Maximum sum of values in a given range of an Array for Q queries when shuffling is allowed

Given an array arr[] and K subarrays in the form (Li, Ri), the task is to find the maximum possible value of 

. It is allowed to shuffle the array before calculating this value to get the maximum sum.
Examples: 

Input: arr[] = {1, 2, 4, 2, 1}, subarrays = {{1, 2}, {2, 4}, {1, 3}, {5, 5}, {3, 5}} 
Output: 26 
Explanation: 
Shuffled Array to get the maximum sum – {2, 4, 2, 1, 1} 
Subarray Sum = arr[1:2] + arr[2:4] + arr[1:3] + arr[5:5] + arr[3:5] 
=> 6 + 7 + 8 + 1 + 4 = 26

Input: arr[] = {4, 1, 2, 1, 9, 2}, subarrays = {{1, 2}, {1, 3}, {1, 4}, {3, 4}} 
Output: 49 
Explanation: 
Shuffled Array to get the maximum sum – {2, 4, 9, 2, 1, 1} 
Subarray Sum = arr[1:2] + arr[1:3] + arr[1:4] + arr[3:4] 
=> 6 + 15 + 17 + 11 = 49 



Naive Approach: A simple solution is to compute the maximum sum of all possible permutations of the given array and check which sequence gives us the maximum summation.

Efficient Approach: The idea is to use Prefix Arrays to find out the frequency of indices over all subarrays. We can do this as follows: 

//Initialize an array 
prefSum[n] = {0, 0, ...0}
for each Li, Ri:
    prefSum[Li]++
    prefSum[Ri+1]--

// Find Prefix sum
for i=1 to N:
    prefSum[i] += prefSum[i-1]

// prefSum contains frequency of 
// each index over all subarrays

Finally, greedily choose the index with the highest frequency and put the largest element of the array at that index. This way we will get the largest possible sum.

Below is the implementation of the above approach: 

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// C++ implementation to find the
// maximum sum of K subarrays
// when shuffling is allowed
 
#include <bits/stdc++.h>
using namespace std;
 
// Function to find the
// maximum sum of all subarrays
int maximumSubarraySum(
    int a[], int n,
    vector<pair<int, int> >& subarrays)
{
    // Initialize maxsum
    // and prefixArray
    int i, maxsum = 0;
    int prefixArray[n] = { 0 };
 
    // Find the frequency
    // using prefix Array
    for (i = 0; i < subarrays.size(); ++i) {
        prefixArray[subarrays[i].first - 1]++;
        prefixArray[subarrays[i].second]--;
    }
 
    // Perform prefix sum
    for (i = 1; i < n; i++) {
        prefixArray[i] += prefixArray[i - 1];
    }
 
    // Sort both arrays to get a greedy result
    sort(prefixArray,
         prefixArray + n,
         greater<int>());
    sort(a, a + n, greater<int>());
 
    // Finally multiply largest frequency with
    // largest array element.
    for (i = 0; i < n; i++)
        maxsum += a[i] * prefixArray[i];
 
    // Return the answer
    return maxsum;
}
 
// Driver Code
int main()
{
    int n = 6;
 
    // Initial Array
    int a[] = { 4, 1, 2, 1, 9, 2 };
 
    // Subarrays
    vector<pair<int, int> > subarrays;
    subarrays.push_back({ 1, 2 });
    subarrays.push_back({ 1, 3 });
    subarrays.push_back({ 1, 4 });
    subarrays.push_back({ 3, 4 });
 
    // Function Call
    cout << maximumSubarraySum(a, n,
                               subarrays);
}
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Output: 
49


 

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