p>The two-dimensional <span class="arithmatex">\(dp\)</span> table is of size <span class="arithmatex">\((n+1) \times (amt+1)\)</span>.</p>
<p><strong>Second step: Identify the optimal substructure and derive the state transition equation</strong></p>
<p>This problem differs from the unbounded knapsack problem in two aspects of the state transition equation.</p>
<ul>
<li>This problem seeks the minimum, so the operator <span class="arithmatex">\(\max()\)</span> needs to be changed to <span class="arithmatex">\(\min()\)</span>.</li>
<li>The optimization is focused on the number of coins, so simply add <span class="arithmatex">\(+1\)</span> when a coin is chosen.</li>
</ul>
<div class="arithmatex">\[
dp[i, a] = \min(dp[i-1, a], dp[i, a - coins[i-1]] + 1)
\]</div>
<p><strong>Third step: Define boundary conditions and state transition order</strong></p>
<p>When the target amount is <span class="arithmatex">\(0\)</span>, the minimum number of coins needed to make it up is <span class="arithmatex">\(0\)</span>, so all <span class="arithmatex">\(dp[i, 0]\)</span> in the first column are <span class="arithmatex">\(0\)</span>.</p>
<p>When there are no coins, <strong>it is impossible to make up any amount &gt;0</strong>, which is an invalid solution. To allow the <span class="arithmatex">\(\min()\)</span> function in the state transition equation to recognize and filter out invalid solutions, consider using <span class="arithmatex">\(+\infty\)</span> to represent them, i.e., set all <span class="arithmatex">\(dp[0, a]\)</span> in the first row to <span class="arithmatex">\(+\infty\)</span>.</p>
<h3 id="2-code-implementation_1">2. &nbsp; Code implementation<a class="headerlink" href="#2-code-implementation_1" title="Permanent link">&para;</a></h3>
<p>Most programming languages do not provide a <span class="arithmatex">\(+\infty\)</span> variable, only the maximum value of an integer <code>int</code> can be used as a substitute. This can lead to overflow: the <span class="arithmatex">\(+1\)</span> operation in the state transition equation may overflow.</p>
<p>For this reason, we use the number <span class="arithmatex">\(amt + 1\)</span> to represent an invalid solution, because the maximum number of coins needed to make up <span class="arithmatex">\(amt\)</span> is at most <span class="arithmatex">\(amt\)</span>. Before returning the result, check if <span class="arithmatex">\(dp[n, amt]\)</span> equals <span class="arithmatex">\(amt + 1\)</span>, and if so, return <span class="arithmatex">\(-1\)</span>, indicating that the target amount cannot be made up. The code is as follows:</p>
<div class="tabbed-set tabbed-alternate" data-tabs="4:14"><input checked="checked" id="__tabbed_4_1" name="__tabbed_4" type="radio" /><input id="__tabbed_4_2" name="__tabbed_4" type="radio" /><input id="__tabbed_4_3" name="__tabbed_4" type="radio" /><input id="__tabbed_4_4" name="__tabbed_4" type="radio" /><input id="__tabbed_4_5" name="__tabbed_4" type="radio" /><input id="__tabbed_4_6" name="__tabbed_4" type="radio" /><input id="__tabbed_4_7" name="__tabbed_4" type="radio" /><input id="__tabbed_4_8" name="__tabbed_4" type="radio" /><input id="__tabbed_4_9" name="__tabbed_4" type="radio" /><input id="__tabbed_4_10" name="__tabbed_4" type="radio" /><input id="__tabbed_4_11" name="__tabbed_4" type="radio" /><input id="__tabbed_4_12" name="__tabbed_4" type="radio" /><input id="__tabbed_4_13" name="__tabbed_4" type="radio" /><input id="__tabbed_4_14" name="__tabbed_4" type="radio" /><div class="tabbed-labels"><label for="__tabbed_4_1">Python</label><label for="__tabbed_4_2">C++</label><label for="__tabbed_4_3">Java</label><label for="__tabbed_4_4">C#</label><label for="__tabbed_4_5">Go</label><label for="__tabbed_4_6">Swift</label><label for="__tabbed_4_7">JS</label><label for="__tabbed_4_8">TS</label><label for="__tabbed_4_9">Dart</label><label for="__tabbed_4_10">Rust</label><label for="__tabbed_4_11">C</label><label for="__tabbed_4_12">Kotlin</label><label for="__tabbed_4_13">Ruby</label><label for="__tabbed_4_14">Zig</label></div>
<div class="tabbed-content">
<div class="tabbed-block">
<div class="highlight"><span class="filename">coin_change.py</span><pre><span></span><code><a id="__codelineno-28-1" name="__codelineno-28-1" href="#__codelineno-28-1"></a><span class="k">def</span> <span class="nf">coin_change_dp</span><span class="p">(</span><span class="n">coins</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">int</span><span class="p">],</span> <span class="n">amt</span><span class="p">:</span> <span class="nb">int</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="nb">int</span><span class="p">:</span>
<a id="__codelineno-28-2" name="__codelineno-28-2" href="#__codelineno-28-2"></a><span class="w"> </span><span class="sd">&quot;&quot;&quot;Coin change: Dynamic programming&quot;&quot;&quot;</span>
<a id="__codelineno-28-3" name="__codelineno-28-3" href="#__codelineno-28-3"></a> <span class="n">n</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">coins</span><span class="p">)</span>
<a id="__codelineno-28-4" name="__codelineno-28-4" href="#__codelineno-28-4"></a> <span class="n">MAX</span> <span class="o">=</span> <span class="n">amt</span> <span class="o">+</span> <span class="mi">1</span>
<a id="__codelineno-28-5" name="__codelineno-28-5" href="#__codelineno-28-5"></a> <span class="c1"># Initialize dp table</span>
<a id="__codelineno-28-6" name="__codelineno-28-6" href="#__codelineno-28-6"></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">amt</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-28-7" name="__codelineno-28-7" href="#__codelineno-28-7"></a> <span class="c1"># State transition: first row and first column</span>
<a id="__codelineno-28-8" name="__codelineno-28-8" href="#__codelineno-28-8"></a> <span class="k">for</span> <span class="n">a</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">amt</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
<a id="__codelineno-28-9" name="__codelineno-28-9" href="#__codelineno-28-9"></a> <span class="n">dp</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="n">a</span><span class="p">]</span> <span class="o">=</span> <span class="n">MAX</span>
<a id="__codelineno-28-10" name="__codelineno-28-10" href="#__codelineno-28-10"></a> <span class="c1"># State transition: the rest of the rows and columns</span>
<a id="__codelineno-28-11" name="__codelineno-28-11" href="#__codelineno-28-11"></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-28-12" name="__codelineno-28-12" href="#__codelineno-28-12"></a> <span class="k">for</span> <span class="n">a</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">amt</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
<a id="__codelineno-28-13" name="__codelineno-28-13" href="#__codelineno-28-13"></a> <span class="k">if</span> <span class="n">coins</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">&gt;</span> <span class="n">a</span><span class="p">:</span>
<a id="__codelineno-28-14" name="__codelineno-28-14" href="#__codelineno-28-14"></a> <span class="c1"># If exceeding the target amount, do not choose coin i</span>
<a id="__codelineno-28-15" name="__codelineno-28-15" href="#__codelineno-28-15"></a> <span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">a</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">a</span><span class="p">]</span>
<a id="__codelineno-28-16" name="__codelineno-28-16" href="#__codelineno-28-16"></a> <span class="k">else</span><span class="p">:</span>
<a id="__codelineno-28-17" name="__codelineno-28-17" href="#__codelineno-28-17"></a> <span class="c1"># The smaller value between not choosing and choosing coin i</span>
<a id="__codelineno-28-18" name="__codelineno-28-18" href="#__codelineno-28-18"></a> <span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">a</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="o">-</span> <span class="mi">1</span><span class="p">][</span><span class="n">a</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">a</span> <span class="o">-</span> <span class="n">coins</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="mi">1</span><span class="p">)</span>
<a id="__codelineno-28-19" name="__codelineno-28-19" href="#__codelineno-28-19"></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">amt</span><span class="p">]</span> <span class="k">if</span> <span class="n">dp</span><span class="p">[</span><span class="n">n</span><span class="p">][</span><span class="n">amt</span><span class="p">]</span> <span class="o">!=</span> <span class="n">MAX</span> <span class="k">else</span> <span class="o">-</span><span class="mi">1</span>
</code></pre></div>
</div>
<div class="tabbed-block">
<div class="highlight"><span class="filename">coin_change.cpp</span><pre><span></span><code><a id="__codelineno-29-1" name="__codelineno-29-1" href="#__codelineno-29-1"></a><span class="cm">/* Coin change: Dynamic programming */</span>
<a id="__codelineno-29-2" name="__codelineno-29-2" href="#__codelineno-29-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">coinChangeDP</span><span class="p">(</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="o">&amp;</span><span class="n">coins</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">amt</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-29-3" name="__codelineno-29-3" href="#__codelineno-29-3"></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">coins</span><span class="p">.</span><span class="n">size</span><span class="p">();</span>
<a id="__codelineno-29-4" name="__codelineno-29-4" href="#__codelineno-29-4"></a><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">MAX</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">amt</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-29-5" name="__codelineno-29-5" href="#__codelineno-29-5"></a><span class="w"> </span><span class="c1">// Initialize dp table</span>
<a id="__codelineno-29-6" name="__codelineno-29-6" href="#__codelineno-29-6"></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">amt</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-29-7" name="__codelineno-29-7" href="#__codelineno-29-7"></a><span class="w"> </span><span class="c1">// State transition: first row and first column</span>
<a id="__codelineno-29-8" name="__codelineno-29-8" href="#__codelineno-29-8"></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">a</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">a</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">amt</span><span class="p">;</span><span class="w"> </span><span class="n">a</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-29-9" name="__codelineno-29-9" href="#__codelineno-29-9"></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">a</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">MAX</span><span class="p">;</span>
<a id="__codelineno-29-10" name="__codelineno-29-10" href="#__codelineno-29-10"></a><span class="w"> </span><span class="p">}</span>
<a id="__codelineno-29-11" name="__codelineno-29-11" href="#__codelineno-29-11"></a><span class="w"> </span><span class="c1">// State transition: the rest of the rows and columns</span>
