algo/python/40_dynamic_programming/knapsack.py
2018-12-26 13:26:22 +00:00

31 lines
1.0 KiB
Python

"""
Author: Wenru Dong
"""
from typing import List
def knapsack01(weights: List[int], values: List[int], capacity: int) -> int:
# Denote the state as (i, c), where i is the stage number,
# and c is the capacity available. Denote f(i, c) to be the
# maximum value when the capacity available is c, and Item 0
# to Item i-1 are to be packed.
# The goal is to find f(n-1, W), where W is the total capacity.
# Then the DP functional equation is:
# f(i, c) = max(xᵢvᵢ + f(i-1, c-xᵢwᵢ)), xᵢ ∈ D, i ≥ 0,
# f(-1, c) = 0, 0 ≤ c ≤ W,
# where
# / {0}, if wᵢ > c
# D = D(i, c) =
# \ {0, 1}, if wᵢ ≤ c
prev = [0] * (capacity + 1)
for w, v in zip(weights, values):
prev = [c >= w and max(prev[c], prev[c-w] + v) for c in range(capacity + 1)]
return prev[-1]
if __name__ == "__main__":
# To find the maximum weight that can be packed,
# set values equal to the weights
print(knapsack01([2, 2, 4, 6, 3], [2, 2, 4, 6, 3], 9))