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* Retranslate the complete English edition from Simplified Chinese * Polish three English explanations after reader-focused review
102 lines
3.6 KiB
Python
102 lines
3.6 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 knapsack: brute-force search"""
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# If all items have been considered or no capacity remains, return value 0
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if i == 0 or c == 0:
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return 0
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# If the item exceeds the remaining capacity, it must be excluded
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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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# Calculate the maximum values obtained by excluding and including item i
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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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# Return the larger value from the two choices
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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 knapsack: memoized search"""
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# If all items have been considered or no capacity remains, return value 0
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if i == 0 or c == 0:
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return 0
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# If a result has already been recorded, return it directly
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if mem[i][c] != -1:
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return mem[i][c]
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# If the item exceeds the remaining capacity, it must be excluded
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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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# Calculate the maximum values obtained by excluding and including item i
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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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# Record and return the larger value from the two choices
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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 knapsack: dynamic programming"""
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n = len(wgt)
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# Initialize the dp table
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dp = [[0] * (cap + 1) for _ in range(n + 1)]
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# State transitions
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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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# If item i exceeds the remaining capacity, do not choose it
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dp[i][c] = dp[i - 1][c]
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else:
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# Take the larger value obtained by excluding and including item i
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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 knapsack: space-optimized dynamic programming"""
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n = len(wgt)
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# Initialize the dp table
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dp = [0] * (cap + 1)
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# State transitions
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for i in range(1, n + 1):
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# Traverse in decreasing order
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for c in range(cap, 0, -1):
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if wgt[i - 1] > c:
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# If item i exceeds the remaining capacity, do not choose it
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dp[c] = dp[c]
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else:
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# Take the larger value obtained by excluding and including item i
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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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# Brute-force search
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res = knapsack_dfs(wgt, val, n, cap)
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print(f"Maximum total value within the knapsack capacity: {res}")
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# Memoized search
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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"Maximum total value within the knapsack capacity: {res}")
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# Dynamic programming
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res = knapsack_dp(wgt, val, cap)
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print(f"Maximum total value within the knapsack capacity: {res}")
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# Space-optimized dynamic programming
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res = knapsack_dp_comp(wgt, val, cap)
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print(f"Maximum total value within the knapsack capacity: {res}")
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