<a id="__codelineno-29-12" name="__codelineno-29-12" href="#__codelineno-29-12"></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-29-13" name="__codelineno-29-13" href="#__codelineno-29-13"></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">a</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">a</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">amt</span><span class="p">;</span><span class="w"> </span><span class="n">a</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-29-14" name="__codelineno-29-14" href="#__codelineno-29-14"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">coins</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">&gt;</span><span class="w"> </span><span class="n">a</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-29-15" name="__codelineno-29-15" href="#__codelineno-29-15"></a><span class="w"> </span><span class="c1">// If exceeding the target amount, do not choose coin i</span>
<a id="__codelineno-29-16" name="__codelineno-29-16" href="#__codelineno-29-16"></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">a</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">a</span><span class="p">];</span>
<a id="__codelineno-29-17" name="__codelineno-29-17" href="#__codelineno-29-17"></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-29-18" name="__codelineno-29-18" href="#__codelineno-29-18"></a><span class="w"> </span><span class="c1">// The smaller value between not choosing and choosing coin i</span>
<a id="__codelineno-29-19" name="__codelineno-29-19" href="#__codelineno-29-19"></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">a</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">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">a</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="p">][</span><span class="n">a</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">coins</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="mi">1</span><span class="p">);</span>
<a id="__codelineno-29-20" name="__codelineno-29-20" href="#__codelineno-29-20"></a><span class="w"> </span><span class="p">}</span>
<a id="__codelineno-29-21" name="__codelineno-29-21" href="#__codelineno-29-21"></a><span class="w"> </span><span class="p">}</span>
<a id="__codelineno-29-22" name="__codelineno-29-22" href="#__codelineno-29-22"></a><span class="w"> </span><span class="p">}</span>
<a id="__codelineno-29-23" name="__codelineno-29-23" href="#__codelineno-29-23"></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">amt</span><span class="p">]</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="n">MAX</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">amt</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-29-24" name="__codelineno-29-24" href="#__codelineno-29-24"></a><span class="p">}</span>
</code></pre></div>
</div>
<div class="tabbed-block">
<div class="highlight"><span class="filename">coin_change.java</span><pre><span></span><code><a id="__codelineno-30-1" name="__codelineno-30-1" href="#__codelineno-30-1"></a><span class="cm">/* Coin change: Dynamic programming */</span>
<a id="__codelineno-30-2" name="__codelineno-30-2" href="#__codelineno-30-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">coinChangeDP</span><span class="p">(</span><span class="kt">int</span><span class="o">[]</span><span class="w"> </span><span class="n">coins</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">amt</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-30-3" name="__codelineno-30-3" href="#__codelineno-30-3"></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">coins</span><span class="p">.</span><span class="na">length</span><span class="p">;</span>
<a id="__codelineno-30-4" name="__codelineno-30-4" href="#__codelineno-30-4"></a><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">MAX</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">amt</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-30-5" name="__codelineno-30-5" href="#__codelineno-30-5"></a><span class="w"> </span><span class="c1">// Initialize dp table</span>
<a id="__codelineno-30-6" name="__codelineno-30-6" href="#__codelineno-30-6"></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">amt</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-30-7" name="__codelineno-30-7" href="#__codelineno-30-7"></a><span class="w"> </span><span class="c1">// State transition: first row and first column</span>
<a id="__codelineno-30-8" name="__codelineno-30-8" href="#__codelineno-30-8"></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">a</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">a</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">amt</span><span class="p">;</span><span class="w"> </span><span class="n">a</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-30-9" name="__codelineno-30-9" href="#__codelineno-30-9"></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">a</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">MAX</span><span class="p">;</span>
<a id="__codelineno-30-10" name="__codelineno-30-10" href="#__codelineno-30-10"></a><span class="w"> </span><span class="p">}</span>
<a id="__codelineno-30-11" name="__codelineno-30-11" href="#__codelineno-30-11"></a><span class="w"> </span><span class="c1">// State transition: the rest of the rows and columns</span>
<a id="__codelineno-30-12" name="__codelineno-30-12" href="#__codelineno-30-12"></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-30-13" name="__codelineno-30-13" href="#__codelineno-30-13"></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">a</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">a</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">amt</span><span class="p">;</span><span class="w"> </span><span class="n">a</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-30-14" name="__codelineno-30-14" href="#__codelineno-30-14"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">coins</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="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">a</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-30-15" name="__codelineno-30-15" href="#__codelineno-30-15"></a><span class="w"> </span><span class="c1">// If exceeding the target amount, do not choose coin i</span>
<a id="__codelineno-30-16" name="__codelineno-30-16" href="#__codelineno-30-16"></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">a</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">a</span><span class="o">]</span><span class="p">;</span>
<a id="__codelineno-30-17" name="__codelineno-30-17" href="#__codelineno-30-17"></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-30-18" name="__codelineno-30-18" href="#__codelineno-30-18"></a><span class="w"> </span><span class="c1">// The smaller value between not choosing and choosing coin i</span>
<a id="__codelineno-30-19" name="__codelineno-30-19" href="#__codelineno-30-19"></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">a</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">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">a</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="o">][</span><span class="n">a</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">coins</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="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">);</span>
<a id="__codelineno-30-20" name="__codelineno-30-20" href="#__codelineno-30-20"></a><span class="w"> </span><span class="p">}</span>
<a id="__codelineno-30-21" name="__codelineno-30-21" href="#__codelineno-30-21"></a><span class="w"> </span><span class="p">}</span>
<a id="__codelineno-30-22" name="__codelineno-30-22" href="#__codelineno-30-22"></a><span class="w"> </span><span class="p">}</span>
<a id="__codelineno-30-23" name="__codelineno-30-23" href="#__codelineno-30-23"></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">amt</span><span class="o">]</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="n">MAX</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">n</span><span class="o">][</span><span class="n">amt</span><span class="o">]</span><span class="w"> </span><span class="p">:</span><span class="w"> </span><span class="o">-</span><span class="mi">1</span><span class="p">;</span>
<a id="__codelineno-30-24" name="__codelineno-30-24" href="#__codelineno-30-24"></a><span class="p">}</span>
</code></pre></div>
</div>
<div class="tabbed-block">
<div class="highlight"><span class="filename">coin_change.cs</span><pre><span></span><code><a id="__codelineno-31-1" name="__codelineno-31-1" href="#__codelineno-31-1"></a><span class="na">[class]</span><span class="p">{</span><span class="n">coin_change</span><span class="p">}</span><span class="o">-</span><span class="p">[</span><span class="n">func</span><span class="p">]{</span><span class="n">CoinChangeDP</span><span class="p">}</span>
</code></pre></div>
</div>
<div class="tabbed-block">
<div class="highlight"><span class="filename">coin_change.go</span><pre><span></span><code><a id="__codelineno-32-1" name="__codelineno-32-1" href="#__codelineno-32-1"></a><span class="p">[</span><span class="nx">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="kd">func</span><span class="p">]{</span><span class="nx">coinChangeDP</span><span class="p">}</span>
</code></pre></div>
</div>
<div class="tabbed-block">
<div class="highlight"><span class="filename">coin_change.swift</span><pre><span></span><code><a id="__codelineno-33-1" name="__codelineno-33-1" href="#__codelineno-33-1"></a><span class="p">[</span><span class="kd">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="kd">func</span><span class="p">]{</span><span class="n">coinChangeDP</span><span class="p">}</span>
</code></pre></div>
</div>
<div class="tabbed-block">
<div class="highlight"><span class="filename">coin_change.js</span><pre><span></span><code><a id="__codelineno-34-1" name="__codelineno-34-1" href="#__codelineno-34-1"></a><span class="p">[</span><span class="kd">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="nx">func</span><span class="p">]{</span><span class="nx">coinChangeDP</span><span class="p">}</span>
</code></pre></div>
</div>
<div class="tabbed-block">
<div class="highlight"><span class="filename">coin_change.ts</span><pre><span></span><code><a id="__codelineno-35-1" name="__codelineno-35-1" href="#__codelineno-35-1"></a><span class="p">[</span><span class="kd">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="nx">func</span><span class="p">]{</span><span class="nx">coinChangeDP</span><span class="p">}</span>
</code></pre></div>
</div>
<div class="tabbed-block">
<div class="highlight"><span class="filename">coin_change.dart</span><pre><span></span><code><a id="__codelineno-36-1" name="__codelineno-36-1" href="#__codelineno-36-1"></a><span class="p">[</span><span class="n">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="n">func</span><span class="p">]{</span><span class="n">coinChangeDP</span><span class="p">}</span>
</code></pre></div>
</div>
<div class="tabbed-block">
<div class="highlight"><span class="filename">coin_change.rs</span><pre><span></span><code><a id="__codelineno-37-1" name="__codelineno-37-1" href="#__codelineno-37-1"></a><span class="p">[</span><span class="n">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="n">func</span><span class="p">]{</span><span class="n">coin_change_dp</span><span class="p">}</span>
</code></pre></div>
</div>
<div class="tabbed-block">
<div class="highlight"><span class="filename">coin_change.c</span><pre><span></span><code><a id="__codelineno-38-1" name="__codelineno-38-1" href="#__codelineno-38-1"></a><span class="p">[</span><span class="n">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="n">func</span><span class="p">]{</span><span class="n">coinChangeDP</span><span class="p">}</span>
