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< input class = "md-nav__toggle md-toggle " type = "checkbox" id = "__nav_4" >
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< a href = "../../chapter_data_structure/" > 3. 数据结构< / a >
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3. 数据结构
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< a href = "../../chapter_data_structure/classification_of_data_structure/" class = "md-nav__link" >
3.1. 数据结构分类
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3.2. 基本数据类型
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3.3. 数字编码 *
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< a href = "../../chapter_data_structure/character_encoding/" class = "md-nav__link" >
3.4. 字符编码 *
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3.5. 小结
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4. 数组与链表
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4.1. 数组
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< a href = "../../chapter_array_and_linkedlist/linked_list/" class = "md-nav__link" >
4.2. 链表
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4.3. 列表
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4.4. 小结
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< a href = "../../chapter_stack_and_queue/" > 5. 栈与队列< / a >
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5. 栈与队列
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< a href = "../../chapter_stack_and_queue/stack/" class = "md-nav__link" >
5.1. 栈
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< a href = "../../chapter_stack_and_queue/queue/" class = "md-nav__link" >
5.2. 队列
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< a href = "../../chapter_stack_and_queue/deque/" class = "md-nav__link" >
5.3. 双向队列
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5.4. 小结
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< a href = "../../chapter_hashing/" > 6. 散列表< / a >
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6. 散列表
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6.1. 哈希表（ ）
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6.2. 哈希冲突（ ）
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< a href = "../../chapter_hashing/hash_algorithm/" class = "md-nav__link" >
6.3. 哈希算法（ ）
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6.4. 小结
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< input class = "md-nav__toggle md-toggle " type = "checkbox" id = "__nav_8" >
< div class = "md-nav__link md-nav__link--index " >
< a href = "../../chapter_tree/" > 7. 树< / a >
< label for = "__nav_8" >
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< / label >
< / div >
< nav class = "md-nav" data-md-level = "1" aria-labelledby = "__nav_8_label" aria-expanded = "false" >
< label class = "md-nav__title" for = "__nav_8" >
< span class = "md-nav__icon md-icon" > < / span >
7. 树
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7.1. 二叉树
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< a href = "../../chapter_tree/binary_tree_traversal/" class = "md-nav__link" >
7.2. 二叉树遍历
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< a href = "../../chapter_tree/array_representation_of_tree/" class = "md-nav__link" >
7.3. 二叉树数组表示
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< a href = "../../chapter_tree/binary_search_tree/" class = "md-nav__link" >
7.4. 二叉搜索树
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< a href = "../../chapter_tree/avl_tree/" class = "md-nav__link" >
7.5. AVL 树 *
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< a href = "../../chapter_tree/summary/" class = "md-nav__link" >
7.6. 小结
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< div class = "md-nav__link md-nav__link--index " >
< a href = "../../chapter_heap/" > 8. 堆< / a >
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< / label >
< / div >
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< label class = "md-nav__title" for = "__nav_9" >
< span class = "md-nav__icon md-icon" > < / span >
8. 堆
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< a href = "../../chapter_heap/heap/" class = "md-nav__link" >
8.1. 堆
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< a href = "../../chapter_heap/build_heap/" class = "md-nav__link" >
8.2. 建堆操作
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< a href = "../../chapter_heap/top_k/" class = "md-nav__link" >
8.3. Top-K 问题（ ）
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< a href = "../../chapter_heap/summary/" class = "md-nav__link" >
8.4. 小结
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< a href = "../../chapter_graph/" > 9. 图< / a >
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9. 图
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9.1. 图
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< a href = "../../chapter_graph/graph_operations/" class = "md-nav__link" >
9.2. 图基础操作
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9.3. 图的遍历
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9.4. 小结
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< a href = "../../chapter_searching/" > 10. 搜索< / a >
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10. 搜索
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10.1. 二分查找
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< a href = "../../chapter_searching/binary_search_edge/" class = "md-nav__link" >
10.2. 二分查找边界
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10.3. 哈希优化策略
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10.4. 重识搜索算法
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10.5. 小结
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< a href = "../../chapter_sorting/" > 11. 排序< / a >
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11. 排序
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11.1. 排序算法
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11.2. 选择排序
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< a href = "../../chapter_sorting/bubble_sort/" class = "md-nav__link" >
11.3. 冒泡排序
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11.4. 插入排序
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11.5. 快速排序
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11.6. 归并排序
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11.7. 堆排序
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11.8. 桶排序
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11.9. 计数排序
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11.10. 基数排序
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11.11. 小结
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12. 回溯
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12.1. 回溯算法
