48 lines
1.3 KiB
Python
48 lines
1.3 KiB
Python
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"""
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0/1 背包问题 — 动态规划经典问题
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给定容量 W 和价值/重量数组，求最大价值
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"""
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def knapsack_01(weights, values, W):
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"""0/1背包 DP 解法"""
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n = len(weights)
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# dp[i][w] 表示前 i 个物品在容量 w 下的最大价值
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dp = [[0] * (W + 1) for _ in range(n + 1)]
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for i in range(1, n + 1):
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for w in range(1, W + 1):
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if weights[i - 1] <= w:
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# 选或不选当前物品
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dp[i][w] = max(
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dp[i - 1][w],
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dp[i - 1][w - weights[i - 1]] + values[i - 1]
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)
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else:
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dp[i][w] = dp[i - 1][w]
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return dp[n][W]
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def knapsack_01_optimized(weights, values, W):
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"""空间优化版：一维数组"""
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dp = [0] * (W + 1)
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for i in range(len(weights)):
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for w in range(W, weights[i] - 1, -1):
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dp[w] = max(dp[w], dp[w - weights[i]] + values[i])
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return dp[W]
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if __name__ == "__main__":
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weights = [2, 3, 4, 5]
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values = [3, 4, 5, 6]
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W = 8
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result1 = knapsack_01(weights, values, W)
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result2 = knapsack_01_optimized(weights, values, W)
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print(f"物品重量: {weights}")
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print(f"物品价值: {values}")
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print(f"背包容量: {W}")
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print(f"最大价值 (二维DP): {result1}")
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print(f"最大价值 (一维优化): {result2}")
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