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102 lines
3.3 KiB
102 lines
3.3 KiB
8 months ago
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"""
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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: list[int], val: list[int], i: int, c: int) -> int:
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"""0-1 背包：暴力搜尋"""
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# 若已選完所有物品或背包無剩餘容量，則返回價值 0
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if i == 0 or c == 0:
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return 0
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# 若超過背包容量，則只能選擇不放入背包
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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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# 計算不放入和放入物品 i 的最大價值
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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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# 返回兩種方案中價值更大的那一個
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return max(no, yes)
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def knapsack_dfs_mem(
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wgt: list[int], val: list[int], mem: list[list[int]], i: int, c: int
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) -> int:
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"""0-1 背包：記憶化搜尋"""
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# 若已選完所有物品或背包無剩餘容量，則返回價值 0
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if i == 0 or c == 0:
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return 0
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# 若已有記錄，則直接返回
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if mem[i][c] != -1:
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return mem[i][c]
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# 若超過背包容量，則只能選擇不放入背包
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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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# 計算不放入和放入物品 i 的最大價值
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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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# 記錄並返回兩種方案中價值更大的那一個
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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: list[int], val: list[int], cap: int) -> int:
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"""0-1 背包：動態規劃"""
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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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# 狀態轉移
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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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# 若超過背包容量，則不選物品 i
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dp[i][c] = dp[i - 1][c]
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else:
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# 不選和選物品 i 這兩種方案的較大值
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dp[i][c] = max(dp[i - 1][c], dp[i - 1][c - wgt[i - 1]] + val[i - 1])
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return dp[n][cap]
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def knapsack_dp_comp(wgt: list[int], val: list[int], cap: int) -> int:
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"""0-1 背包：空間最佳化後的動態規劃"""
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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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# 狀態轉移
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for i in range(1, n + 1):
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# 倒序走訪
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for c in range(cap, 0, -1):
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if wgt[i - 1] > c:
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# 若超過背包容量，則不選物品 i
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dp[c] = dp[c]
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else:
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# 不選和選物品 i 這兩種方案的較大值
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dp[c] = max(dp[c], dp[c - wgt[i - 1]] + val[i - 1])
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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 = [50, 120, 150, 210, 240]
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cap = 50
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n = len(wgt)
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# 暴力搜尋
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res = knapsack_dfs(wgt, val, n, cap)
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print(f"不超過背包容量的最大物品價值為 {res}")
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# 記憶化搜尋
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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(f"不超過背包容量的最大物品價值為 {res}")
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# 動態規劃
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res = knapsack_dp(wgt, val, cap)
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print(f"不超過背包容量的最大物品價值為 {res}")
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# 空間最佳化後的動態規劃
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res = knapsack_dp_comp(wgt, val, cap)
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print(f"不超過背包容量的最大物品價值為 {res}")
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