[UCSD白板题] Fractional Knapsack

Problem Introduction

Given a set of items and total capacity of a knapsack,find the maximal value of fractions of items that fit into the knapsack.

Problem Description

Task.The goal of this code problem is to implement an algorithm for the fractional knapsack problem.

Input Format.The first line of the input contains the number \(n\) of items and the capacity \(W\) of a knapsack.The next \(n\) lines define the values and weights of the items. The \(i\)-th line contain integers \(v_i\) and \(w_i\)—the value and the weight of \(i\)-th item,respectively.

Constraints.\(1 \leq n \leq 10^3, 0 \leq W \leq 2 \cdot 10^6; 0 \leq v_i \leq 2 \cdot 10^6, 0 < w_i \leq 2 \cdot 10^6\) for all \(1 \leq i \leq n.\) All the numbers are integers.

Output Format.Output the maximal value of fractions of items that fit into the knapsack.The absolution value of the difference between the answer with at least four digits after the decimal point(otherwise your answer,while being computed correctly,can turn out to be wrong because of rounding issues).

Sample 1.
Input:

3 50
60 20
100 50
120 30

Output:

180.0000

Sample 2.
Input:

1 10
500 30

Output:

166.6667

Solution

# Uses python3
import sys
import numpy as np

def get_optimal_value(capacity, weights, values):
    value = 0.
    indices = np.argsort([-v/w for w,v in zip(weights,values)])
    for idx in indices:
        if capacity <= 0:
            break
        weight = min(capacity, weights[idx])
        capacity -= weight
        value +=  weight * (values[idx] / weights[idx])
    return value

if __name__ == "__main__":
    data = list(map(int, sys.stdin.read().split()))
    n, capacity = data[0:2]
    values = data[2:(2 * n + 2):2]
    weights = data[3:(2 * n + 2):2]
    opt_value = get_optimal_value(capacity, weights, values)
    print("{:.10f}".format(opt_value))