</code></pre></div>
</div>
<div class="tabbed-block">
<div class="highlight"><span class="filename">coin_change.kt</span><pre><span></span><code><a id="__codelineno-39-1" name="__codelineno-39-1" href="#__codelineno-39-1"></a><span class="o">[</span><span class="n">class</span><span class="o">]</span><span class="p">{}</span><span class="o">-[</span><span class="n">func</span><span class="o">]</span><span class="p">{</span><span class="n">coinChangeDP</span><span class="p">}</span>
</code></pre></div>
</div>
<div class="tabbed-block">
<div class="highlight"><span class="filename">coin_change.rb</span><pre><span></span><code><a id="__codelineno-40-1" name="__codelineno-40-1" href="#__codelineno-40-1"></a><span class="o">[</span><span class="n">class</span><span class="o">]</span><span class="p">{}</span><span class="o">-[</span><span class="n">func</span><span class="o">]</span><span class="p">{</span><span class="n">coin_change_dp</span><span class="p">}</span>
</code></pre></div>
</div>
<div class="tabbed-block">
<div class="highlight"><span class="filename">coin_change.zig</span><pre><span></span><code><a id="__codelineno-41-1" name="__codelineno-41-1" href="#__codelineno-41-1"></a><span class="p">[</span><span class="n">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="n">func</span><span class="p">]{</span><span class="n">coinChangeDP</span><span class="p">}</span>
</code></pre></div>
</div>
</div>
</div>
<p>Figure 14-25 show the dynamic programming process for the coin change problem, which is very similar to the unbounded knapsack problem.</p>
<div class="tabbed-set tabbed-alternate" data-tabs="5:15"><input checked="checked" id="__tabbed_5_1" name="__tabbed_5" type="radio" /><input id="__tabbed_5_2" name="__tabbed_5" type="radio" /><input id="__tabbed_5_3" name="__tabbed_5" type="radio" /><input id="__tabbed_5_4" name="__tabbed_5" type="radio" /><input id="__tabbed_5_5" name="__tabbed_5" type="radio" /><input id="__tabbed_5_6" name="__tabbed_5" type="radio" /><input id="__tabbed_5_7" name="__tabbed_5" type="radio" /><input id="__tabbed_5_8" name="__tabbed_5" type="radio" /><input id="__tabbed_5_9" name="__tabbed_5" type="radio" /><input id="__tabbed_5_10" name="__tabbed_5" type="radio" /><input id="__tabbed_5_11" name="__tabbed_5" type="radio" /><input id="__tabbed_5_12" name="__tabbed_5" type="radio" /><input id="__tabbed_5_13" name="__tabbed_5" type="radio" /><input id="__tabbed_5_14" name="__tabbed_5" type="radio" /><input id="__tabbed_5_15" name="__tabbed_5" type="radio" /><div class="tabbed-labels"><label for="__tabbed_5_1">&lt;1&gt;</label><label for="__tabbed_5_2">&lt;2&gt;</label><label for="__tabbed_5_3">&lt;3&gt;</label><label for="__tabbed_5_4">&lt;4&gt;</label><label for="__tabbed_5_5">&lt;5&gt;</label><label for="__tabbed_5_6">&lt;6&gt;</label><label for="__tabbed_5_7">&lt;7&gt;</label><label for="__tabbed_5_8">&lt;8&gt;</label><label for="__tabbed_5_9">&lt;9&gt;</label><label for="__tabbed_5_10">&lt;10&gt;</label><label for="__tabbed_5_11">&lt;11&gt;</label><label for="__tabbed_5_12">&lt;12&gt;</label><label for="__tabbed_5_13">&lt;13&gt;</label><label for="__tabbed_5_14">&lt;14&gt;</label><label for="__tabbed_5_15">&lt;15&gt;</label></div>
<div class="tabbed-content">
<div class="tabbed-block">
<p><a class="glightbox" href="../unbounded_knapsack_problem.assets/coin_change_dp_step1.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Dynamic programming process for the coin change problem" class="animation-figure" src="../unbounded_knapsack_problem.assets/coin_change_dp_step1.png" /></a></p>
</div>
<div class="tabbed-block">
<p><a class="glightbox" href="../unbounded_knapsack_problem.assets/coin_change_dp_step2.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="coin_change_dp_step2" class="animation-figure" src="../unbounded_knapsack_problem.assets/coin_change_dp_step2.png" /></a></p>
</div>
<div class="tabbed-block">
<p><a class="glightbox" href="../unbounded_knapsack_problem.assets/coin_change_dp_step3.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="coin_change_dp_step3" class="animation-figure" src="../unbounded_knapsack_problem.assets/coin_change_dp_step3.png" /></a></p>
</div>
<div class="tabbed-block">
<p><a class="glightbox" href="../unbounded_knapsack_problem.assets/coin_change_dp_step4.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="coin_change_dp_step4" class="animation-figure" src="../unbounded_knapsack_problem.assets/coin_change_dp_step4.png" /></a></p>
</div>
<div class="tabbed-block">
<p><a class="glightbox" href="../unbounded_knapsack_problem.assets/coin_change_dp_step5.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="coin_change_dp_step5" class="animation-figure" src="../unbounded_knapsack_problem.assets/coin_change_dp_step5.png" /></a></p>
</div>
<div class="tabbed-block">
<p><a class="glightbox" href="../unbounded_knapsack_problem.assets/coin_change_dp_step6.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="coin_change_dp_step6" class="animation-figure" src="../unbounded_knapsack_problem.assets/coin_change_dp_step6.png" /></a></p>
</div>
<div class="tabbed-block">
<p><a class="glightbox" href="../unbounded_knapsack_problem.assets/coin_change_dp_step7.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="coin_change_dp_step7" class="animation-figure" src="../unbounded_knapsack_problem.assets/coin_change_dp_step7.png" /></a></p>
</div>
<div class="tabbed-block">
<p><a class="glightbox" href="../unbounded_knapsack_problem.assets/coin_change_dp_step8.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="coin_change_dp_step8" class="animation-figure" src="../unbounded_knapsack_problem.assets/coin_change_dp_step8.png" /></a></p>
</div>
<div class="tabbed-block">
<p><a class="glightbox" href="../unbounded_knapsack_problem.assets/coin_change_dp_step9.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="coin_change_dp_step9" class="animation-figure" src="../unbounded_knapsack_problem.assets/coin_change_dp_step9.png" /></a></p>
</div>
<div class="tabbed-block">
<p><a class="glightbox" href="../unbounded_knapsack_problem.assets/coin_change_dp_step10.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="coin_change_dp_step10" class="animation-figure" src="../unbounded_knapsack_problem.assets/coin_change_dp_step10.png" /></a></p>
</div>
<div class="tabbed-block">
<p><a class="glightbox" href="../unbounded_knapsack_problem.assets/coin_change_dp_step11.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="coin_change_dp_step11" class="animation-figure" src="../unbounded_knapsack_problem.assets/coin_change_dp_step11.png" /></a></p>
</div>
<div class="tabbed-block">
<p><a class="glightbox" href="../unbounded_knapsack_problem.assets/coin_change_dp_step12.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="coin_change_dp_step12" class="animation-figure" src="../unbounded_knapsack_problem.assets/coin_change_dp_step12.png" /></a></p>
</div>
<div class="tabbed-block">
<p><a class="glightbox" href="../unbounded_knapsack_problem.assets/coin_change_dp_step13.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="coin_change_dp_step13" class="animation-figure" src="../unbounded_knapsack_problem.assets/coin_change_dp_step13.png" /></a></p>
</div>
<div class="tabbed-block">
<p><a class="glightbox" href="../unbounded_knapsack_problem.assets/coin_change_dp_step14.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="coin_change_dp_step14" class="animation-figure" src="../unbounded_knapsack_problem.assets/coin_change_dp_step14.png" /></a></p>
</div>
<div class="tabbed-block">
<p><a class="glightbox" href="../unbounded_knapsack_problem.assets/coin_change_dp_step15.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="coin_change_dp_step15" class="animation-figure" src="../unbounded_knapsack_problem.assets/coin_change_dp_step15.png" /></a></p>
</div>
</div>
</div>
<p align="center"> Figure 14-25 &nbsp; Dynamic programming process for the coin change problem </p>
<h3 id="3-space-optimization_1">3. &nbsp; Space optimization<a class="headerlink" href="#3-space-optimization_1" title="Permanent link">&para;</a></h3>
<p>The space optimization for the coin change problem is handled in the same way as for the unbounded knapsack problem:</p>
<div class="tabbed-set tabbed-alternate" data-tabs="6:14"><input checked="checked" id="__tabbed_6_1" name="__tabbed_6" type="radio" /><input id="__tabbed_6_2" name="__tabbed_6" type="radio" /><input id="__tabbed_6_3" name="__tabbed_6" type="radio" /><input id="__tabbed_6_4" name="__tabbed_6" type="radio" /><input id="__tabbed_6_5" name="__tabbed_6" type="radio" /><input id="__tabbed_6_6" name="__tabbed_6" type="radio" /><input id="__tabbed_6_7" name="__tabbed_6" type="radio" /><input id="__tabbed_6_8" name="__tabbed_6" type="radio" /><input id="__tabbed_6_9" name="__tabbed_6" type="radio" /><input id="__tabbed_6_10" name="__tabbed_6" type="radio" /><input id="__tabbed_6_11" name="__tabbed_6" type="radio" /><input id="__tabbed_6_12" name="__tabbed_6" type="radio" /><input id="__tabbed_6_13" name="__tabbed_6" type="radio" /><input id="__tabbed_6_14" name="__tabbed_6" type="radio" /><div class="tabbed-labels"><label for="__tabbed_6_1">Python</label><label for="__tabbed_6_2">C++</label><label for="__tabbed_6_3">Java</label><label for="__tabbed_6_4">C#</label><label for="__tabbed_6_5">Go</label><label for="__tabbed_6_6">Swift</label><label for="__tabbed_6_7">JS</label><label for="__tabbed_6_8">TS</label><label for="__tabbed_6_9">Dart</label><label for="__tabbed_6_10">Rust</label><label for="__tabbed_6_11">C</label><label for="__tabbed_6_12">Kotlin</label><label for="__tabbed_6_13">Ruby</label><label for="__tabbed_6_14">Zig</label></div>
<div class="tabbed-content">
<div class="tabbed-block">
<div class="highlight"><span class="filename">coin_change.py</span><pre><span></span><code><a id="__codelineno-42-1" name="__codelineno-42-1" href="#__codelineno-42-1"></a><span class="k">def</span> <span class="nf">coin_change_dp_comp</span><span class="p">(</span><span class="n">coins</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">int</span><span class="p">],</span> <span class="n">amt</span><span class="p">:</span> <span class="nb">int</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="nb">int</span><span class="p">:</span>
<a id="__codelineno-42-2" name="__codelineno-42-2" href="#__codelineno-42-2"></a><span class="w"> </span><span class="sd">&quot;&quot;&quot;Coin change: Space-optimized dynamic programming&quot;&quot;&quot;</span>
<a id="__codelineno-42-3" name="__codelineno-42-3" href="#__codelineno-42-3"></a> <span class="n">n</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">coins</span><span class="p">)</span>
<a id="__codelineno-42-4" name="__codelineno-42-4" href="#__codelineno-42-4"></a> <span class="n">MAX</span> <span class="o">=</span> <span class="n">amt</span> <span class="o">+</span> <span class="mi">1</span>
<a id="__codelineno-42-5" name="__codelineno-42-5" href="#__codelineno-42-5"></a> <span class="c1"># Initialize dp table</span>
<a id="__codelineno-42-6" name="__codelineno-42-6" href="#__codelineno-42-6"></a> <span class="n">dp</span> <span class="o">=</span> <span class="p">[</span><span class="n">MAX</span><span class="p">]</span> <span class="o">*</span> <span class="p">(</span><span class="n">amt</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