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12.2. 全排列问题
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12.3. 子集和问题（ ）
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12.4. N 皇后问题
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12.5. 小结
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< a href = "../" > 13. 动态规划< / a >
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13. 动态规划
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13.1. 初探动态规划（ ）
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13.2. DP 问题特性（ ）
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< a href = "../dp_solution_pipeline.md" class = "md-nav__link" >
13.3. DP 解题步骤（ ）
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13.4. 0-1 背包问题（ ）
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< a href = "./" class = "md-nav__link md-nav__link--active" >
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13.4. 0-1 背包问题（ ）
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< / a >
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目录
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< li class = "md-nav__item" >
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13.4.1. 方法一：暴力搜索
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13.4.2. 方法二：记忆化搜索
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< / a >
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< a href = "#1343" class = "md-nav__link" >
13.4.3. 方法三：动态规划
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14. 附录
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14.1. 编程环境安装
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14.2. 一起参与创作
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参考文献
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13.4.1. 方法一：暴力搜索
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13.4.2. 方法二：记忆化搜索
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13.4.3. 方法三：动态规划
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< svg xmlns = "http://www.w3.org/2000/svg" viewBox = "0 0 24 24" > < path d = "M10 20H6V4h7v5h5v3.1l2-2V8l-6-6H6c-1.1 0-2 .9-2 2v16c0 1.1.9 2 2 2h4v-2m10.2-7c.1 0 .3.1.4.2l1.3 1.3c.2.2.2.6 0 .8l-1 1-2.1-2.1 1-1c.1-.1.2-.2.4-.2m0 3.9L14.1 23H12v-2.1l6.1-6.1 2.1 2.1Z" / > < / svg >
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< h1 id = "134-0-1" > 13.4. 0-1 背包问题< a class = "headerlink" href = "#134-0-1" title = "Permanent link" > ¶ < / a > < / h1 >
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< p > 背包问题是一个非常好的动态规划入门题目，是动态规划中最常见的问题形式。其具有很多变种，例如 0-1 背包问题、完全背包问题、多重背包问题等。< / p >
< p > 在本节中，我们先来学习基础的的 0-1 背包问题。< / p >
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< div class = "admonition question" >
< 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 > ，现在有个容量为 < span class = "arithmatex" > \(cap\)< / span > 的背包，请求解在不超过背包容量下背包中物品的最大价值。< / p >
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< p > 请注意，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 > 下图给出了一个 0-1 背包的示例数据，背包内的最大价值为 < span class = "arithmatex" > \(220\)< / span > 。< / p >
< p > < img alt = "0-1 背包的示例数据" src = "../knapsack_problem.assets/knapsack_example.png" / > < / p >
< p align = "center" > Fig. 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。此外，该问题的目标是求解“在限定背包容量下的最大价值”，因此较大概率是个动态规划问题。我们接下来尝试求解它。< / p >
< p > < strong > 第一步：思考每一轮的决策是什么，从而得到状态定义< / strong > < / p >
< p > 在 0-1 背包问题中，不放入背包，背包容量不变；放入背包，背包容量减小。由此可得状态定义：物品编号 < span class = "arithmatex" > \(i\)< / span > 和背包容量 < span class = "arithmatex" > \(c\)< / span > ，记为 < span class = "arithmatex" > \([i, c]\)< / span > 。< / p >
< p > < strong > 第二步：明确子问题是什么，从而得到 < span class = "arithmatex" > \(dp\)< / span > 列表< / strong > < / p >
< p > 状态 < span class = "arithmatex" > \([i, c]\)< / span > 对应的子问题为：< strong > 前 < span class = "arithmatex" > \(i\)< / span > 个物品在容量为 < span class = "arithmatex" > \(c\)< / span > 背包中的最大价值< / strong > ，记为 < span class = "arithmatex" > \(dp[i, c]\)< / span > 。< / p >
< p > 至此，我们得到一个尺寸为 < span class = "arithmatex" > \(n \times cap\)< / span > 的二维 < span class = "arithmatex" > \(dp\)< / span > 矩阵。< / p >
< p > < strong > 第三步：找出最优子结构，进而推导出状态转移方程< / strong > < / p >
< p > 当我们做出物品 < span class = "arithmatex" > \(i\)< / span > 的决策后，剩余的是前 < span class = "arithmatex" > \(i-1\)< / span > 个物品的决策。因此，状态转移分为两种情况：< / p >
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< ul >
< li > < strong > 不放入物品 < span class = "arithmatex" > \(i\)< / span > < / strong > ：背包容量不变，状态转移至 < span class = "arithmatex" > \([i-1, c]\)< / span > ； < / li >
< li > < strong > 放入物品 < span class = "arithmatex" > \(i\)< / span > < / strong > ：背包容量减小 < span class = "arithmatex" > \(wgt[i-1]\)< / span > ，价值增加 < span class = "arithmatex" > \(val[i-1]\)< / span > ，状态转移至 < span class = "arithmatex" > \([i-1, c-wgt[i-1]]\)< / span > ； < / li >
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< p > 上述的状态转移向我们揭示了本题的最优子结构：< strong > 最大价值 < span class = "arithmatex" > \(dp[i, c]\)< / span > 等于不放入物品 < span class = "arithmatex" > \(i\)< / span > 和放入物品 < span class = "arithmatex" > \(i\)< / span > 两种方案中的价值更大的那一个< / strong > 。由此可推出状态转移方程：< / p >
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< div class = "arithmatex" > \[
dp[i, c] = \max(dp[i-1, c], dp[i-1, c - wgt[i-1]] + val[i-1])
\]< / div >
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< p > 需要注意的是，若当前物品重量 < span class = "arithmatex" > \(wgt[i - 1]\)< / span > 超出剩余背包容量 < span class = "arithmatex" > \(c\)< / span > ，则只能选择不放入背包。< / p >
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< h2 id = "1341" > 13.4.1. 方法一：暴力搜索< a class = "headerlink" href = "#1341" title = "Permanent link" > ¶ < / a > < / h2 >
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< p > 搜索代码包含以下要素：< / p >
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< ul >
< li > < strong > 递归参数< / strong > ：状态 < span class = "arithmatex" > \([i, c]\)< / span > ； < strong > 返回值< / strong > ：子问题的解 < span class = "arithmatex" > \(dp[i, c]\)< / span > 。< / li >
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< li > < strong > 终止条件< / strong > ：当物品编号越界 < span class = "arithmatex" > \(i = 0\)< / span > 或背包剩余容量为 < span class = "arithmatex" > \(0\)< / span > 时，终止递归并返回价值 < span class = "arithmatex" > \(0\)< / span > 。< / li >
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< li > < strong > 剪枝< / strong > ：若当前物品重量超出背包剩余容量，则只能不放入背包。< / li >
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< / ul >
