# 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:

 // C++ implementation to find the // maximum sum of K subarrays // when shuffling is allowed   #include  using namespace std;   // Function to find the // maximum sum of all subarrays int maximumSubarraySum(     int a[], int n,     vector >& 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 > 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); }

Output:
49



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