<a id="__codelineno-42-7" name="__codelineno-42-7" href="#__codelineno-42-7"></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">0</span>
<a id="__codelineno-42-8" name="__codelineno-42-8" href="#__codelineno-42-8"></a> <span class="c1"># State transition</span>
<a id="__codelineno-42-9" name="__codelineno-42-9" href="#__codelineno-42-9"></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-42-10" name="__codelineno-42-10" href="#__codelineno-42-10"></a> <span class="c1"># Traverse in order</span>
<a id="__codelineno-42-11" name="__codelineno-42-11" href="#__codelineno-42-11"></a> <span class="k">for</span> <span class="n">a</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">amt</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
<a id="__codelineno-42-12" name="__codelineno-42-12" href="#__codelineno-42-12"></a> <span class="k">if</span> <span class="n">coins</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">&gt;</span> <span class="n">a</span><span class="p">:</span>
<a id="__codelineno-42-13" name="__codelineno-42-13" href="#__codelineno-42-13"></a> <span class="c1"># If exceeding the target amount, do not choose coin i</span>
<a id="__codelineno-42-14" name="__codelineno-42-14" href="#__codelineno-42-14"></a> <span class="n">dp</span><span class="p">[</span><span class="n">a</span><span class="p">]</span> <span class="o">=</span> <span class="n">dp</span><span class="p">[</span><span class="n">a</span><span class="p">]</span>
<a id="__codelineno-42-15" name="__codelineno-42-15" href="#__codelineno-42-15"></a> <span class="k">else</span><span class="p">:</span>
<a id="__codelineno-42-16" name="__codelineno-42-16" href="#__codelineno-42-16"></a> <span class="c1"># The smaller value between not choosing and choosing coin i</span>
<a id="__codelineno-42-17" name="__codelineno-42-17" href="#__codelineno-42-17"></a> <span class="n">dp</span><span class="p">[</span><span class="n">a</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">a</span><span class="p">],</span> <span class="n">dp</span><span class="p">[</span><span class="n">a</span> <span class="o">-</span> <span class="n">coins</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="mi">1</span><span class="p">)</span>
<a id="__codelineno-42-18" name="__codelineno-42-18" href="#__codelineno-42-18"></a> <span class="k">return</span> <span class="n">dp</span><span class="p">[</span><span class="n">amt</span><span class="p">]</span> <span class="k">if</span> <span class="n">dp</span><span class="p">[</span><span class="n">amt</span><span class="p">]</span> <span class="o">!=</span> <span class="n">MAX</span> <span class="k">else</span> <span class="o">-</span><span class="mi">1</span>
</code></pre></div>
</div>
<div class="tabbed-block">
<div class="highlight"><span class="filename">coin_change.cpp</span><pre><span></span><code><a id="__codelineno-43-1" name="__codelineno-43-1" href="#__codelineno-43-1"></a><span class="cm">/* Coin change: Space-optimized dynamic programming */</span>
<a id="__codelineno-43-2" name="__codelineno-43-2" href="#__codelineno-43-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">coinChangeDPComp</span><span class="p">(</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="o">&amp;</span><span class="n">coins</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">amt</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-43-3" name="__codelineno-43-3" href="#__codelineno-43-3"></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">coins</span><span class="p">.</span><span class="n">size</span><span class="p">();</span>
<a id="__codelineno-43-4" name="__codelineno-43-4" href="#__codelineno-43-4"></a><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">MAX</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">amt</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-43-5" name="__codelineno-43-5" href="#__codelineno-43-5"></a><span class="w"> </span><span class="c1">// Initialize dp table</span>
<a id="__codelineno-43-6" name="__codelineno-43-6" href="#__codelineno-43-6"></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">amt</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">MAX</span><span class="p">);</span>
<a id="__codelineno-43-7" name="__codelineno-43-7" href="#__codelineno-43-7"></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="mi">0</span><span class="p">;</span>
<a id="__codelineno-43-8" name="__codelineno-43-8" href="#__codelineno-43-8"></a><span class="w"> </span><span class="c1">// State transition</span>
<a id="__codelineno-43-9" name="__codelineno-43-9" href="#__codelineno-43-9"></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-43-10" name="__codelineno-43-10" href="#__codelineno-43-10"></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">a</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">a</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">amt</span><span class="p">;</span><span class="w"> </span><span class="n">a</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-43-11" name="__codelineno-43-11" href="#__codelineno-43-11"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">coins</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">&gt;</span><span class="w"> </span><span class="n">a</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-43-12" name="__codelineno-43-12" href="#__codelineno-43-12"></a><span class="w"> </span><span class="c1">// If exceeding the target amount, do not choose coin i</span>
<a id="__codelineno-43-13" name="__codelineno-43-13" href="#__codelineno-43-13"></a><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">a</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">a</span><span class="p">];</span>
<a id="__codelineno-43-14" name="__codelineno-43-14" href="#__codelineno-43-14"></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-43-15" name="__codelineno-43-15" href="#__codelineno-43-15"></a><span class="w"> </span><span class="c1">// The smaller value between not choosing and choosing coin i</span>
<a id="__codelineno-43-16" name="__codelineno-43-16" href="#__codelineno-43-16"></a><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">a</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">dp</span><span class="p">[</span><span class="n">a</span><span class="p">],</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">a</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">coins</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="mi">1</span><span class="p">);</span>
<a id="__codelineno-43-17" name="__codelineno-43-17" href="#__codelineno-43-17"></a><span class="w"> </span><span class="p">}</span>
<a id="__codelineno-43-18" name="__codelineno-43-18" href="#__codelineno-43-18"></a><span class="w"> </span><span class="p">}</span>
<a id="__codelineno-43-19" name="__codelineno-43-19" href="#__codelineno-43-19"></a><span class="w"> </span><span class="p">}</span>
<a id="__codelineno-43-20" name="__codelineno-43-20" href="#__codelineno-43-20"></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">amt</span><span class="p">]</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="n">MAX</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">amt</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-43-21" name="__codelineno-43-21" href="#__codelineno-43-21"></a><span class="p">}</span>
</code></pre></div>
</div>
<div class="tabbed-block">
<div class="highlight"><span class="filename">coin_change.java</span><pre><span></span><code><a id="__codelineno-44-1" name="__codelineno-44-1" href="#__codelineno-44-1"></a><span class="cm">/* Coin change: Space-optimized dynamic programming */</span>
<a id="__codelineno-44-2" name="__codelineno-44-2" href="#__codelineno-44-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">coinChangeDPComp</span><span class="p">(</span><span class="kt">int</span><span class="o">[]</span><span class="w"> </span><span class="n">coins</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">amt</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-44-3" name="__codelineno-44-3" href="#__codelineno-44-3"></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">coins</span><span class="p">.</span><span class="na">length</span><span class="p">;</span>
<a id="__codelineno-44-4" name="__codelineno-44-4" href="#__codelineno-44-4"></a><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">MAX</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">amt</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-44-5" name="__codelineno-44-5" href="#__codelineno-44-5"></a><span class="w"> </span><span class="c1">// Initialize dp table</span>
<a id="__codelineno-44-6" name="__codelineno-44-6" href="#__codelineno-44-6"></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">amt</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-44-7" name="__codelineno-44-7" href="#__codelineno-44-7"></a><span class="w"> </span><span class="n">Arrays</span><span class="p">.</span><span class="na">fill</span><span class="p">(</span><span class="n">dp</span><span class="p">,</span><span class="w"> </span><span class="n">MAX</span><span class="p">);</span>
<a id="__codelineno-44-8" name="__codelineno-44-8" href="#__codelineno-44-8"></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="mi">0</span><span class="p">;</span>
<a id="__codelineno-44-9" name="__codelineno-44-9" href="#__codelineno-44-9"></a><span class="w"> </span><span class="c1">// State transition</span>
<a id="__codelineno-44-10" name="__codelineno-44-10" href="#__codelineno-44-10"></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-44-11" name="__codelineno-44-11" href="#__codelineno-44-11"></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">a</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">a</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">amt</span><span class="p">;</span><span class="w"> </span><span class="n">a</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-44-12" name="__codelineno-44-12" href="#__codelineno-44-12"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">coins</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="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">a</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-44-13" name="__codelineno-44-13" href="#__codelineno-44-13"></a><span class="w"> </span><span class="c1">// If exceeding the target amount, do not choose coin i</span>
<a id="__codelineno-44-14" name="__codelineno-44-14" href="#__codelineno-44-14"></a><span class="w"> </span><span class="n">dp</span><span class="o">[</span><span class="n">a</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">a</span><span class="o">]</span><span class="p">;</span>
<a id="__codelineno-44-15" name="__codelineno-44-15" href="#__codelineno-44-15"></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-44-16" name="__codelineno-44-16" href="#__codelineno-44-16"></a><span class="w"> </span><span class="c1">// The smaller value between not choosing and choosing coin i</span>
<a id="__codelineno-44-17" name="__codelineno-44-17" href="#__codelineno-44-17"></a><span class="w"> </span><span class="n">dp</span><span class="o">[</span><span class="n">a</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">dp</span><span class="o">[</span><span class="n">a</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">a</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">coins</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="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">);</span>