< div class = "tabbed-set tabbed-alternate" data-tabs = "1:11" > < input checked = "checked" id = "__tabbed_1_1" name = "__tabbed_1" type = "radio" / > < input id = "__tabbed_1_2" name = "__tabbed_1" type = "radio" / > < input id = "__tabbed_1_3" name = "__tabbed_1" type = "radio" / > < input id = "__tabbed_1_4" name = "__tabbed_1" type = "radio" / > < input id = "__tabbed_1_5" name = "__tabbed_1" type = "radio" / > < input id = "__tabbed_1_6" name = "__tabbed_1" type = "radio" / > < input id = "__tabbed_1_7" name = "__tabbed_1" type = "radio" / > < input id = "__tabbed_1_8" name = "__tabbed_1" type = "radio" / > < input id = "__tabbed_1_9" name = "__tabbed_1" type = "radio" / > < input id = "__tabbed_1_10" name = "__tabbed_1" type = "radio" / > < input id = "__tabbed_1_11" name = "__tabbed_1" type = "radio" / > < div class = "tabbed-labels" > < label for = "__tabbed_1_1" > Java< / label > < label for = "__tabbed_1_2" > C++< / label > < label for = "__tabbed_1_3" > Python< / label > < label for = "__tabbed_1_4" > Go< / label > < label for = "__tabbed_1_5" > JavaScript< / label > < label for = "__tabbed_1_6" > TypeScript< / label > < label for = "__tabbed_1_7" > C< / label > < label for = "__tabbed_1_8" > C#< / label > < label for = "__tabbed_1_9" > Swift< / label > < label for = "__tabbed_1_10" > Zig< / label > < label for = "__tabbed_1_11" > Dart< / label > < / div >
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< div class = "highlight" > < span class = "filename" > knapsack.java< / span > < pre > < span > < / span > < code > < a id = "__codelineno-0-1" name = "__codelineno-0-1" href = "#__codelineno-0-1" > < / a > < span class = "o" > [< / span > < span class = "kd" > class< / span > < span class = "err" > ]{< / span > < span class = "nc" > knapsack< / 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" > knapsackDFS< / span > < span class = "p" > }< / span >
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< div class = "highlight" > < span class = "filename" > knapsack.cpp< / span > < pre > < span > < / span > < code > < a id = "__codelineno-1-1" name = "__codelineno-1-1" href = "#__codelineno-1-1" > < / a > < span class = "p" > [< / span > < span class = "k" > 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" > knapsackDFS< / span > < span class = "p" > }< / span >
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< div class = "highlight" > < span class = "filename" > knapsack.py< / span > < pre > < span > < / span > < code > < a id = "__codelineno-2-1" name = "__codelineno-2-1" href = "#__codelineno-2-1" > < / a > < span class = "k" > def< / span > < span class = "nf" > knapsack_dfs< / span > < span class = "p" > (< / span > < span class = "n" > wgt< / span > < span class = "p" > ,< / span > < span class = "n" > val< / span > < span class = "p" > ,< / span > < span class = "n" > i< / span > < span class = "p" > ,< / span > < span class = "n" > c< / span > < span class = "p" > ):< / span >
< a id = "__codelineno-2-2" name = "__codelineno-2-2" href = "#__codelineno-2-2" > < / a > < span class = "w" > < / span > < span class = "sd" > " " " 0-1 背包：暴力搜索" " " < / span >
< a id = "__codelineno-2-3" name = "__codelineno-2-3" href = "#__codelineno-2-3" > < / a > < span class = "c1" > # 若已选完所有物品或背包无容量，则返回价值 0< / span >
< a id = "__codelineno-2-4" name = "__codelineno-2-4" href = "#__codelineno-2-4" > < / a > < span class = "k" > if< / span > < span class = "n" > i< / span > < span class = "o" > ==< / span > < span class = "mi" > 0< / span > < span class = "ow" > or< / span > < span class = "n" > c< / span > < span class = "o" > ==< / span > < span class = "mi" > 0< / span > < span class = "p" > :< / span >
< a id = "__codelineno-2-5" name = "__codelineno-2-5" href = "#__codelineno-2-5" > < / a > < span class = "k" > return< / span > < span class = "mi" > 0< / span >
< a id = "__codelineno-2-6" name = "__codelineno-2-6" href = "#__codelineno-2-6" > < / a > < span class = "c1" > # 若超过背包容量，则只能不放入背包< / span >
< a id = "__codelineno-2-7" name = "__codelineno-2-7" href = "#__codelineno-2-7" > < / 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" > > < / span > < span class = "n" > c< / span > < span class = "p" > :< / span >
< a id = "__codelineno-2-8" name = "__codelineno-2-8" href = "#__codelineno-2-8" > < / a > < span class = "k" > return< / span > < span class = "n" > knapsack_dfs< / span > < span class = "p" > (< / span > < span class = "n" > wgt< / span > < span class = "p" > ,< / 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 > < span class = "n" > c< / span > < span class = "p" > )< / span >
< a id = "__codelineno-2-9" name = "__codelineno-2-9" href = "#__codelineno-2-9" > < / a > < span class = "c1" > # 计算不放入和放入物品 i 的最大价值< / span >
< a id = "__codelineno-2-10" name = "__codelineno-2-10" href = "#__codelineno-2-10" > < / a > < span class = "n" > no< / span > < span class = "o" > =< / span > < span class = "n" > knapsack_dfs< / span > < span class = "p" > (< / span > < span class = "n" > wgt< / span > < span class = "p" > ,< / 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 > < span class = "n" > c< / span > < span class = "p" > )< / span >
< a id = "__codelineno-2-11" name = "__codelineno-2-11" href = "#__codelineno-2-11" > < / a > < span class = "n" > yes< / span > < span class = "o" > =< / span > < span class = "n" > knapsack_dfs< / span > < span class = "p" > (< / span > < span class = "n" > wgt< / span > < span class = "p" > ,< / 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 > < 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-2-12" name = "__codelineno-2-12" href = "#__codelineno-2-12" > < / a > < span class = "c1" > # 返回两种方案中价值更大的那一个< / span >
< a id = "__codelineno-2-13" name = "__codelineno-2-13" href = "#__codelineno-2-13" > < / a > < span class = "k" > return< / span > < span class = "nb" > max< / span > < span class = "p" > (< / span > < span class = "n" > no< / span > < span class = "p" > ,< / span > < span class = "n" > yes< / span > < span class = "p" > )< / span >
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< div class = "highlight" > < span class = "filename" > knapsack.go< / span > < pre > < span > < / span > < code > < a id = "__codelineno-3-1" name = "__codelineno-3-1" href = "#__codelineno-3-1" > < / 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" > knapsackDFS< / span > < span class = "p" > }< / span >
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< div class = "highlight" > < span class = "filename" > knapsack.js< / 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 = "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" > knapsackDFS< / span > < span class = "p" > }< / span >
