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@@ -3534,7 +3534,7 @@
<p class="admonition-title">Question</p>
<p>给定 <span class="arithmatex">\(n\)</span> 个物品，第 <span class="arithmatex">\(i\)</span> 个物品的重量为 <span class="arithmatex">\(wgt[i-1]\)</span>、价值为 <span class="arithmatex">\(val[i-1]\)</span> ，和一个容量为 <span class="arithmatex">\(cap\)</span> 的背包。<strong>每个物品可以重复选取</strong>，问在不超过背包容量下能放入物品的最大价值。</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="完全背包问题的示例数据" src="../unbounded_knapsack_problem.assets/unbounded_knapsack_example.png" /></a></p>
<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="完全背包问题的示例数据" class="animation-figure" src="../unbounded_knapsack_problem.assets/unbounded_knapsack_example.png" /></a></p>
<p align="center"> 图 14-22 &nbsp; 完全背包问题的示例数据 </p>
<h3 id="1">1. &nbsp; 动态规划思路<a class="headerlink" href="#1" title="Permanent link">&para;</a></h3>
@@ -3842,22 +3842,22 @@ dp[i, c] = \max(dp[i-1, c], dp[i, c - wgt[i-1]] + val[i-1])
<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="完全背包的空间优化后的动态规划过程" src="../unbounded_knapsack_problem.assets/unbounded_knapsack_dp_comp_step1.png" /></a></p>
<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="完全背包的空间优化后的动态规划过程" class="animation-figure" src="../unbounded_knapsack_problem.assets/unbounded_knapsack_dp_comp_step1.png" /></a></p>
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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" src="../unbounded_knapsack_problem.assets/unbounded_knapsack_dp_comp_step2.png" /></a></p>
<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" src="../unbounded_knapsack_problem.assets/unbounded_knapsack_dp_comp_step3.png" /></a></p>
<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" src="../unbounded_knapsack_problem.assets/unbounded_knapsack_dp_comp_step4.png" /></a></p>
<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" src="../unbounded_knapsack_problem.assets/unbounded_knapsack_dp_comp_step5.png" /></a></p>
<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" src="../unbounded_knapsack_problem.assets/unbounded_knapsack_dp_comp_step6.png" /></a></p>
<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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@@ -4140,7 +4140,7 @@ dp[i, c] = \max(dp[i-1, c], dp[i, c - wgt[i-1]] + val[i-1])
<p class="admonition-title">Question</p>
<p>给定 <span class="arithmatex">\(n\)</span> 种硬币，第 <span class="arithmatex">\(i\)</span> 种硬币的面值为 <span class="arithmatex">\(coins[i - 1]\)</span> ，目标金额为 <span class="arithmatex">\(amt\)</span> ，<strong>每种硬币可以重复选取</strong>，问能够凑出目标金额的最少硬币个数。如果无法凑出目标金额则返回 <span class="arithmatex">\(-1\)</span> 。</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="零钱兑换问题的示例数据" src="../unbounded_knapsack_problem.assets/coin_change_example.png" /></a></p>
<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="零钱兑换问题的示例数据" class="animation-figure" src="../unbounded_knapsack_problem.assets/coin_change_example.png" /></a></p>
<p align="center"> 图 14-24 &nbsp; 零钱兑换问题的示例数据 </p>
<h3 id="1_1">1. &nbsp; 动态规划思路<a class="headerlink" href="#1_1" title="Permanent link">&para;</a></h3>
@@ -4511,49 +4511,49 @@ dp[i, a] = \min(dp[i-1, a], dp[i, a - coins[i-1]] + 1)
<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>
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<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="零钱兑换问题的动态规划过程" src="../unbounded_knapsack_problem.assets/coin_change_dp_step1.png" /></a></p>
<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="零钱兑换问题的动态规划过程" class="animation-figure" src="../unbounded_knapsack_problem.assets/coin_change_dp_step1.png" /></a></p>
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<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" src="../unbounded_knapsack_problem.assets/coin_change_dp_step2.png" /></a></p>
<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>
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<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" src="../unbounded_knapsack_problem.assets/coin_change_dp_step3.png" /></a></p>
<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>
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<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" src="../unbounded_knapsack_problem.assets/coin_change_dp_step4.png" /></a></p>
<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>
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<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" src="../unbounded_knapsack_problem.assets/coin_change_dp_step5.png" /></a></p>
<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>
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<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" src="../unbounded_knapsack_problem.assets/coin_change_dp_step6.png" /></a></p>
<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>
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<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" src="../unbounded_knapsack_problem.assets/coin_change_dp_step7.png" /></a></p>
<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>
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<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" src="../unbounded_knapsack_problem.assets/coin_change_dp_step8.png" /></a></p>
<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>
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<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" src="../unbounded_knapsack_problem.assets/coin_change_dp_step9.png" /></a></p>
<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>
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<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" src="../unbounded_knapsack_problem.assets/coin_change_dp_step10.png" /></a></p>
<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>
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<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" src="../unbounded_knapsack_problem.assets/coin_change_dp_step11.png" /></a></p>
<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>
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<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" src="../unbounded_knapsack_problem.assets/coin_change_dp_step12.png" /></a></p>
<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>
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<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" src="../unbounded_knapsack_problem.assets/coin_change_dp_step13.png" /></a></p>
<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>
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<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" src="../unbounded_knapsack_problem.assets/coin_change_dp_step14.png" /></a></p>
<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>
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<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" src="../unbounded_knapsack_problem.assets/coin_change_dp_step15.png" /></a></p>
<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>
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@@ -4870,7 +4870,7 @@ dp[i, a] = \min(dp[i-1, a], dp[i, a - coins[i-1]] + 1)
<p class="admonition-title">Question</p>
<p>给定 <span class="arithmatex">\(n\)</span> 种硬币，第 <span class="arithmatex">\(i\)</span> 种硬币的面值为 <span class="arithmatex">\(coins[i - 1]\)</span> ，目标金额为 <span class="arithmatex">\(amt\)</span> ，每种硬币可以重复选取，<strong>问在凑出目标金额的硬币组合数量</strong>。</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="零钱兑换问题 II 的示例数据" src="../unbounded_knapsack_problem.assets/coin_change_ii_example.png" /></a></p>
<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="零钱兑换问题 II 的示例数据" class="animation-figure" src="../unbounded_knapsack_problem.assets/coin_change_ii_example.png" /></a></p>
<p align="center"> 图 14-26 &nbsp; 零钱兑换问题 II 的示例数据 </p>
<h3 id="1_2">1. &nbsp; 动态规划思路<a class="headerlink" href="#1_2" title="Permanent link">&para;</a></h3>