<a id="__codelineno-44-18" name="__codelineno-44-18" href="#__codelineno-44-18"></a><span class="w"> </span><span class="p">}</span>
<a id="__codelineno-44-19" name="__codelineno-44-19" href="#__codelineno-44-19"></a><span class="w"> </span><span class="p">}</span>
<a id="__codelineno-44-20" name="__codelineno-44-20" href="#__codelineno-44-20"></a><span class="w"> </span><span class="p">}</span>
<a id="__codelineno-44-21" name="__codelineno-44-21" href="#__codelineno-44-21"></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">amt</span><span class="o">]</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="n">MAX</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">amt</span><span class="o">]</span><span class="w"> </span><span class="p">:</span><span class="w"> </span><span class="o">-</span><span class="mi">1</span><span class="p">;</span>
<a id="__codelineno-44-22" name="__codelineno-44-22" href="#__codelineno-44-22"></a><span class="p">}</span>
</code></pre></div>
</div>
<div class="tabbed-block">
<div class="highlight"><span class="filename">coin_change.cs</span><pre><span></span><code><a id="__codelineno-45-1" name="__codelineno-45-1" href="#__codelineno-45-1"></a><span class="na">[class]</span><span class="p">{</span><span class="n">coin_change</span><span class="p">}</span><span class="o">-</span><span class="p">[</span><span class="n">func</span><span class="p">]{</span><span class="n">CoinChangeDPComp</span><span class="p">}</span>
</code></pre></div>
</div>
<div class="tabbed-block">
<div class="highlight"><span class="filename">coin_change.go</span><pre><span></span><code><a id="__codelineno-46-1" name="__codelineno-46-1" href="#__codelineno-46-1"></a><span class="p">[</span><span class="nx">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="kd">func</span><span class="p">]{</span><span class="nx">coinChangeDPComp</span><span class="p">}</span>
</code></pre></div>
</div>
<div class="tabbed-block">
<div class="highlight"><span class="filename">coin_change.swift</span><pre><span></span><code><a id="__codelineno-47-1" name="__codelineno-47-1" href="#__codelineno-47-1"></a><span class="p">[</span><span class="kd">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="kd">func</span><span class="p">]{</span><span class="n">coinChangeDPComp</span><span class="p">}</span>
</code></pre></div>
</div>
<div class="tabbed-block">
<div class="highlight"><span class="filename">coin_change.js</span><pre><span></span><code><a id="__codelineno-48-1" name="__codelineno-48-1" href="#__codelineno-48-1"></a><span class="p">[</span><span class="kd">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="nx">func</span><span class="p">]{</span><span class="nx">coinChangeDPComp</span><span class="p">}</span>
</code></pre></div>
</div>
<div class="tabbed-block">
<div class="highlight"><span class="filename">coin_change.ts</span><pre><span></span><code><a id="__codelineno-49-1" name="__codelineno-49-1" href="#__codelineno-49-1"></a><span class="p">[</span><span class="kd">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="nx">func</span><span class="p">]{</span><span class="nx">coinChangeDPComp</span><span class="p">}</span>
</code></pre></div>
</div>
<div class="tabbed-block">
<div class="highlight"><span class="filename">coin_change.dart</span><pre><span></span><code><a id="__codelineno-50-1" name="__codelineno-50-1" href="#__codelineno-50-1"></a><span class="p">[</span><span class="n">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="n">func</span><span class="p">]{</span><span class="n">coinChangeDPComp</span><span class="p">}</span>
</code></pre></div>
</div>
<div class="tabbed-block">
<div class="highlight"><span class="filename">coin_change.rs</span><pre><span></span><code><a id="__codelineno-51-1" name="__codelineno-51-1" href="#__codelineno-51-1"></a><span class="p">[</span><span class="n">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="n">func</span><span class="p">]{</span><span class="n">coin_change_dp_comp</span><span class="p">}</span>
</code></pre></div>
</div>
<div class="tabbed-block">
<div class="highlight"><span class="filename">coin_change.c</span><pre><span></span><code><a id="__codelineno-52-1" name="__codelineno-52-1" href="#__codelineno-52-1"></a><span class="p">[</span><span class="n">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="n">func</span><span class="p">]{</span><span class="n">coinChangeDPComp</span><span class="p">}</span>
</code></pre></div>
</div>
<div class="tabbed-block">
<div class="highlight"><span class="filename">coin_change.kt</span><pre><span></span><code><a id="__codelineno-53-1" name="__codelineno-53-1" href="#__codelineno-53-1"></a><span class="o">[</span><span class="n">class</span><span class="o">]</span><span class="p">{}</span><span class="o">-[</span><span class="n">func</span><span class="o">]</span><span class="p">{</span><span class="n">coinChangeDPComp</span><span class="p">}</span>
</code></pre></div>
</div>
<div class="tabbed-block">
<div class="highlight"><span class="filename">coin_change.rb</span><pre><span></span><code><a id="__codelineno-54-1" name="__codelineno-54-1" href="#__codelineno-54-1"></a><span class="o">[</span><span class="n">class</span><span class="o">]</span><span class="p">{}</span><span class="o">-[</span><span class="n">func</span><span class="o">]</span><span class="p">{</span><span class="n">coin_change_dp_comp</span><span class="p">}</span>
</code></pre></div>
</div>
<div class="tabbed-block">
<div class="highlight"><span class="filename">coin_change.zig</span><pre><span></span><code><a id="__codelineno-55-1" name="__codelineno-55-1" href="#__codelineno-55-1"></a><span class="p">[</span><span class="n">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="n">func</span><span class="p">]{</span><span class="n">coinChangeDPComp</span><span class="p">}</span>
</code></pre></div>
</div>
</div>
</div>
<h2 id="1453-coin-change-problem-ii">14.5.3 &nbsp; Coin change problem II<a class="headerlink" href="#1453-coin-change-problem-ii" title="Permanent link">&para;</a></h2>
<div class="admonition question">
<p class="admonition-title">Question</p>
<p>Given <span class="arithmatex">\(n\)</span> types of coins, where the denomination of the <span class="arithmatex">\(i^{th}\)</span> type of coin is <span class="arithmatex">\(coins[i - 1]\)</span>, and the target amount is <span class="arithmatex">\(amt\)</span>. Each type of coin can be selected multiple times, <strong>ask how many combinations of coins can make up the target amount</strong>. See the example below.</p>
</div>
<p><a class="glightbox" href="../unbounded_knapsack_problem.assets/coin_change_ii_example.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Example data for Coin Change Problem II" class="animation-figure" src="../unbounded_knapsack_problem.assets/coin_change_ii_example.png" /></a></p>
<p align="center"> Figure 14-26 &nbsp; Example data for Coin Change Problem II </p>
<h3 id="1-dynamic-programming-approach_2">1. &nbsp; Dynamic programming approach<a class="headerlink" href="#1-dynamic-programming-approach_2" title="Permanent link">&para;</a></h3>
<p>Compared to the previous problem, the goal of this problem is to determine the number of combinations, so the sub-problem becomes: <strong>the number of combinations that can make up amount <span class="arithmatex">\(a\)</span> using the first <span class="arithmatex">\(i\)</span> types of coins</strong>. The <span class="arithmatex">\(dp\)</span> table remains a two-dimensional matrix of size <span class="arithmatex">\((n+1) \times (amt + 1)\)</span>.</p>
<p>The number of combinations for the current state is the sum of the combinations from not selecting the current coin and selecting the current coin. The state transition equation is:</p>
<div class="arithmatex">\[
dp[i, a] = dp[i-1, a] + dp[i, a - coins[i-1]]
\]</div>
<p>When the target amount is <span class="arithmatex">\(0\)</span>, no coins are needed to make up the target amount, so all <span class="arithmatex">\(dp[i, 0]\)</span> in the first column should be initialized to <span class="arithmatex">\(1\)</span>. When there are no coins, it is impossible to make up any amount &gt;0, so all <span class="arithmatex">\(dp[0, a]\)</span> in the first row should be set to <span class="arithmatex">\(0\)</span>.</p>
<h3 id="2-code-implementation_2">2. &nbsp; Code implementation<a class="headerlink" href="#2-code-implementation_2" title="Permanent link">&para;</a></h3>
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<div class="highlight"><span class="filename">coin_change_ii.py</span><pre><span></span><code><a id="__codelineno-56-1" name="__codelineno-56-1" href="#__codelineno-56-1"></a><span class="k">def</span> <span class="nf">coin_change_ii_dp</span><span class="p">(</span><span class="n">coins</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">int</span><span class="p">],</span> <span class="n">amt</span><span class="p">:</span> <span class="nb">int</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="nb">int</span><span class="p">:</span>
<a id="__codelineno-56-2" name="__codelineno-56-2" href="#__codelineno-56-2"></a><span class="w"> </span><span class="sd">&quot;&quot;&quot;Coin change II: Dynamic programming&quot;&quot;&quot;</span>
<a id="__codelineno-56-3" name="__codelineno-56-3" href="#__codelineno-56-3"></a> <span class="n">n</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">coins</span><span class="p">)</span>
<a id="__codelineno-56-4" name="__codelineno-56-4" href="#__codelineno-56-4"></a> <span class="c1"># Initialize dp table</span>
<a id="__codelineno-56-5" name="__codelineno-56-5" href="#__codelineno-56-5"></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">amt</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-56-6" name="__codelineno-56-6" href="#__codelineno-56-6"></a> <span class="c1"># Initialize first column</span>
<a id="__codelineno-56-7" name="__codelineno-56-7" href="#__codelineno-56-7"></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="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
<a id="__codelineno-56-8" name="__codelineno-56-8" href="#__codelineno-56-8"></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="o">=</span> <span class="mi">1</span>
<a id="__codelineno-56-9" name="__codelineno-56-9" href="#__codelineno-56-9"></a> <span class="c1"># State transition</span>
<a id="__codelineno-56-10" name="__codelineno-56-10" href="#__codelineno-56-10"></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-56-11" name="__codelineno-56-11" href="#__codelineno-56-11"></a> <span class="k">for</span> <span class="n">a</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">amt</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
<a id="__codelineno-56-12" name="__codelineno-56-12" href="#__codelineno-56-12"></a> <span class="k">if</span> <span class="n">coins</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">&gt;</span> <span class="n">a</span><span class="p">:</span>
<a id="__codelineno-56-13" name="__codelineno-56-13" href="#__codelineno-56-13"></a> <span class="c1"># If exceeding the target amount, do not choose coin i</span>
