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https://github.com/krahets/hello-algo.git
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e720aa2d24
* Sync recent changes to the revised Word. * Revised the preface chapter * Revised the introduction chapter * Revised the computation complexity chapter * Revised the chapter data structure * Revised the chapter array and linked list * Revised the chapter stack and queue * Revised the chapter hashing * Revised the chapter tree * Revised the chapter heap * Revised the chapter graph * Revised the chapter searching * Reivised the sorting chapter * Revised the divide and conquer chapter * Revised the chapter backtacking * Revised the DP chapter * Revised the greedy chapter * Revised the appendix chapter * Revised the preface chapter doubly * Revised the figures
101 lines
3.3 KiB
Python
101 lines
3.3 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: list[int], val: list[int], i: int, c: int) -> int:
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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剩余容量，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，则只能选择不放入背包
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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(
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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：记忆化搜索"""
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# 若已选完所有物品或背包无剩余容量，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，则只能选择不放入背包
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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: list[int], val: list[int], cap: int) -> int:
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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], 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：空间优化后的动态规划"""
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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], 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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# 暴力搜索
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res = knapsack_dfs(wgt, val, n, cap)
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print(f"不超过背包容量的最大物品价值为 {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(f"不超过背包容量的最大物品价值为 {res}")
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# 动态规划
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res = knapsack_dp(wgt, val, cap)
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print(f"不超过背包容量的最大物品价值为 {res}")
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# 空间优化后的动态规划
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res = knapsack_dp_comp(wgt, val, cap)
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print(f"不超过背包容量的最大物品价值为 {res}")
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