Fractional Knapsack Queries

Given an integer array, consisting of positive weights “W” and there values “V” respectively as a pair and some queries consisting of an integer ‘C’ specifying the capacity of the knapsack, find the maximum value of products that can be put in the knapsack if the breaking of items is allowed.

Examples:

Input: arr[] = { {1, 2}, {1, 3}, {3, 7} }, q = {1, 2, 3, 4, 5}
Output: {3, 5.33333, 7.66667, 10, 12}
For ‘C’ = 1, we will fill the knap-sack with element of value 3.
For ‘C’ = 2, first, we fill with element of value 3, then remaining 1 capacity
with element with value 7.
For ‘C’ = 3, first we fill with element of value 3, then remaining 2 capacities
with element with value 7.
For ‘C’ = 4, first we fill with element of value 3, then remaining 3 capacities
with element with value 7.
For ‘C’ = 5, first we fill with element of value 3, next 3 capacities
with element with value 7 and remaining 1 with element of value 1.

Its recommended you go through this article on Fractional knapsack before going through this article.

Naive Approach: A simple approach will be to sort the array in decreasing order of V/W values i.e. there value/weight ratio. Then, for each query, we will iterate the array sum up the required integer values while the knap-sack isn’t completely full.
Time Complexity : O(q*n*log(n))

Efficient approach: Queries can be optimized by performing prefix_sum of a sorted array on both weight and value.
Below is the algorithm:

  1. Sort the array by there Value/Weight ratio in descending order.
  2. Find prefix-sum of Weight and Value respectively of the array.
    • For answering a query:

    • Perform a binary search to find the first element with prefix_sum on weight(W) larger than ‘C’. Strictly speaking, find upper bound on the value of ‘C’ in a prefix_sum array of ‘W’. Let’s say this element is at an index ‘X’.
    • Include sum of values from index {0, X-1} and the fractional value from index ‘X’ that can be accommodated in remaining capacity.

Below is the implementation of the above approach:

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// C++ program to implement above approach
#include <bits/stdc++.h>
using namespace std;
  
// Function on the basis of which we will
// perform the sorting
bool comp(pair<int, int> a, pair<int, int> b)
{
    return (double)a.second / a.first >
                   (double)b.second / b.first;
}
  
// Function to sort the array on its value/weight
// and perform-sum on both weight and value
void preProcess(pair<int, int> arr[], int n)
{
    sort(arr, arr + n, comp);
    for (int i = 1; i < n; i++) {
        arr[i].first += arr[i - 1].first;
        arr[i].second += arr[i - 1].second;
    }
}
  
// Function to answer queries
double maxValue(int w, pair<int, int> arr[], int n)
{
    // If w is large enough
    // to cover all weights
    if (arr[n - 1].first <= w)
        return arr[n - 1].second;
  
    // Value to search on arr
    pair<int, int> search_bound = { w, INT_MAX };
  
    // Index of the item which we we will put
    // partially in our knap-sack
    int x = upper_bound(arr, arr + n, search_bound) - arr;
  
    // Required value
    if (x == 0)
        return (double)w * arr[0].second / arr[0].first;
    else
        return arr[x - 1].second +
              (double)(w - arr[x - 1].first) * 
              (arr[x].second - arr[x - 1].second) /
              (arr[x].first - arr[x - 1].first);
}
  
void PrintQueries(int query[], pair<int, int> arr[],
                                        int n, int m)
{
    for (int i = 0; i < m; i += 1) {
        cout << maxValue(query[i], arr, n) << endl;
    }
}
  
// Driver code
int main()
{
    // Input array representing the data of w[] and v[]
    pair<int, int> arr[] = { { 1, 2 }, { 1, 3 }, { 3, 7 } };
    int query[5] = { 1, 2, 3, 4, 5 };
  
    // Size of the array
    int n = sizeof(arr) / sizeof(pair<int, int>);
    int m = sizeof(query) / sizeof(query[0]);
  
    // Pre-processing
    preProcess(arr, n);
  
    PrintQueries(query, arr, n, m);
  
    return 0;
}

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Output:

3
5.33333
7.66667
10
12

Time Complexity : O((n+q)*log(n))



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