<a id="__codelineno-56-14" name="__codelineno-56-14" href="#__codelineno-56-14"></a> <span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">a</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">a</span><span class="p">]</span>
<a id="__codelineno-56-15" name="__codelineno-56-15" href="#__codelineno-56-15"></a> <span class="k">else</span><span class="p">:</span>
<a id="__codelineno-56-16" name="__codelineno-56-16" href="#__codelineno-56-16"></a> <span class="c1"># The sum of the two options of not choosing and choosing coin i</span>
<a id="__codelineno-56-17" name="__codelineno-56-17" href="#__codelineno-56-17"></a> <span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">a</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">a</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="p">][</span><span class="n">a</span> <span class="o">-</span> <span class="n">coins</span><span class="p">[</span><span class="n">i</span> <span class="o">-</span> <span class="mi">1</span><span class="p">]]</span>
<a id="__codelineno-56-18" name="__codelineno-56-18" href="#__codelineno-56-18"></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">amt</span><span class="p">]</span>
</code></pre></div>
</div>
<div class="tabbed-block">
<div class="highlight"><span class="filename">coin_change_ii.cpp</span><pre><span></span><code><a id="__codelineno-57-1" name="__codelineno-57-1" href="#__codelineno-57-1"></a><span class="cm">/* Coin change II: Dynamic programming */</span>
<a id="__codelineno-57-2" name="__codelineno-57-2" href="#__codelineno-57-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">coinChangeIIDP</span><span class="p">(</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="o">&amp;</span><span class="n">coins</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">amt</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-57-3" name="__codelineno-57-3" href="#__codelineno-57-3"></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">coins</span><span class="p">.</span><span class="n">size</span><span class="p">();</span>
<a id="__codelineno-57-4" name="__codelineno-57-4" href="#__codelineno-57-4"></a><span class="w"> </span><span class="c1">// Initialize dp table</span>
<a id="__codelineno-57-5" name="__codelineno-57-5" href="#__codelineno-57-5"></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">amt</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-57-6" name="__codelineno-57-6" href="#__codelineno-57-6"></a><span class="w"> </span><span class="c1">// Initialize first column</span>
<a id="__codelineno-57-7" name="__codelineno-57-7" href="#__codelineno-57-7"></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-57-8" name="__codelineno-57-8" href="#__codelineno-57-8"></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="mi">1</span><span class="p">;</span>
<a id="__codelineno-57-9" name="__codelineno-57-9" href="#__codelineno-57-9"></a><span class="w"> </span><span class="p">}</span>
<a id="__codelineno-57-10" name="__codelineno-57-10" href="#__codelineno-57-10"></a><span class="w"> </span><span class="c1">// State transition</span>
<a id="__codelineno-57-11" name="__codelineno-57-11" href="#__codelineno-57-11"></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">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-57-12" name="__codelineno-57-12" href="#__codelineno-57-12"></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">a</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">a</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">amt</span><span class="p">;</span><span class="w"> </span><span class="n">a</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-57-13" name="__codelineno-57-13" href="#__codelineno-57-13"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">coins</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">&gt;</span><span class="w"> </span><span class="n">a</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-57-14" name="__codelineno-57-14" href="#__codelineno-57-14"></a><span class="w"> </span><span class="c1">// If exceeding the target amount, do not choose coin i</span>
<a id="__codelineno-57-15" name="__codelineno-57-15" href="#__codelineno-57-15"></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">a</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">a</span><span class="p">];</span>
<a id="__codelineno-57-16" name="__codelineno-57-16" href="#__codelineno-57-16"></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-57-17" name="__codelineno-57-17" href="#__codelineno-57-17"></a><span class="w"> </span><span class="c1">// The sum of the two options of not choosing and choosing coin i</span>
<a id="__codelineno-57-18" name="__codelineno-57-18" href="#__codelineno-57-18"></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">a</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">a</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="p">][</span><span class="n">a</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">coins</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>
<a id="__codelineno-57-19" name="__codelineno-57-19" href="#__codelineno-57-19"></a><span class="w"> </span><span class="p">}</span>
<a id="__codelineno-57-20" name="__codelineno-57-20" href="#__codelineno-57-20"></a><span class="w"> </span><span class="p">}</span>
<a id="__codelineno-57-21" name="__codelineno-57-21" href="#__codelineno-57-21"></a><span class="w"> </span><span class="p">}</span>
<a id="__codelineno-57-22" name="__codelineno-57-22" href="#__codelineno-57-22"></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">amt</span><span class="p">];</span>
<a id="__codelineno-57-23" name="__codelineno-57-23" href="#__codelineno-57-23"></a><span class="p">}</span>
</code></pre></div>
</div>
<div class="tabbed-block">
<div class="highlight"><span class="filename">coin_change_ii.java</span><pre><span></span><code><a id="__codelineno-58-1" name="__codelineno-58-1" href="#__codelineno-58-1"></a><span class="cm">/* Coin change II: Dynamic programming */</span>
<a id="__codelineno-58-2" name="__codelineno-58-2" href="#__codelineno-58-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">coinChangeIIDP</span><span class="p">(</span><span class="kt">int</span><span class="o">[]</span><span class="w"> </span><span class="n">coins</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">amt</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-58-3" name="__codelineno-58-3" href="#__codelineno-58-3"></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">coins</span><span class="p">.</span><span class="na">length</span><span class="p">;</span>
<a id="__codelineno-58-4" name="__codelineno-58-4" href="#__codelineno-58-4"></a><span class="w"> </span><span class="c1">// Initialize dp table</span>
<a id="__codelineno-58-5" name="__codelineno-58-5" href="#__codelineno-58-5"></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">amt</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-58-6" name="__codelineno-58-6" href="#__codelineno-58-6"></a><span class="w"> </span><span class="c1">// Initialize first column</span>
<a id="__codelineno-58-7" name="__codelineno-58-7" href="#__codelineno-58-7"></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-58-8" name="__codelineno-58-8" href="#__codelineno-58-8"></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="mi">1</span><span class="p">;</span>
<a id="__codelineno-58-9" name="__codelineno-58-9" href="#__codelineno-58-9"></a><span class="w"> </span><span class="p">}</span>
<a id="__codelineno-58-10" name="__codelineno-58-10" href="#__codelineno-58-10"></a><span class="w"> </span><span class="c1">// State transition</span>
<a id="__codelineno-58-11" name="__codelineno-58-11" href="#__codelineno-58-11"></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">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-58-12" name="__codelineno-58-12" href="#__codelineno-58-12"></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">a</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">a</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">amt</span><span class="p">;</span><span class="w"> </span><span class="n">a</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-58-13" name="__codelineno-58-13" href="#__codelineno-58-13"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">coins</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="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">a</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-58-14" name="__codelineno-58-14" href="#__codelineno-58-14"></a><span class="w"> </span><span class="c1">// If exceeding the target amount, do not choose coin i</span>
<a id="__codelineno-58-15" name="__codelineno-58-15" href="#__codelineno-58-15"></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">a</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">a</span><span class="o">]</span><span class="p">;</span>
<a id="__codelineno-58-16" name="__codelineno-58-16" href="#__codelineno-58-16"></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-58-17" name="__codelineno-58-17" href="#__codelineno-58-17"></a><span class="w"> </span><span class="c1">// The sum of the two options of not choosing and choosing coin i</span>
<a id="__codelineno-58-18" name="__codelineno-58-18" href="#__codelineno-58-18"></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">a</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">a</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="o">][</span><span class="n">a</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">coins</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="p">;</span>
<a id="__codelineno-58-19" name="__codelineno-58-19" href="#__codelineno-58-19"></a><span class="w"> </span><span class="p">}</span>
<a id="__codelineno-58-20" name="__codelineno-58-20" href="#__codelineno-58-20"></a><span class="w"> </span><span class="p">}</span>
<a id="__codelineno-58-21" name="__codelineno-58-21" href="#__codelineno-58-21"></a><span class="w"> </span><span class="p">}</span>
<a id="__codelineno-58-22" name="__codelineno-58-22" href="#__codelineno-58-22"></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">amt</span><span class="o">]</span><span class="p">;</span>
<a id="__codelineno-58-23" name="__codelineno-58-23" href="#__codelineno-58-23"></a><span class="p">}</span>
</code></pre></div>
</div>
<div class="tabbed-block">
<div class="highlight"><span class="filename">coin_change_ii.cs</span><pre><span></span><code><a id="__codelineno-59-1" name="__codelineno-59-1" href="#__codelineno-59-1"></a><span class="na">[class]</span><span class="p">{</span><span class="n">coin_change_ii</span><span class="p">}</span><span class="o">-</span><span class="p">[</span><span class="n">func</span><span class="p">]{</span><span class="n">CoinChangeIIDP</span><span class="p">}</span>
</code></pre></div>
</div>
<div class="tabbed-block">
<div class="highlight"><span class="filename">coin_change_ii.go</span><pre><span></span><code><a id="__codelineno-60-1" name="__codelineno-60-1" href="#__codelineno-60-1"></a><span class="p">[</span><span class="nx">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="kd">func</span><span class="p">]{</span><span class="nx">coinChangeIIDP</span><span class="p">}</span>
</code></pre></div>
</div>
<div class="tabbed-block">