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< div class = "highlight" > < span class = "filename" > knapsack.ts< / 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 = "nx" > func< / span > < span class = "p" > ]{< / span > < span class = "nx" > knapsackDFS< / span > < span class = "p" > }< / span >
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< div class = "highlight" > < span class = "filename" > knapsack.c< / 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 = "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" > knapsackDFS< / span > < span class = "p" > }< / span >
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< div class = "highlight" > < span class = "filename" > knapsack.cs< / span > < pre > < span > < / span > < code > < a id = "__codelineno-7-1" name = "__codelineno-7-1" href = "#__codelineno-7-1" > < / a > < span class = "na" > [class]< / span > < span class = "p" > {< / span > < span class = "n" > 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" > knapsackDFS< / span > < span class = "p" > }< / span >
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< div class = "highlight" > < span class = "filename" > knapsack.swift< / 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 = "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" > knapsackDFS< / span > < span class = "p" > }< / span >
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< div class = "highlight" > < span class = "filename" > knapsack.zig< / 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" > knapsackDFS< / span > < span class = "p" > }< / span >
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< div class = "highlight" > < span class = "filename" > knapsack.dart< / 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" > knapsackDFS< / span > < span class = "p" > }< / span >
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< p > 如下图所示，由于每个物品都会产生不选和选两条搜索分支，因此最差时间复杂度为 < span class = "arithmatex" > \(O(2^n)\)< / span > 。< / p >
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< p > 观察递归树，容易发现其中存在一些「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，尤其是当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 > < img alt = "0-1 背包的暴力搜索递归树" src = "../knapsack_problem.assets/knapsack_dfs.png" / > < / p >
< p align = "center" > Fig. 0-1 背包的暴力搜索递归树 < / p >
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< h2 id = "1342" > 13.4.2. 方法二：记忆化搜索< a class = "headerlink" href = "#1342" title = "Permanent link" > ¶ < / a > < / h2 >
< p > 为了防止重复求解重叠子问题，我们借助一个记忆列表 < code > mem< / code > 来记录子问题的解，其中 < code > mem[i][c]< / code > 对应解 < span class = "arithmatex" > \(dp[i, c]\)< / span > 。< / p >
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< div class = "tabbed-set tabbed-alternate" data-tabs = "2:11" > < input checked = "checked" id = "__tabbed_2_1" name = "__tabbed_2" type = "radio" / > < input id = "__tabbed_2_2" name = "__tabbed_2" type = "radio" / > < input id = "__tabbed_2_3" name = "__tabbed_2" type = "radio" / > < input id = "__tabbed_2_4" name = "__tabbed_2" type = "radio" / > < input id = "__tabbed_2_5" name = "__tabbed_2" type = "radio" / > < input id = "__tabbed_2_6" name = "__tabbed_2" type = "radio" / > < input id = "__tabbed_2_7" name = "__tabbed_2" type = "radio" / > < input id = "__tabbed_2_8" name = "__tabbed_2" type = "radio" / > < input id = "__tabbed_2_9" name = "__tabbed_2" type = "radio" / > < input id = "__tabbed_2_10" name = "__tabbed_2" type = "radio" / > < input id = "__tabbed_2_11" name = "__tabbed_2" type = "radio" / > < div class = "tabbed-labels" > < label for = "__tabbed_2_1" > Java< / label > < label for = "__tabbed_2_2" > C++< / label > < label for = "__tabbed_2_3" > Python< / label > < label for = "__tabbed_2_4" > Go< / label > < label for = "__tabbed_2_5" > JavaScript< / label > < label for = "__tabbed_2_6" > TypeScript< / label > < label for = "__tabbed_2_7" > C< / label > < label for = "__tabbed_2_8" > C#< / label > < label for = "__tabbed_2_9" > Swift< / label > < label for = "__tabbed_2_10" > Zig< / label > < label for = "__tabbed_2_11" > Dart< / label > < / div >
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< div class = "highlight" > < span class = "filename" > knapsack.java< / 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 = "kd" > class< / span > < span class = "err" > ]{< / span > < span class = "nc" > knapsack< / 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" > knapsackDFSMem< / span > < span class = "p" > }< / span >
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< div class = "highlight" > < span class = "filename" > knapsack.cpp< / span > < pre > < span > < / span > < code > < a id = "__codelineno-12-1" name = "__codelineno-12-1" href = "#__codelineno-12-1" > < / a > < span class = "p" > [< / span > < span class = "k" > 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" > knapsackDFSMem< / span > < span class = "p" > }< / span >
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< div class = "highlight" > < span class = "filename" > knapsack.py< / span > < pre > < span > < / span > < code > < a id = "__codelineno-13-1" name = "__codelineno-13-1" href = "#__codelineno-13-1" > < / a > < span class = "k" > def< / span > < span class = "nf" > knapsack_dfs_mem< / span > < span class = "p" > (< / span > < span class = "n" > wgt< / span > < span class = "p" > ,< / span > < span class = "n" > val< / span > < span class = "p" > ,< / span > < span class = "n" > mem< / span > < span class = "p" > ,< / span > < span class = "n" > i< / span > < span class = "p" > ,< / span > < span class = "n" > c< / span > < span class = "p" > ):< / span >
< a id = "__codelineno-13-2" name = "__codelineno-13-2" href = "#__codelineno-13-2" > < / a > < span class = "w" > < / span > < span class = "sd" > " " " 0-1 背包：记忆化搜索" " " < / span >
< a id = "__codelineno-13-3" name = "__codelineno-13-3" href = "#__codelineno-13-3" > < / a > < span class = "c1" > # 若已选完所有物品或背包无容量，则返回价值 0< / span >
< a id = "__codelineno-13-4" name = "__codelineno-13-4" href = "#__codelineno-13-4" > < / a > < span class = "k" > if< / span > < span class = "n" > i< / span > < span class = "o" > ==< / span > < span class = "mi" > 0< / span > < span class = "ow" > or< / span > < span class = "n" > c< / span > < span class = "o" > ==< / span > < span class = "mi" > 0< / span > < span class = "p" > :< / span >
< a id = "__codelineno-13-5" name = "__codelineno-13-5" href = "#__codelineno-13-5" > < / a > < span class = "k" > return< / span > < span class = "mi" > 0< / span >
< a id = "__codelineno-13-6" name = "__codelineno-13-6" href = "#__codelineno-13-6" > < / a > < span class = "c1" > # 若已有记录，则直接返回< / span >
