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@ -3387,7 +3387,7 @@
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<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。每个物品只能选择一次，问在不超过背包容量下能放入物品的最大价值。</p>
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</div>
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<p>观察图 14-17 ，由于物品编号 <span class="arithmatex">\(i\)</span> 从 <span class="arithmatex">\(1\)</span> 开始计数，数组索引从 <span class="arithmatex">\(0\)</span> 开始计数，因此物品 <span class="arithmatex">\(i\)</span> 对应重量 <span class="arithmatex">\(wgt[i-1]\)</span> 和价值 <span class="arithmatex">\(val[i-1]\)</span> 。</p>
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<p><a class="glightbox" href="../knapsack_problem.assets/knapsack_example.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="0-1 背包的示例数据" src="../knapsack_problem.assets/knapsack_example.png" /></a></p>
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<p><a class="glightbox" href="../knapsack_problem.assets/knapsack_example.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="0-1 背包的示例数据" class="animation-figure" src="../knapsack_problem.assets/knapsack_example.png" /></a></p>
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<p align="center"> 图 14-17 0-1 背包的示例数据 </p>
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<p>我们可以将 0-1 背包问题看作是一个由 <span class="arithmatex">\(n\)</span> 轮决策组成的过程，每个物体都有不放入和放入两种决策，因此该问题是满足决策树模型的。</p>
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@ -3655,7 +3655,7 @@ dp[i, c] = \max(dp[i-1, c], dp[i-1, c - wgt[i-1]] + val[i-1])
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</div>
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<p>如图 14-18 所示，由于每个物品都会产生不选和选两条搜索分支，因此时间复杂度为 <span class="arithmatex">\(O(2^n)\)</span> 。</p>
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<p>观察递归树，容易发现其中存在重叠子问题，例如 <span class="arithmatex">\(dp[1, 10]\)</span> 等。而当物品较多、背包容量较大，尤其是相同重量的物品较多时，重叠子问题的数量将会大幅增多。</p>
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<p><a class="glightbox" href="../knapsack_problem.assets/knapsack_dfs.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="0-1 背包的暴力搜索递归树" src="../knapsack_problem.assets/knapsack_dfs.png" /></a></p>
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<p><a class="glightbox" href="../knapsack_problem.assets/knapsack_dfs.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="0-1 背包的暴力搜索递归树" class="animation-figure" src="../knapsack_problem.assets/knapsack_dfs.png" /></a></p>
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<p align="center"> 图 14-18 0-1 背包的暴力搜索递归树 </p>
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<h3 id="2">2. 方法二：记忆化搜索<a class="headerlink" href="#2" title="Permanent link">¶</a></h3>
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@ -3966,7 +3966,7 @@ dp[i, c] = \max(dp[i-1, c], dp[i-1, c - wgt[i-1]] + val[i-1])
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</div>
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</div>
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<p>图 14-19 展示了在记忆化递归中被剪掉的搜索分支。</p>
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<p><a class="glightbox" href="../knapsack_problem.assets/knapsack_dfs_mem.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="0-1 背包的记忆化搜索递归树" src="../knapsack_problem.assets/knapsack_dfs_mem.png" /></a></p>
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<p><a class="glightbox" href="../knapsack_problem.assets/knapsack_dfs_mem.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="0-1 背包的记忆化搜索递归树" class="animation-figure" src="../knapsack_problem.assets/knapsack_dfs_mem.png" /></a></p>
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<p align="center"> 图 14-19 0-1 背包的记忆化搜索递归树 </p>
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<h3 id="3">3. 方法三：动态规划<a class="headerlink" href="#3" title="Permanent link">¶</a></h3>
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@ -4257,46 +4257,46 @@ dp[i, c] = \max(dp[i-1, c], dp[i-1, c - wgt[i-1]] + val[i-1])
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<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"><1></label><label for="__tabbed_4_2"><2></label><label for="__tabbed_4_3"><3></label><label for="__tabbed_4_4"><4></label><label for="__tabbed_4_5"><5></label><label for="__tabbed_4_6"><6></label><label for="__tabbed_4_7"><7></label><label for="__tabbed_4_8"><8></label><label for="__tabbed_4_9"><9></label><label for="__tabbed_4_10"><10></label><label for="__tabbed_4_11"><11></label><label for="__tabbed_4_12"><12></label><label for="__tabbed_4_13"><13></label><label for="__tabbed_4_14"><14></label></div>
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<p><a class="glightbox" href="../knapsack_problem.assets/knapsack_dp_step1.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="0-1 背包的动态规划过程" src="../knapsack_problem.assets/knapsack_dp_step1.png" /></a></p>
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<p><a class="glightbox" href="../knapsack_problem.assets/knapsack_dp_step1.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="0-1 背包的动态规划过程" class="animation-figure" src="../knapsack_problem.assets/knapsack_dp_step1.png" /></a></p>
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<p><a class="glightbox" href="../knapsack_problem.assets/knapsack_dp_step2.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="knapsack_dp_step2" src="../knapsack_problem.assets/knapsack_dp_step2.png" /></a></p>