<div class="highlight"><span class="filename">coin_change_ii.swift</span><pre><span></span><code><a id="__codelineno-61-1" name="__codelineno-61-1" href="#__codelineno-61-1"></a><span class="p">[</span><span class="kd">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="kd">func</span><span class="p">]{</span><span class="n">coinChangeIIDP</span><span class="p">}</span>
</code></pre></div>
</div>
<div class="tabbed-block">
<div class="highlight"><span class="filename">coin_change_ii.js</span><pre><span></span><code><a id="__codelineno-62-1" name="__codelineno-62-1" href="#__codelineno-62-1"></a><span class="p">[</span><span class="kd">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="nx">func</span><span class="p">]{</span><span class="nx">coinChangeIIDP</span><span class="p">}</span>
</code></pre></div>
</div>
<div class="tabbed-block">
<div class="highlight"><span class="filename">coin_change_ii.ts</span><pre><span></span><code><a id="__codelineno-63-1" name="__codelineno-63-1" href="#__codelineno-63-1"></a><span class="p">[</span><span class="kd">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="nx">func</span><span class="p">]{</span><span class="nx">coinChangeIIDP</span><span class="p">}</span>
</code></pre></div>
</div>
<div class="tabbed-block">
<div class="highlight"><span class="filename">coin_change_ii.dart</span><pre><span></span><code><a id="__codelineno-64-1" name="__codelineno-64-1" href="#__codelineno-64-1"></a><span class="p">[</span><span class="n">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="n">func</span><span class="p">]{</span><span class="n">coinChangeIIDP</span><span class="p">}</span>
</code></pre></div>
</div>
<div class="tabbed-block">
<div class="highlight"><span class="filename">coin_change_ii.rs</span><pre><span></span><code><a id="__codelineno-65-1" name="__codelineno-65-1" href="#__codelineno-65-1"></a><span class="p">[</span><span class="n">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="n">func</span><span class="p">]{</span><span class="n">coin_change_ii_dp</span><span class="p">}</span>
</code></pre></div>
</div>
<div class="tabbed-block">
<div class="highlight"><span class="filename">coin_change_ii.c</span><pre><span></span><code><a id="__codelineno-66-1" name="__codelineno-66-1" href="#__codelineno-66-1"></a><span class="p">[</span><span class="n">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="n">func</span><span class="p">]{</span><span class="n">coinChangeIIDP</span><span class="p">}</span>
</code></pre></div>
</div>
<div class="tabbed-block">
<div class="highlight"><span class="filename">coin_change_ii.kt</span><pre><span></span><code><a id="__codelineno-67-1" name="__codelineno-67-1" href="#__codelineno-67-1"></a><span class="o">[</span><span class="n">class</span><span class="o">]</span><span class="p">{}</span><span class="o">-[</span><span class="n">func</span><span class="o">]</span><span class="p">{</span><span class="n">coinChangeIIDP</span><span class="p">}</span>
</code></pre></div>
</div>
<div class="tabbed-block">
<div class="highlight"><span class="filename">coin_change_ii.rb</span><pre><span></span><code><a id="__codelineno-68-1" name="__codelineno-68-1" href="#__codelineno-68-1"></a><span class="o">[</span><span class="n">class</span><span class="o">]</span><span class="p">{}</span><span class="o">-[</span><span class="n">func</span><span class="o">]</span><span class="p">{</span><span class="n">coin_change_ii_dp</span><span class="p">}</span>
</code></pre></div>
</div>
<div class="tabbed-block">
<div class="highlight"><span class="filename">coin_change_ii.zig</span><pre><span></span><code><a id="__codelineno-69-1" name="__codelineno-69-1" href="#__codelineno-69-1"></a><span class="p">[</span><span class="n">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="n">func</span><span class="p">]{</span><span class="n">coinChangeIIDP</span><span class="p">}</span>
</code></pre></div>
</div>
</div>
</div>
<h3 id="3-space-optimization_2">3. &nbsp; Space optimization<a class="headerlink" href="#3-space-optimization_2" title="Permanent link">&para;</a></h3>
<p>The space optimization approach is the same, just remove the coin dimension:</p>
<div class="tabbed-set tabbed-alternate" data-tabs="8:14"><input checked="checked" id="__tabbed_8_1" name="__tabbed_8" type="radio" /><input id="__tabbed_8_2" name="__tabbed_8" type="radio" /><input id="__tabbed_8_3" name="__tabbed_8" type="radio" /><input id="__tabbed_8_4" name="__tabbed_8" type="radio" /><input id="__tabbed_8_5" name="__tabbed_8" type="radio" /><input id="__tabbed_8_6" name="__tabbed_8" type="radio" /><input id="__tabbed_8_7" name="__tabbed_8" type="radio" /><input id="__tabbed_8_8" name="__tabbed_8" type="radio" /><input id="__tabbed_8_9" name="__tabbed_8" type="radio" /><input id="__tabbed_8_10" name="__tabbed_8" type="radio" /><input id="__tabbed_8_11" name="__tabbed_8" type="radio" /><input id="__tabbed_8_12" name="__tabbed_8" type="radio" /><input id="__tabbed_8_13" name="__tabbed_8" type="radio" /><input id="__tabbed_8_14" name="__tabbed_8" type="radio" /><div class="tabbed-labels"><label for="__tabbed_8_1">Python</label><label for="__tabbed_8_2">C++</label><label for="__tabbed_8_3">Java</label><label for="__tabbed_8_4">C#</label><label for="__tabbed_8_5">Go</label><label for="__tabbed_8_6">Swift</label><label for="__tabbed_8_7">JS</label><label for="__tabbed_8_8">TS</label><label for="__tabbed_8_9">Dart</label><label for="__tabbed_8_10">Rust</label><label for="__tabbed_8_11">C</label><label for="__tabbed_8_12">Kotlin</label><label for="__tabbed_8_13">Ruby</label><label for="__tabbed_8_14">Zig</label></div>
<div class="tabbed-content">
<div class="tabbed-block">
<div class="highlight"><span class="filename">coin_change_ii.py</span><pre><span></span><code><a id="__codelineno-70-1" name="__codelineno-70-1" href="#__codelineno-70-1"></a><span class="k">def</span> <span class="nf">coin_change_ii_dp_comp</span><span class="p">(</span><span class="n">coins</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">int</span><span class="p">],</span> <span class="n">amt</span><span class="p">:</span> <span class="nb">int</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="nb">int</span><span class="p">:</span>
<a id="__codelineno-70-2" name="__codelineno-70-2" href="#__codelineno-70-2"></a><span class="w"> </span><span class="sd">&quot;&quot;&quot;Coin change II: Space-optimized dynamic programming&quot;&quot;&quot;</span>
<a id="__codelineno-70-3" name="__codelineno-70-3" href="#__codelineno-70-3"></a> <span class="n">n</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">coins</span><span class="p">)</span>
<a id="__codelineno-70-4" name="__codelineno-70-4" href="#__codelineno-70-4"></a> <span class="c1"># Initialize dp table</span>
<a id="__codelineno-70-5" name="__codelineno-70-5" href="#__codelineno-70-5"></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">amt</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
<a id="__codelineno-70-6" name="__codelineno-70-6" href="#__codelineno-70-6"></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-70-7" name="__codelineno-70-7" href="#__codelineno-70-7"></a> <span class="c1"># State transition</span>
<a id="__codelineno-70-8" name="__codelineno-70-8" href="#__codelineno-70-8"></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-70-9" name="__codelineno-70-9" href="#__codelineno-70-9"></a> <span class="c1"># Traverse in order</span>
<a id="__codelineno-70-10" name="__codelineno-70-10" href="#__codelineno-70-10"></a> <span class="k">for</span> <span class="n">a</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">amt</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
<a id="__codelineno-70-11" name="__codelineno-70-11" href="#__codelineno-70-11"></a> <span class="k">if</span> <span class="n">coins</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">&gt;</span> <span class="n">a</span><span class="p">:</span>
<a id="__codelineno-70-12" name="__codelineno-70-12" href="#__codelineno-70-12"></a> <span class="c1"># If exceeding the target amount, do not choose coin i</span>
<a id="__codelineno-70-13" name="__codelineno-70-13" href="#__codelineno-70-13"></a> <span class="n">dp</span><span class="p">[</span><span class="n">a</span><span class="p">]</span> <span class="o">=</span> <span class="n">dp</span><span class="p">[</span><span class="n">a</span><span class="p">]</span>
<a id="__codelineno-70-14" name="__codelineno-70-14" href="#__codelineno-70-14"></a> <span class="k">else</span><span class="p">:</span>
<a id="__codelineno-70-15" name="__codelineno-70-15" href="#__codelineno-70-15"></a> <span class="c1"># The sum of the two options of not choosing and choosing coin i</span>
<a id="__codelineno-70-16" name="__codelineno-70-16" href="#__codelineno-70-16"></a> <span class="n">dp</span><span class="p">[</span><span class="n">a</span><span class="p">]</span> <span class="o">=</span> <span class="n">dp</span><span class="p">[</span><span class="n">a</span><span class="p">]</span> <span class="o">+</span> <span class="n">dp</span><span class="p">[</span><span class="n">a</span> <span class="o">-</span> <span class="n">coins</span><span class="p">[</span><span class="n">i</span> <span class="o">-</span> <span class="mi">1</span><span class="p">]]</span>
<a id="__codelineno-70-17" name="__codelineno-70-17" href="#__codelineno-70-17"></a> <span class="k">return</span> <span class="n">dp</span><span class="p">[</span><span class="n">amt</span><span class="p">]</span>
</code></pre></div>
</div>
<div class="tabbed-block">
<div class="highlight"><span class="filename">coin_change_ii.cpp</span><pre><span></span><code><a id="__codelineno-71-1" name="__codelineno-71-1" href="#__codelineno-71-1"></a><span class="cm">/* Coin change II: Space-optimized dynamic programming */</span>
<a id="__codelineno-71-2" name="__codelineno-71-2" href="#__codelineno-71-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">coinChangeIIDPComp</span><span class="p">(</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="o">&amp;</span><span class="n">coins</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">amt</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-71-3" name="__codelineno-71-3" href="#__codelineno-71-3"></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">coins</span><span class="p">.</span><span class="n">size</span><span class="p">();</span>
<a id="__codelineno-71-4" name="__codelineno-71-4" href="#__codelineno-71-4"></a><span class="w"> </span><span class="c1">// Initialize dp table</span>
<a id="__codelineno-71-5" name="__codelineno-71-5" href="#__codelineno-71-5"></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">amt</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-71-6" name="__codelineno-71-6" href="#__codelineno-71-6"></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="mi">1</span><span class="p">;</span>
<a id="__codelineno-71-7" name="__codelineno-71-7" href="#__codelineno-71-7"></a><span class="w"> </span><span class="c1">// State transition</span>
