DP E - NISHIO Hirokazu's Scrapbox (Auto-translated from Japanese)

DP E

Maximize value below a specified weight limit backpack Issue

But the space of weight limit is too large to make DP with weight as the defining region.

Exchange of value range and definition range, define a value range, and do a DP where the value is the minimum weight that achieves that value.

from dynamic programming

DP_E

Problem statement E - Knapsack 2

Exchange of value range and definition range

The space of weights is 10^11

The space of values is 10^5

In DP_D, a function of value was created using weight as the domain of definition, and the maximum value was obtained

That approach is too broad a definitional area for this issue.

So, conversely, we create a function of weight with value as the domain of definition.

Then find the largest subscript in the domain whose weight satisfies the constraint

https://gyazo.com/8f6820c6fa358d709bd62c806b3166fb

code:python

def solve(N, W, WV):

MAX_VALUE = N * 10 ** 3

weights = INF * (MAX_VALUE + 1)

weights0 = 0

for i in range(N):

next_weights = weights:

weight, value = WVi

for j in range(MAX_VALUE - value + 1):

next_weightsj + value = min(

weightsj + value,

weightsj + weight)

weights = next_weights

for i in range(MAX_VALUE, -1, -1):

if weightsi <= W:

return i

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