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99 lines
2.9 KiB
Python
99 lines
2.9 KiB
Python
"""
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File: knapsack.py
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Created Time: 2023-07-03
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Author: Krahets (krahets@163.com)
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"""
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def knapsack_dfs(wgt, val, i, c):
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"""0-1 背包：暴力搜索"""
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# 若已选完所有物品或背包无容量，则返回价值 0
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if i == 0 or c == 0:
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return 0
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# 若超过背包容量，则只能不放入背包
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if wgt[i - 1] > c:
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return knapsack_dfs(wgt, val, i - 1, c)
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# 计算不放入和放入物品 i 的最大价值
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no = knapsack_dfs(wgt, val, i - 1, c)
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yes = knapsack_dfs(wgt, val, i - 1, c - wgt[i - 1]) + val[i - 1]
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# 返回两种方案中价值更大的那一个
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return max(no, yes)
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def knapsack_dfs_mem(wgt, val, mem, i, c):
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"""0-1 背包：记忆化搜索"""
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# 若已选完所有物品或背包无容量，则返回价值 0
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if i == 0 or c == 0:
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return 0
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# 若已有记录，则直接返回
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if mem[i][c] != -1:
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return mem[i][c]
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# 若超过背包容量，则只能不放入背包
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if wgt[i - 1] > c:
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return knapsack_dfs_mem(wgt, val, mem, i - 1, c)
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# 计算不放入和放入物品 i 的最大价值
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no = knapsack_dfs_mem(wgt, val, mem, i - 1, c)
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yes = knapsack_dfs_mem(wgt, val, mem, i - 1, c - wgt[i - 1]) + val[i - 1]
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# 记录并返回两种方案中价值更大的那一个
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mem[i][c] = max(no, yes)
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return mem[i][c]
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def knapsack_dp(wgt, val, cap):
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"""0-1 背包：动态规划"""
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n = len(wgt)
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# 初始化 dp 表
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dp = [[0] * (cap + 1) for _ in range(n + 1)]
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# 状态转移
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for i in range(1, n + 1):
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for c in range(1, cap + 1):
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if wgt[i - 1] > c:
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# 若超过背包容量，则不选物品 i
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dp[i][c] = dp[i - 1][c]
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else:
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# 不选和选物品 i 这两种方案的较大值
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dp[i][c] = max(dp[i - 1][c - wgt[i - 1]] + val[i - 1], dp[i - 1][c])
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return dp[n][cap]
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def knapsack_dp_comp(wgt, val, cap):
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"""0-1 背包：状态压缩后的动态规划"""
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n = len(wgt)
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# 初始化 dp 表
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dp = [0] * (cap + 1)
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# 状态转移
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for i in range(1, n + 1):
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# 倒序遍历
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for c in range(cap, 0, -1):
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if wgt[i - 1] > c:
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# 若超过背包容量，则不选物品 i
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dp[c] = dp[c]
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else:
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# 不选和选物品 i 这两种方案的较大值
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dp[c] = max(dp[c - wgt[i - 1]] + val[i - 1], dp[c])
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return dp[cap]
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"""Driver Code"""
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if __name__ == "__main__":
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wgt = [10, 20, 30, 40, 50]
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val = [60, 100, 120, 160, 200]
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cap = 50
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n = len(wgt)
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# 暴力搜索
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res = knapsack_dfs(wgt, val, n, cap)
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print(res)
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# 记忆化搜索
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mem = [[-1] * (cap + 1) for _ in range(n + 1)]
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res = knapsack_dfs_mem(wgt, val, mem, n, cap)
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print(res)
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# 动态规划
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res = knapsack_dp(wgt, val, cap)
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print(res)
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# 状态压缩后的动态规划
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res = knapsack_dp_comp(wgt, val, cap)
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print(res)
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