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<p><a class="glightbox" href="../knapsack_problem.assets/knapsack_dp_step2.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="knapsack_dp_step2" class="animation-figure" src="../knapsack_problem.assets/knapsack_dp_step2.png" /></a></p>
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<p><a class="glightbox" href="../knapsack_problem.assets/knapsack_dp_step3.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="knapsack_dp_step3" src="../knapsack_problem.assets/knapsack_dp_step3.png" /></a></p>
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<p><a class="glightbox" href="../knapsack_problem.assets/knapsack_dp_step3.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="knapsack_dp_step3" class="animation-figure" src="../knapsack_problem.assets/knapsack_dp_step3.png" /></a></p>
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<p><a class="glightbox" href="../knapsack_problem.assets/knapsack_dp_step4.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="knapsack_dp_step4" src="../knapsack_problem.assets/knapsack_dp_step4.png" /></a></p>
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<p><a class="glightbox" href="../knapsack_problem.assets/knapsack_dp_step4.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="knapsack_dp_step4" class="animation-figure" src="../knapsack_problem.assets/knapsack_dp_step4.png" /></a></p>
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<p><a class="glightbox" href="../knapsack_problem.assets/knapsack_dp_step5.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="knapsack_dp_step5" src="../knapsack_problem.assets/knapsack_dp_step5.png" /></a></p>
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<p><a class="glightbox" href="../knapsack_problem.assets/knapsack_dp_step5.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="knapsack_dp_step5" class="animation-figure" src="../knapsack_problem.assets/knapsack_dp_step5.png" /></a></p>
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<p><a class="glightbox" href="../knapsack_problem.assets/knapsack_dp_step6.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="knapsack_dp_step6" src="../knapsack_problem.assets/knapsack_dp_step6.png" /></a></p>
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<p><a class="glightbox" href="../knapsack_problem.assets/knapsack_dp_step6.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="knapsack_dp_step6" class="animation-figure" src="../knapsack_problem.assets/knapsack_dp_step6.png" /></a></p>
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<p><a class="glightbox" href="../knapsack_problem.assets/knapsack_dp_step7.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="knapsack_dp_step7" src="../knapsack_problem.assets/knapsack_dp_step7.png" /></a></p>
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<p><a class="glightbox" href="../knapsack_problem.assets/knapsack_dp_step7.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="knapsack_dp_step7" class="animation-figure" src="../knapsack_problem.assets/knapsack_dp_step7.png" /></a></p>
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<p><a class="glightbox" href="../knapsack_problem.assets/knapsack_dp_step8.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="knapsack_dp_step8" src="../knapsack_problem.assets/knapsack_dp_step8.png" /></a></p>
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<p><a class="glightbox" href="../knapsack_problem.assets/knapsack_dp_step8.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="knapsack_dp_step8" class="animation-figure" src="../knapsack_problem.assets/knapsack_dp_step8.png" /></a></p>
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<p><a class="glightbox" href="../knapsack_problem.assets/knapsack_dp_step9.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="knapsack_dp_step9" src="../knapsack_problem.assets/knapsack_dp_step9.png" /></a></p>
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<p><a class="glightbox" href="../knapsack_problem.assets/knapsack_dp_step9.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="knapsack_dp_step9" class="animation-figure" src="../knapsack_problem.assets/knapsack_dp_step9.png" /></a></p>
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<p><a class="glightbox" href="../knapsack_problem.assets/knapsack_dp_step10.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="knapsack_dp_step10" src="../knapsack_problem.assets/knapsack_dp_step10.png" /></a></p>
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<p><a class="glightbox" href="../knapsack_problem.assets/knapsack_dp_step10.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="knapsack_dp_step10" class="animation-figure" src="../knapsack_problem.assets/knapsack_dp_step10.png" /></a></p>
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<p><a class="glightbox" href="../knapsack_problem.assets/knapsack_dp_step11.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="knapsack_dp_step11" src="../knapsack_problem.assets/knapsack_dp_step11.png" /></a></p>
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<p><a class="glightbox" href="../knapsack_problem.assets/knapsack_dp_step11.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="knapsack_dp_step11" class="animation-figure" src="../knapsack_problem.assets/knapsack_dp_step11.png" /></a></p>
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<p><a class="glightbox" href="../knapsack_problem.assets/knapsack_dp_step12.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="knapsack_dp_step12" src="../knapsack_problem.assets/knapsack_dp_step12.png" /></a></p>