< a id = "__codelineno-13-7" name = "__codelineno-13-7" href = "#__codelineno-13-7" > < / a > < span class = "k" > if< / span > < span class = "n" > mem< / 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 = "o" > -< / span > < span class = "mi" > 1< / span > < span class = "p" > :< / span >
< a id = "__codelineno-13-8" name = "__codelineno-13-8" href = "#__codelineno-13-8" > < / a > < span class = "k" > return< / span > < span class = "n" > mem< / span > < span class = "p" > [< / span > < span class = "n" > i< / span > < span class = "p" > ][< / span > < span class = "n" > c< / span > < span class = "p" > ]< / span >
< a id = "__codelineno-13-9" name = "__codelineno-13-9" href = "#__codelineno-13-9" > < / a > < span class = "c1" > # 若超过背包容量，则只能不放入背包< / span >
< a id = "__codelineno-13-10" name = "__codelineno-13-10" href = "#__codelineno-13-10" > < / 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" > > < / span > < span class = "n" > c< / span > < span class = "p" > :< / span >
< a id = "__codelineno-13-11" name = "__codelineno-13-11" href = "#__codelineno-13-11" > < / a > < span class = "k" > return< / span > < span class = "n" > knapsack_dfs_mem< / span > < span class = "p" > (< / span > < span class = "n" > wgt< / span > < span class = "p" > ,< / span > < span class = "n" > val< / span > < span class = "p" > ,< / span > < span class = "n" > mem< / 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-13-12" name = "__codelineno-13-12" href = "#__codelineno-13-12" > < / a > < span class = "c1" > # 计算不放入和放入物品 i 的最大价值< / span >
< a id = "__codelineno-13-13" name = "__codelineno-13-13" href = "#__codelineno-13-13" > < / a > < span class = "n" > no< / span > < span class = "o" > =< / span > < span class = "n" > knapsack_dfs_mem< / span > < span class = "p" > (< / span > < span class = "n" > wgt< / span > < span class = "p" > ,< / span > < span class = "n" > val< / span > < span class = "p" > ,< / span > < span class = "n" > mem< / 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-13-14" name = "__codelineno-13-14" href = "#__codelineno-13-14" > < / a > < span class = "n" > yes< / span > < span class = "o" > =< / span > < span class = "n" > knapsack_dfs_mem< / span > < span class = "p" > (< / span > < span class = "n" > wgt< / span > < span class = "p" > ,< / span > < span class = "n" > val< / span > < span class = "p" > ,< / span > < span class = "n" > mem< / 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 = "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-13-15" name = "__codelineno-13-15" href = "#__codelineno-13-15" > < / a > < span class = "c1" > # 记录并返回两种方案中价值更大的那一个< / span >
< a id = "__codelineno-13-16" name = "__codelineno-13-16" href = "#__codelineno-13-16" > < / a > < span class = "n" > mem< / 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" > no< / span > < span class = "p" > ,< / span > < span class = "n" > yes< / span > < span class = "p" > )< / span >
< a id = "__codelineno-13-17" name = "__codelineno-13-17" href = "#__codelineno-13-17" > < / a > < span class = "k" > return< / span > < span class = "n" > mem< / span > < span class = "p" > [< / span > < span class = "n" > i< / span > < span class = "p" > ][< / span > < span class = "n" > c< / span > < span class = "p" > ]< / span >
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< div class = "highlight" > < span class = "filename" > knapsack.go< / span > < pre > < span > < / span > < code > < a id = "__codelineno-14-1" name = "__codelineno-14-1" href = "#__codelineno-14-1" > < / 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" > knapsackDFSMem< / span > < span class = "p" > }< / span >
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< div class = "highlight" > < span class = "filename" > knapsack.js< / span > < pre > < span > < / span > < code > < a id = "__codelineno-15-1" name = "__codelineno-15-1" href = "#__codelineno-15-1" > < / 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" > knapsackDFSMem< / span > < span class = "p" > }< / span >
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< div class = "highlight" > < span class = "filename" > knapsack.ts< / span > < pre > < span > < / span > < code > < a id = "__codelineno-16-1" name = "__codelineno-16-1" href = "#__codelineno-16-1" > < / 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" > knapsackDFSMem< / span > < span class = "p" > }< / span >
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< div class = "highlight" > < span class = "filename" > knapsack.c< / span > < pre > < span > < / span > < code > < a id = "__codelineno-17-1" name = "__codelineno-17-1" href = "#__codelineno-17-1" > < / 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" > knapsackDFSMem< / span > < span class = "p" > }< / span >
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< div class = "highlight" > < span class = "filename" > knapsack.cs< / span > < pre > < span > < / span > < code > < a id = "__codelineno-18-1" name = "__codelineno-18-1" href = "#__codelineno-18-1" > < / a > < span class = "na" > [class]< / span > < span class = "p" > {< / span > < span class = "n" > 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" > knapsackDFSMem< / span > < span class = "p" > }< / span >
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< div class = "highlight" > < span class = "filename" > 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" > knapsackDFSMem< / span > < span class = "p" > }< / span >
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< div class = "highlight" > < span class = "filename" > knapsack.zig< / 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 = "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" > knapsackDFSMem< / span > < span class = "p" > }< / span >
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< div class = "highlight" > < span class = "filename" > knapsack.dart< / 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 = "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" > knapsackDFSMem< / span > < span class = "p" > }< / span >
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< p > 引入记忆化之后，所有子问题都只被计算一次，< strong > 因此时间复杂度取决于子问题数量< / strong > ，也就是 < span class = "arithmatex" > \(O(n \times cap)\)< / span > 。< / p >
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< p > < img alt = "0-1 背包的记忆化搜索递归树" src = "../knapsack_problem.assets/knapsack_dfs_mem.png" / > < / p >
< p align = "center" > Fig. 0-1 背包的记忆化搜索递归树 < / p >
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< h2 id = "1343" > 13.4.3. 方法三：动态规划< a class = "headerlink" href = "#1343" title = "Permanent link" > ¶ < / a > < / h2 >
< p > 动态规划解法本质上就是在状态转移中填充 < code > dp< / code > 矩阵的过程，代码如下所示。< / p >
