From 55d8b710fad76adda433af4ad8b5b0ee8d90e3d1 Mon Sep 17 00:00:00 2001 From: Yudong Jin Date: Thu, 6 Jul 2023 00:06:14 +0800 Subject: [PATCH] Add the section of knapsack problem. (#580) --- .../chapter_dynamic_programming/knapsack.py | 99 +++++ .../knapsack_problem.assets/knapsack_dfs.png | Bin 0 -> 105862 bytes .../knapsack_dfs_mem.png | Bin 0 -> 105434 bytes .../knapsack_dp_comp_step1.png | Bin 0 -> 64074 bytes .../knapsack_dp_comp_step2.png | Bin 0 -> 68113 bytes .../knapsack_dp_comp_step3.png | Bin 0 -> 69230 bytes .../knapsack_dp_comp_step4.png | Bin 0 -> 68402 bytes .../knapsack_dp_comp_step5.png | Bin 0 -> 66215 bytes .../knapsack_dp_comp_step6.png | Bin 0 -> 67079 bytes .../knapsack_dp_step1.png | Bin 0 -> 62880 bytes .../knapsack_dp_step10.png | Bin 0 -> 66144 bytes .../knapsack_dp_step11.png | Bin 0 -> 66009 bytes .../knapsack_dp_step12.png | Bin 0 -> 70384 bytes .../knapsack_dp_step13.png | Bin 0 -> 70695 bytes .../knapsack_dp_step14.png | Bin 0 -> 59214 bytes .../knapsack_dp_step2.png | Bin 0 -> 67470 bytes .../knapsack_dp_step3.png | Bin 0 -> 67887 bytes .../knapsack_dp_step4.png | Bin 0 -> 67682 bytes .../knapsack_dp_step5.png | Bin 0 -> 67313 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docs/chapter_dynamic_programming/knapsack_problem.assets/knapsack_dp_comp_step4.png create mode 100644 docs/chapter_dynamic_programming/knapsack_problem.assets/knapsack_dp_comp_step5.png create mode 100644 docs/chapter_dynamic_programming/knapsack_problem.assets/knapsack_dp_comp_step6.png create mode 100644 docs/chapter_dynamic_programming/knapsack_problem.assets/knapsack_dp_step1.png create mode 100644 docs/chapter_dynamic_programming/knapsack_problem.assets/knapsack_dp_step10.png create mode 100644 docs/chapter_dynamic_programming/knapsack_problem.assets/knapsack_dp_step11.png create mode 100644 docs/chapter_dynamic_programming/knapsack_problem.assets/knapsack_dp_step12.png create mode 100644 docs/chapter_dynamic_programming/knapsack_problem.assets/knapsack_dp_step13.png create mode 100644 docs/chapter_dynamic_programming/knapsack_problem.assets/knapsack_dp_step14.png create mode 100644 docs/chapter_dynamic_programming/knapsack_problem.assets/knapsack_dp_step2.png create mode 100644 docs/chapter_dynamic_programming/knapsack_problem.assets/knapsack_dp_step3.png create mode 100644 docs/chapter_dynamic_programming/knapsack_problem.assets/knapsack_dp_step4.png create mode 100644 docs/chapter_dynamic_programming/knapsack_problem.assets/knapsack_dp_step5.png create mode 100644 docs/chapter_dynamic_programming/knapsack_problem.assets/knapsack_dp_step6.png create mode 100644 docs/chapter_dynamic_programming/knapsack_problem.assets/knapsack_dp_step7.png create mode 100644 docs/chapter_dynamic_programming/knapsack_problem.assets/knapsack_dp_step8.png create mode 100644 docs/chapter_dynamic_programming/knapsack_problem.assets/knapsack_dp_step9.png create mode 100644 docs/chapter_dynamic_programming/knapsack_problem.assets/knapsack_example.png create mode 100644 docs/chapter_dynamic_programming/knapsack_problem.md diff --git a/codes/python/chapter_dynamic_programming/knapsack.py