<a id="__codelineno-71-8" name="__codelineno-71-8" href="#__codelineno-71-8"></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-71-9" name="__codelineno-71-9" href="#__codelineno-71-9"></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">a</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">a</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">amt</span><span class="p">;</span><span class="w"> </span><span class="n">a</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-71-10" name="__codelineno-71-10" href="#__codelineno-71-10"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">coins</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">&gt;</span><span class="w"> </span><span class="n">a</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-71-11" name="__codelineno-71-11" href="#__codelineno-71-11"></a><span class="w"> </span><span class="c1">// If exceeding the target amount, do not choose coin i</span>
<a id="__codelineno-71-12" name="__codelineno-71-12" href="#__codelineno-71-12"></a><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">a</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">a</span><span class="p">];</span>
<a id="__codelineno-71-13" name="__codelineno-71-13" href="#__codelineno-71-13"></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-71-14" name="__codelineno-71-14" href="#__codelineno-71-14"></a><span class="w"> </span><span class="c1">// The sum of the two options of not choosing and choosing coin i</span>
<a id="__codelineno-71-15" name="__codelineno-71-15" href="#__codelineno-71-15"></a><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">a</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">a</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">a</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">coins</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>
<a id="__codelineno-71-16" name="__codelineno-71-16" href="#__codelineno-71-16"></a><span class="w"> </span><span class="p">}</span>
<a id="__codelineno-71-17" name="__codelineno-71-17" href="#__codelineno-71-17"></a><span class="w"> </span><span class="p">}</span>
<a id="__codelineno-71-18" name="__codelineno-71-18" href="#__codelineno-71-18"></a><span class="w"> </span><span class="p">}</span>
<a id="__codelineno-71-19" name="__codelineno-71-19" href="#__codelineno-71-19"></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">amt</span><span class="p">];</span>
<a id="__codelineno-71-20" name="__codelineno-71-20" href="#__codelineno-71-20"></a><span class="p">}</span>
</code></pre></div>
</div>
<div class="tabbed-block">
<div class="highlight"><span class="filename">coin_change_ii.java</span><pre><span></span><code><a id="__codelineno-72-1" name="__codelineno-72-1" href="#__codelineno-72-1"></a><span class="cm">/* Coin change II: Space-optimized dynamic programming */</span>
<a id="__codelineno-72-2" name="__codelineno-72-2" href="#__codelineno-72-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">coinChangeIIDPComp</span><span class="p">(</span><span class="kt">int</span><span class="o">[]</span><span class="w"> </span><span class="n">coins</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">amt</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-72-3" name="__codelineno-72-3" href="#__codelineno-72-3"></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">coins</span><span class="p">.</span><span class="na">length</span><span class="p">;</span>
<a id="__codelineno-72-4" name="__codelineno-72-4" href="#__codelineno-72-4"></a><span class="w"> </span><span class="c1">// Initialize dp table</span>
<a id="__codelineno-72-5" name="__codelineno-72-5" href="#__codelineno-72-5"></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">amt</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-72-6" name="__codelineno-72-6" href="#__codelineno-72-6"></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="mi">1</span><span class="p">;</span>
<a id="__codelineno-72-7" name="__codelineno-72-7" href="#__codelineno-72-7"></a><span class="w"> </span><span class="c1">// State transition</span>
<a id="__codelineno-72-8" name="__codelineno-72-8" href="#__codelineno-72-8"></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-72-9" name="__codelineno-72-9" href="#__codelineno-72-9"></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">a</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">a</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">amt</span><span class="p">;</span><span class="w"> </span><span class="n">a</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-72-10" name="__codelineno-72-10" href="#__codelineno-72-10"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">coins</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="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">a</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-72-11" name="__codelineno-72-11" href="#__codelineno-72-11"></a><span class="w"> </span><span class="c1">// If exceeding the target amount, do not choose coin i</span>
<a id="__codelineno-72-12" name="__codelineno-72-12" href="#__codelineno-72-12"></a><span class="w"> </span><span class="n">dp</span><span class="o">[</span><span class="n">a</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">a</span><span class="o">]</span><span class="p">;</span>
<a id="__codelineno-72-13" name="__codelineno-72-13" href="#__codelineno-72-13"></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-72-14" name="__codelineno-72-14" href="#__codelineno-72-14"></a><span class="w"> </span><span class="c1">// The sum of the two options of not choosing and choosing coin i</span>
<a id="__codelineno-72-15" name="__codelineno-72-15" href="#__codelineno-72-15"></a><span class="w"> </span><span class="n">dp</span><span class="o">[</span><span class="n">a</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">a</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">a</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">coins</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="p">;</span>
<a id="__codelineno-72-16" name="__codelineno-72-16" href="#__codelineno-72-16"></a><span class="w"> </span><span class="p">}</span>
<a id="__codelineno-72-17" name="__codelineno-72-17" href="#__codelineno-72-17"></a><span class="w"> </span><span class="p">}</span>
<a id="__codelineno-72-18" name="__codelineno-72-18" href="#__codelineno-72-18"></a><span class="w"> </span><span class="p">}</span>
<a id="__codelineno-72-19" name="__codelineno-72-19" href="#__codelineno-72-19"></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">amt</span><span class="o">]</span><span class="p">;</span>
<a id="__codelineno-72-20" name="__codelineno-72-20" href="#__codelineno-72-20"></a><span class="p">}</span>
</code></pre></div>
</div>
<div class="tabbed-block">
<div class="highlight"><span class="filename">coin_change_ii.cs</span><pre><span></span><code><a id="__codelineno-73-1" name="__codelineno-73-1" href="#__codelineno-73-1"></a><span class="na">[class]</span><span class="p">{</span><span class="n">coin_change_ii</span><span class="p">}</span><span class="o">-</span><span class="p">[</span><span class="n">func</span><span class="p">]{</span><span class="n">CoinChangeIIDPComp</span><span class="p">}</span>
</code></pre></div>
</div>
<div class="tabbed-block">
<div class="highlight"><span class="filename">coin_change_ii.go</span><pre><span></span><code><a id="__codelineno-74-1" name="__codelineno-74-1" href="#__codelineno-74-1"></a><span class="p">[</span><span class="nx">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="kd">func</span><span class="p">]{</span><span class="nx">coinChangeIIDPComp</span><span class="p">}</span>
</code></pre></div>
</div>
<div class="tabbed-block">
<div class="highlight"><span class="filename">coin_change_ii.swift</span><pre><span></span><code><a id="__codelineno-75-1" name="__codelineno-75-1" href="#__codelineno-75-1"></a><span class="p">[</span><span class="kd">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="kd">func</span><span class="p">]{</span><span class="n">coinChangeIIDPComp</span><span class="p">}</span>
</code></pre></div>
</div>
<div class="tabbed-block">
<div class="highlight"><span class="filename">coin_change_ii.js</span><pre><span></span><code><a id="__codelineno-76-1" name="__codelineno-76-1" href="#__codelineno-76-1"></a><span class="p">[</span><span class="kd">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="nx">func</span><span class="p">]{</span><span class="nx">coinChangeIIDPComp</span><span class="p">}</span>
</code></pre></div>
</div>
<div class="tabbed-block">
<div class="highlight"><span class="filename">coin_change_ii.ts</span><pre><span></span><code><a id="__codelineno-77-1" name="__codelineno-77-1" href="#__codelineno-77-1"></a><span class="p">[</span><span class="kd">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="nx">func</span><span class="p">]{</span><span class="nx">coinChangeIIDPComp</span><span class="p">}</span>
</code></pre></div>
</div>
<div class="tabbed-block">
<div class="highlight"><span class="filename">coin_change_ii.dart</span><pre><span></span><code><a id="__codelineno-78-1" name="__codelineno-78-1" href="#__codelineno-78-1"></a><span class="p">[</span><span class="n">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="n">func</span><span class="p">]{</span><span class="n">coinChangeIIDPComp</span><span class="p">}</span>
</code></pre></div>
</div>
<div class="tabbed-block">
<div class="highlight"><span class="filename">coin_change_ii.rs</span><pre><span></span><code><a id="__codelineno-79-1" name="__codelineno-79-1" href="#__codelineno-79-1"></a><span class="p">[</span><span class="n">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="n">func</span><span class="p">]{</span><span class="n">coin_change_ii_dp_comp</span><span class="p">}</span>
</code></pre></div>
</div>
<div class="tabbed-block">
<div class="highlight"><span class="filename">coin_change_ii.c</span><pre><span></span><code><a id="__codelineno-80-1" name="__codelineno-80-1" href="#__codelineno-80-1"></a><span class="p">[</span><span class="n">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="n">func</span><span class="p">]{</span><span class="n">coinChangeIIDPComp</span><span class="p">}</span>
</code></pre></div>
</div>
<div class="tabbed-block">
<div class="highlight"><span class="filename">coin_change_ii.kt</span><pre><span></span><code><a id="__codelineno-81-1" name="__codelineno-81-1" href="#__codelineno-81-1"></a><span class="o">[</span><span class="n">class</span><span class="o">]</span><span class="p">{}</span><span class="o">-[</span><span class="n">func</span><span class="o">]</span><span class="p">{</span><span class="n">coinChangeIIDPComp</span><span class="p">}</span>
</code></pre></div>
</div>
<div class="tabbed-block">
<div class="highlight"><span class="filename">coin_change_ii.rb</span><pre><span></span><code><a id="__codelineno-82-1" name="__codelineno-82-1" href="#__codelineno-82-1"></a><span class="o">[</span><span class="n">class</span><span class="o">]</span><span class="p">{}</span><span class="o">-[</span><span class="n">func</span><span class="o">]</span><span class="p">{</span><span class="n">coin_change_ii_dp_comp</span><span class="p">}</span>
</code></pre></div>
</div>
<div class="tabbed-block">
<div class="highlight"><span class="filename">coin_change_ii.zig</span><pre><span></span><code><a id="__codelineno-83-1" name="__codelineno-83-1" href="#__codelineno-83-1"></a><span class="p">[</span><span class="n">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="n">func</span><span class="p">]{</span><span class="n">coinChangeIIDPComp</span><span class="p">}</span>
</code></pre></div>
</div>
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