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< div class = "tabbed-set tabbed-alternate" data-tabs = "3:11" > < input checked = "checked" id = "__tabbed_3_1" name = "__tabbed_3" type = "radio" / > < input id = "__tabbed_3_2" name = "__tabbed_3" type = "radio" / > < input id = "__tabbed_3_3" name = "__tabbed_3" type = "radio" / > < input id = "__tabbed_3_4" name = "__tabbed_3" type = "radio" / > < input id = "__tabbed_3_5" name = "__tabbed_3" type = "radio" / > < input id = "__tabbed_3_6" name = "__tabbed_3" type = "radio" / > < input id = "__tabbed_3_7" name = "__tabbed_3" type = "radio" / > < input id = "__tabbed_3_8" name = "__tabbed_3" type = "radio" / > < input id = "__tabbed_3_9" name = "__tabbed_3" type = "radio" / > < input id = "__tabbed_3_10" name = "__tabbed_3" type = "radio" / > < input id = "__tabbed_3_11" name = "__tabbed_3" type = "radio" / > < div class = "tabbed-labels" > < label for = "__tabbed_3_1" > Java< / label > < label for = "__tabbed_3_2" > C++< / label > < label for = "__tabbed_3_3" > Python< / label > < label for = "__tabbed_3_4" > Go< / label > < label for = "__tabbed_3_5" > JavaScript< / label > < label for = "__tabbed_3_6" > TypeScript< / label > < label for = "__tabbed_3_7" > C< / label > < label for = "__tabbed_3_8" > C#< / label > < label for = "__tabbed_3_9" > Swift< / label > < label for = "__tabbed_3_10" > Zig< / label > < label for = "__tabbed_3_11" > Dart< / label > < / div >
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< div class = "highlight" > < span class = "filename" > knapsack.java< / span > < pre > < span > < / span > < code > < a id = "__codelineno-22-1" name = "__codelineno-22-1" href = "#__codelineno-22-1" > < / a > < span class = "o" > [< / span > < span class = "kd" > class< / span > < span class = "err" > ]{< / span > < span class = "nc" > knapsack< / 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" > knapsackDP< / span > < span class = "p" > }< / span >
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< div class = "highlight" > < span class = "filename" > knapsack.cpp< / 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 = "k" > 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" > knapsackDP< / span > < span class = "p" > }< / span >
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< div class = "highlight" > < span class = "filename" > knapsack.py< / span > < pre > < span > < / span > < code > < a id = "__codelineno-24-1" name = "__codelineno-24-1" href = "#__codelineno-24-1" > < / a > < span class = "k" > def< / span > < span class = "nf" > knapsack_dp< / span > < span class = "p" > (< / span > < span class = "n" > wgt< / span > < span class = "p" > ,< / span > < span class = "n" > val< / span > < span class = "p" > ,< / span > < span class = "n" > cap< / span > < span class = "p" > ):< / span >
< a id = "__codelineno-24-2" name = "__codelineno-24-2" href = "#__codelineno-24-2" > < / a > < span class = "w" > < / span > < span class = "sd" > " " " 0-1 背包：动态规划" " " < / span >
< a id = "__codelineno-24-3" name = "__codelineno-24-3" href = "#__codelineno-24-3" > < / 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-24-4" name = "__codelineno-24-4" href = "#__codelineno-24-4" > < / a > < span class = "c1" > # 初始化 dp 列表< / span >
< a id = "__codelineno-24-5" name = "__codelineno-24-5" href = "#__codelineno-24-5" > < / 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-24-6" name = "__codelineno-24-6" href = "#__codelineno-24-6" > < / a > < span class = "c1" > # 状态转移< / span >
< a id = "__codelineno-24-7" name = "__codelineno-24-7" href = "#__codelineno-24-7" > < / 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-24-8" name = "__codelineno-24-8" href = "#__codelineno-24-8" > < / 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-24-9" name = "__codelineno-24-9" href = "#__codelineno-24-9" > < / 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" > > < / span > < span class = "n" > c< / span > < span class = "p" > :< / span >
< a id = "__codelineno-24-10" name = "__codelineno-24-10" href = "#__codelineno-24-10" > < / a > < span class = "c1" > # 若超过背包容量，则不选物品 i< / span >
< a id = "__codelineno-24-11" name = "__codelineno-24-11" href = "#__codelineno-24-11" > < / 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-24-12" name = "__codelineno-24-12" href = "#__codelineno-24-12" > < / a > < span class = "k" > else< / span > < span class = "p" > :< / span >
< a id = "__codelineno-24-13" name = "__codelineno-24-13" href = "#__codelineno-24-13" > < / a > < span class = "c1" > # 不选和选物品 i 这两种方案的较大值< / span >
< a id = "__codelineno-24-14" name = "__codelineno-24-14" href = "#__codelineno-24-14" > < / 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 = "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 > < 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-24-15" name = "__codelineno-24-15" href = "#__codelineno-24-15" > < / 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 >
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< div class = "highlight" > < span class = "filename" > knapsack.go< / span > < pre > < span > < / span > < code > < a id = "__codelineno-25-1" name = "__codelineno-25-1" href = "#__codelineno-25-1" > < / 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" > knapsackDP< / span > < span class = "p" > }< / span >
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< div class = "highlight" > < span class = "filename" > knapsack.js< / span > < pre > < span > < / span > < code > < a id = "__codelineno-26-1" name = "__codelineno-26-1" href = "#__codelineno-26-1" > < / 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" > knapsackDP< / span > < span class = "p" > }< / span >
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< div class = "highlight" > < span class = "filename" > knapsack.ts< / 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 = "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" > knapsackDP< / span > < span class = "p" > }< / span >
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< div class = "highlight" > < span class = "filename" > knapsack.c< / span > < pre > < span > < / span > < code > < a id = "__codelineno-28-1" name = "__codelineno-28-1" href = "#__codelineno-28-1" > < / 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" > knapsackDP< / span > < span class = "p" > }< / span >
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< div class = "highlight" > < span class = "filename" > knapsack.cs< / span > < pre > < span > < / span > < code > < a id = "__codelineno-29-1" name = "__codelineno-29-1" href = "#__codelineno-29-1" > < / a > < span class = "na" > [class]< / span > < span class = "p" > {< / span > < span class = "n" > 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" > knapsackDP< / span > < span class = "p" > }< / span >