b/codes/python/chapter_dynamic_programming/knapsack.py new file mode 100644 index 000000000..9f8d3cff4 --- /dev/null +++ b/codes/python/chapter_dynamic_programming/knapsack.py @@ -0,0 +1,99 @@ +""" +File: knapsack.py +Created Time: 2023-07-03 +Author: Krahets (krahets@163.com) +""" + + +def knapsack_dfs(wgt, val, i, c): + """0-1 背包:暴力搜索""" + # 若已选完所有物品或背包无容量,则返回价值 0 + if i == 0 or c == 0: + return 0 + # 若超过背包容量,则只能不放入背包 + if wgt[i - 1] > c: + return knapsack_dfs(wgt, val, i - 1, c) + # 计算不放入和放入物品 i 的最大价值 + no = knapsack_dfs(wgt, val, i - 1, c) + yes = knapsack_dfs(wgt, val, i - 1, c - wgt[i - 1]) + val[i - 1] + # 返回两种方案中价值更大的那一个 + return max(no, yes) + + +def knapsack_dfs_mem(wgt, val, mem, i, c): + """0-1 背包:记忆化搜索""" + # 若已选完所有物品或背包无容量,则返回价值 0 + if i == 0 or c == 0: + return 0 + # 若已有记录,则直接返回 + if mem[i][c] != -1: + return mem[i][c] + # 若超过背包容量,则只能不放入背包 + if wgt[i - 1] > c: + return knapsack_dfs_mem(wgt, val, mem, i - 1, c) + # 计算不放入和放入物品 i 的最大价值 + no = knapsack_dfs_mem(wgt, val, mem, i - 1, c) + yes = knapsack_dfs_mem(wgt, val, mem, i - 1, c - wgt[i - 1]) + val[i - 1] + # 记录并返回两种方案中价值更大的那一个 + mem[i][c] = max(no, yes) + return mem[i][c] + + +def knapsack_dp(wgt, val, cap): + """0-1 背包:动态规划""" + n = len(wgt) + # 初始化 dp 列表 + dp = [[0] * (cap + 1) for _ in range(n + 1)] + # 状态转移 + for i in range(1, n + 1): + for c in range(1, cap + 1): + if wgt[i - 1] > c: + # 若超过背包容量,则不选物品 i + dp[i][c] = dp[i - 1][c] + else: + # 不选和选物品 i 这两种方案的较大值 + dp[i][c] = max(dp[i - 1][c - wgt[i - 1]] + val[i - 1], dp[i - 1][c]) + return dp[n][cap] + + +def knapsack_dp_comp(wgt, val, cap): + """0-1 背包:状态压缩后的动态规划""" + n = len(wgt) + # 初始化 dp 列表 + dp = [0] * (cap + 1) + # 状态转移 + for i in range(1, n + 1): + # 倒序遍历 + for c in range(cap, 0, -1): + if wgt[i - 1] > c: + # 若超过背包容量,则不选物品 i + dp[c] = dp[c] + else: + # 不选和选物品 i 这两种方案的较大值 + dp[c] = max(dp[c - wgt[i - 1]] + val[i - 1], dp[c]) + return dp[cap] + + +"""Driver Code""" +if __name__ == "__main__": + wgt = [10, 20, 30, 40, 50] + val = [60, 100, 120, 160, 200] + cap = 50 + n = len(wgt) + + # 暴力搜索 + res = knapsack_dfs(wgt, val, n, cap) + print(res) + + # 记忆化搜索 + mem = [[-1] * (cap + 1) for _ in range(n + 1)] + res = knapsack_dfs_mem(wgt, val, mem, n, cap) + print(res) + + # 动态规划 + res = knapsack_dp(wgt, val, cap) + print(res) + + # 状态压缩后的动态规划 + res = knapsack_dp_comp(wgt, val, cap) + print(res) diff --git a/docs/chapter_dynamic_programming/knapsack_problem.assets/knapsack_dfs.png b/docs/chapter_dynamic_programming/knapsack_problem.assets/knapsack_dfs.png new file mode 100644 index 0000000000000000000000000000000000000000..fa1e10926d5bb728c368c9073fe8dea3ce4b95cc GIT binary patch literal 105862 zcmd43by!qw_XawHql7flAPv$;cY{cGcXxLqBHc*0ba$t858W-@o#GjM_5FV5`>yl< z88+A6%+6;$EAF+{y$P0;{(u070|x*A5I%|t%L4!}A)8QO7%0e!pFkW4Spgm7KL`TK zM)CF_e|$Dl{b(#D1)zd#!vJ0aQ2{TW4}tsufH=QTem+GUi2LuhJn-G0V<6WIGy{Ns z9}fViL)On9UC8hAIsdF*WCH(l#Z0KbN52GQzWBQh1%@0A@DUd9gsfm~#nc=C0L1R+ zU*K12L|n*o13n51D7pd<)1Vvhdy?{OMg=%_#E=DuVPKI3QG-$~U%c96VBP$9pyuKH zJ#T(Kt>QpyER2Uq*Q1x!*wP-67>+20ABL1P8)^{6Xc=#;?ZYpnLK~M1hp`a~&!4+y 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