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<p><a class="glightbox" href="../knapsack_problem.assets/knapsack_dp_step12.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="knapsack_dp_step12" class="animation-figure" src="../knapsack_problem.assets/knapsack_dp_step12.png" /></a></p>
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<p><a class="glightbox" href="../knapsack_problem.assets/knapsack_dp_step13.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="knapsack_dp_step13" src="../knapsack_problem.assets/knapsack_dp_step13.png" /></a></p>
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<p><a class="glightbox" href="../knapsack_problem.assets/knapsack_dp_step13.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="knapsack_dp_step13" class="animation-figure" src="../knapsack_problem.assets/knapsack_dp_step13.png" /></a></p>
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<p><a class="glightbox" href="../knapsack_problem.assets/knapsack_dp_step14.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="knapsack_dp_step14" src="../knapsack_problem.assets/knapsack_dp_step14.png" /></a></p>
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<p><a class="glightbox" href="../knapsack_problem.assets/knapsack_dp_step14.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="knapsack_dp_step14" class="animation-figure" src="../knapsack_problem.assets/knapsack_dp_step14.png" /></a></p>
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@ -4313,22 +4313,22 @@ dp[i, c] = \max(dp[i-1, c], dp[i-1, c - wgt[i-1]] + val[i-1])
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<div class="tabbed-set tabbed-alternate" data-tabs="5:6"><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" /><div class="tabbed-labels"><label for="__tabbed_5_1"><1></label><label for="__tabbed_5_2"><2></label><label for="__tabbed_5_3"><3></label><label for="__tabbed_5_4"><4></label><label for="__tabbed_5_5"><5></label><label for="__tabbed_5_6"><6></label></div>
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<p><a class="glightbox" href="../knapsack_problem.assets/knapsack_dp_comp_step1.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="0-1 背包的空间优化后的动态规划过程" src="../knapsack_problem.assets/knapsack_dp_comp_step1.png" /></a></p>
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<p><a class="glightbox" href="../knapsack_problem.assets/knapsack_dp_comp_step1.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="0-1 背包的空间优化后的动态规划过程" class="animation-figure" src="../knapsack_problem.assets/knapsack_dp_comp_step1.png" /></a></p>
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<p><a class="glightbox" href="../knapsack_problem.assets/knapsack_dp_comp_step2.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="knapsack_dp_comp_step2" src="../knapsack_problem.assets/knapsack_dp_comp_step2.png" /></a></p>
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<p><a class="glightbox" href="../knapsack_problem.assets/knapsack_dp_comp_step2.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="knapsack_dp_comp_step2" class="animation-figure" src="../knapsack_problem.assets/knapsack_dp_comp_step2.png" /></a></p>
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<p><a class="glightbox" href="../knapsack_problem.assets/knapsack_dp_comp_step3.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="knapsack_dp_comp_step3" src="../knapsack_problem.assets/knapsack_dp_comp_step3.png" /></a></p>
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<p><a class="glightbox" href="../knapsack_problem.assets/knapsack_dp_comp_step3.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="knapsack_dp_comp_step3" class="animation-figure" src="../knapsack_problem.assets/knapsack_dp_comp_step3.png" /></a></p>
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<p><a class="glightbox" href="../knapsack_problem.assets/knapsack_dp_comp_step4.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="knapsack_dp_comp_step4" src="../knapsack_problem.assets/knapsack_dp_comp_step4.png" /></a></p>
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<p><a class="glightbox" href="../knapsack_problem.assets/knapsack_dp_comp_step4.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="knapsack_dp_comp_step4" class="animation-figure" src="../knapsack_problem.assets/knapsack_dp_comp_step4.png" /></a></p>
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<p><a class="glightbox" href="../knapsack_problem.assets/knapsack_dp_comp_step5.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="knapsack_dp_comp_step5" src="../knapsack_problem.assets/knapsack_dp_comp_step5.png" /></a></p>
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<p><a class="glightbox" href="../knapsack_problem.assets/knapsack_dp_comp_step5.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="knapsack_dp_comp_step5" class="animation-figure" src="../knapsack_problem.assets/knapsack_dp_comp_step5.png" /></a></p>
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<p><a class="glightbox" href="../knapsack_problem.assets/knapsack_dp_comp_step6.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="knapsack_dp_comp_step6" src="../knapsack_problem.assets/knapsack_dp_comp_step6.png" /></a></p>
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<p><a class="glightbox" href="../knapsack_problem.assets/knapsack_dp_comp_step6.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="knapsack_dp_comp_step6" class="animation-figure" src="../knapsack_problem.assets/knapsack_dp_comp_step6.png" /></a></p>
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