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< div class = "highlight" > < span class = "filename" > knapsack.swift< / span > < pre > < span > < / span > < code > < a id = "__codelineno-30-1" name = "__codelineno-30-1" href = "#__codelineno-30-1" > < / 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" > knapsackDP< / span > < span class = "p" > }< / span >
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< div class = "highlight" > < span class = "filename" > knapsack.zig< / span > < pre > < span > < / span > < code > < a id = "__codelineno-31-1" name = "__codelineno-31-1" href = "#__codelineno-31-1" > < / 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" > knapsackDP< / span > < span class = "p" > }< / span >
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< div class = "highlight" > < span class = "filename" > knapsack.dart< / 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 = "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" > knapsackDP< / span > < span class = "p" > }< / span >
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< p > 如下图所示，时间复杂度由 < code > dp< / code > 矩阵大小决定，为 < span class = "arithmatex" > \(O(n \times cap)\)< / span > 。< / p >
2023-07-06 00:08:29 +08:00
< 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 > < img alt = "0-1 背包的动态规划过程" src = "../knapsack_problem.assets/knapsack_dp_step1.png" / > < / p >
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< p > < img alt = "knapsack_dp_step2" src = "../knapsack_problem.assets/knapsack_dp_step2.png" / > < / p >
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< p > < img alt = "knapsack_dp_step3" src = "../knapsack_problem.assets/knapsack_dp_step3.png" / > < / p >
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< p > < img alt = "knapsack_dp_step4" src = "../knapsack_problem.assets/knapsack_dp_step4.png" / > < / p >
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< p > < img alt = "knapsack_dp_step5" src = "../knapsack_problem.assets/knapsack_dp_step5.png" / > < / p >
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< p > < img alt = "knapsack_dp_step6" src = "../knapsack_problem.assets/knapsack_dp_step6.png" / > < / p >
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< p > < img alt = "knapsack_dp_step7" src = "../knapsack_problem.assets/knapsack_dp_step7.png" / > < / p >
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< p > < img alt = "knapsack_dp_step8" src = "../knapsack_problem.assets/knapsack_dp_step8.png" / > < / p >
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< p > < img alt = "knapsack_dp_step9" src = "../knapsack_problem.assets/knapsack_dp_step9.png" / > < / p >
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< p > < img alt = "knapsack_dp_step10" src = "../knapsack_problem.assets/knapsack_dp_step10.png" / > < / p >
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< p > < img alt = "knapsack_dp_step11" src = "../knapsack_problem.assets/knapsack_dp_step11.png" / > < / p >
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< p > < img alt = "knapsack_dp_step12" src = "../knapsack_problem.assets/knapsack_dp_step12.png" / > < / p >
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< p > < img alt = "knapsack_dp_step13" src = "../knapsack_problem.assets/knapsack_dp_step13.png" / > < / p >
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< p > < img alt = "knapsack_dp_step14" src = "../knapsack_problem.assets/knapsack_dp_step14.png" / > < / p >
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< p > < strong > 最后考虑状态压缩< / strong > 。以上代码中的 < code > dp< / code > 矩阵占用 < span class = "arithmatex" > \(O(n \times cap)\)< / span > 空间。由于每个状态都只与其上一行的状态有关，因此我们可以使用两个数组滚动前进，将空间复杂度从 < span class = "arithmatex" > \(O(n^2)\)< / span > 将低至 < span class = "arithmatex" > \(O(n)\)< / span > 。代码省略，有兴趣的同学可以自行实现。< / p >
2023-07-06 04:09:15 +08:00
< p > 那么，我们是否可以仅用一个数组实现状态压缩呢？观察可知，每个状态都是由左上方或正上方的格子转移过来的。假设只有一个数组，当遍历到第 < span class = "arithmatex" > \(i\)< / span > 行时，该数组存储的仍然是第 < span class = "arithmatex" > \(i-1\)< / span > 行的状态，为了避免左边区域的格子在状态转移中被覆盖，我们应采取倒序遍历。< / p >
2023-07-06 00:08:29 +08:00
< p > 以下动画展示了在单个数组下从第 < span class = "arithmatex" > \(i=1\)< / span > 行转换至第 < span class = "arithmatex" > \(i=2\)< / span > 行的过程。建议你思考一下正序遍历和倒序遍历的区别。< / p >
< 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 > < img alt = "0-1 背包的状态压缩后的动态规划过程" src = "../knapsack_problem.assets/knapsack_dp_comp_step1.png" / > < / p >
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< p > < img alt = "knapsack_dp_comp_step2" src = "../knapsack_problem.assets/knapsack_dp_comp_step2.png" / > < / p >
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< p > < img alt = "knapsack_dp_comp_step3" src = "../knapsack_problem.assets/knapsack_dp_comp_step3.png" / > < / p >
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< p > < img alt = "knapsack_dp_comp_step4" src = "../knapsack_problem.assets/knapsack_dp_comp_step4.png" / > < / p >
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< p > < img alt = "knapsack_dp_comp_step5" src = "../knapsack_problem.assets/knapsack_dp_comp_step5.png" / > < / p >
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< div class = "tabbed-block" >
< p > < img alt = "knapsack_dp_comp_step6" src = "../knapsack_problem.assets/knapsack_dp_comp_step6.png" / > < / p >
< / div >
< / div >
< / div >
2023-07-07 21:39:04 +08:00
< p > 如以下代码所示，我们仅需将 < code > dp< / code > 矩阵的第一维 < span class = "arithmatex" > \(i\)< / span > 直接删除，并且将内循环修改为倒序遍历即可。< / p >
2023-07-06 00:08:29 +08:00
< div class = "tabbed-set tabbed-alternate" data-tabs = "6:11" > < 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" / > < div class = "tabbed-labels" > < label for = "__tabbed_6_1" > Java< / label > < label for = "__tabbed_6_2" > C++< / label > < label for = "__tabbed_6_3" > Python< / label > < label for = "__tabbed_6_4" > Go< / label > < label for = "__tabbed_6_5" > JavaScript< / label > < label for = "__tabbed_6_6" > TypeScript< / label > < label for = "__tabbed_6_7" > C< / label > < label for = "__tabbed_6_8" > C#< / label > < label for = "__tabbed_6_9" > Swift< / label > < label for = "__tabbed_6_10" > Zig< / label > < label for = "__tabbed_6_11" > Dart< / label > < / div >
< div class = "tabbed-content" >
< div class = "tabbed-block" >
< div class = "highlight" > < span class = "filename" > knapsack.java< / span > < pre > < span > < / span > < code > < a id = "__codelineno-33-1" name = "__codelineno-33-1" href = "#__codelineno-33-1" > < / a > < span class = "o" > [< / span > < span class = "kd" > class< / span > < span class = "err" > ]{< / span > < span class = "nc" > knapsack< / 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" > knapsackDPComp< / span > < span class = "p" > }< / span >
< / code > < / pre > < / div >
< / div >
< div class = "tabbed-block" >
< div class = "highlight" > < span class = "filename" > knapsack.cpp< / 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 = "k" > 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" > knapsackDPComp< / span > < span class = "p" > }< / span >
< / code > < / pre > < / div >
< / div >
< div class = "tabbed-block" >
< div class = "highlight" > < span class = "filename" > knapsack.py< / span > < pre > < span > < / span > < code > < a id = "__codelineno-35-1" name = "__codelineno-35-1" href = "#__codelineno-35-1" > < / a > < span class = "k" > def< / span > < span class = "nf" > knapsack_dp_comp< / span > < span class = "p" > (< / span > < span class = "n" > wgt< / span > < span class = "p" > ,< / span > < span class = "n" > val< / span > < span class = "p" > ,< / span > < span class = "n" > cap< / span > < span class = "p" > ):< / span >
< a id = "__codelineno-35-2" name = "__codelineno-35-2" href = "#__codelineno-35-2" > < / a > < span class = "w" > < / span > < span class = "sd" > " " " 0-1 背包：状态压缩后的动态规划" " " < / span >
< a id = "__codelineno-35-3" name = "__codelineno-35-3" href = "#__codelineno-35-3" > < / 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-35-4" name = "__codelineno-35-4" href = "#__codelineno-35-4" > < / a > < span class = "c1" > # 初始化 dp 列表< / span >
< a id = "__codelineno-35-5" name = "__codelineno-35-5" href = "#__codelineno-35-5" > < / 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-35-6" name = "__codelineno-35-6" href = "#__codelineno-35-6" > < / a > < span class = "c1" > # 状态转移< / span >
< a id = "__codelineno-35-7" name = "__codelineno-35-7" href = "#__codelineno-35-7" > < / 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-35-8" name = "__codelineno-35-8" href = "#__codelineno-35-8" > < / a > < span class = "c1" > # 倒序遍历< / span >
< a id = "__codelineno-35-9" name = "__codelineno-35-9" href = "#__codelineno-35-9" > < / 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 = "n" > cap< / span > < span class = "p" > ,< / span > < span class = "mi" > 0< / span > < span class = "p" > ,< / span > < span class = "o" > -< / span > < span class = "mi" > 1< / span > < span class = "p" > ):< / span >
< a id = "__codelineno-35-10" name = "__codelineno-35-10" href = "#__codelineno-35-10" > < / 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" > > < / span > < span class = "n" > c< / span > < span class = "p" > :< / span >
< a id = "__codelineno-35-11" name = "__codelineno-35-11" href = "#__codelineno-35-11" > < / a > < span class = "c1" > # 若超过背包容量，则不选物品 i< / span >
< a id = "__codelineno-35-12" name = "__codelineno-35-12" href = "#__codelineno-35-12" > < / 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-35-13" name = "__codelineno-35-13" href = "#__codelineno-35-13" > < / a > < span class = "k" > else< / span > < span class = "p" > :< / span >
< a id = "__codelineno-35-14" name = "__codelineno-35-14" href = "#__codelineno-35-14" > < / a > < span class = "c1" > # 不选和选物品 i 这两种方案的较大值< / span >
< a id = "__codelineno-35-15" name = "__codelineno-35-15" href = "#__codelineno-35-15" > < / 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 = "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 > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "n" > c< / span > < span class = "p" > ])< / span >
< a id = "__codelineno-35-16" name = "__codelineno-35-16" href = "#__codelineno-35-16" > < / 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" > knapsack.go< / 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 = "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" > knapsackDPComp< / span > < span class = "p" > }< / span >
< / code > < / pre > < / div >
< / div >
< div class = "tabbed-block" >
< div class = "highlight" > < span class = "filename" > knapsack.js< / 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 = "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" > knapsackDPComp< / span > < span class = "p" > }< / span >
< / code > < / pre > < / div >
< / div >
< div class = "tabbed-block" >
< div class = "highlight" > < span class = "filename" > knapsack.ts< / 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 = "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" > knapsackDPComp< / span > < span class = "p" > }< / span >
< / code > < / pre > < / div >
< / div >
< div class = "tabbed-block" >
< div class = "highlight" > < span class = "filename" > knapsack.c< / span > < pre > < span > < / span > < code > < a id = "__codelineno-39-1" name = "__codelineno-39-1" href = "#__codelineno-39-1" > < / 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" > knapsackDPComp< / span > < span class = "p" > }< / span >
< / code > < / pre > < / div >
< / div >
< div class = "tabbed-block" >
< div class = "highlight" > < span class = "filename" > knapsack.cs< / span > < pre > < span > < / span > < code > < a id = "__codelineno-40-1" name = "__codelineno-40-1" href = "#__codelineno-40-1" > < / a > < span class = "na" > [class]< / span > < span class = "p" > {< / span > < span class = "n" > 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" > knapsackDPComp< / span > < span class = "p" > }< / span >
< / code > < / pre > < / div >
< / div >
< div class = "tabbed-block" >
< div class = "highlight" > < span class = "filename" > knapsack.swift< / 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 = "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" > knapsackDPComp< / span > < span class = "p" > }< / span >
< / code > < / pre > < / div >
< / div >
< div class = "tabbed-block" >
< div class = "highlight" > < span class = "filename" > knapsack.zig< / span > < pre > < span > < / span > < code > < a id = "__codelineno-42-1" name = "__codelineno-42-1" href = "#__codelineno-42-1" > < / 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" > knapsackDPComp< / span > < span class = "p" > }< / span >
< / code > < / pre > < / div >
< / div >
< div class = "tabbed-block" >
< div class = "highlight" > < span class = "filename" > knapsack.dart< / span > < pre > < span > < / span > < code > < a id = "__codelineno-43-1" name = "__codelineno-43-1" href = "#__codelineno-43-1" > < / 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" > knapsackDPComp< / span > < span class = "p" > }< / span >
< / code > < / pre > < / div >
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