Maximum sum after rearranging the array for K queries

• Difficulty Level : Medium
• Last Updated : 14 May, 2021

Given two arrays arr[] containing N integers and Q[][] containing K queries where every query represents a range [L, R]. The task is to rearrange the array and find the maximum possible sum of all the subarrays where each subarray is defined by the elements of the array in the range [L, R] given by each query.
Note: 1 based indexing is used in the Q[][] array to signify the ranges.
Examples:

Input: arr[] = { 2, 6, 10, 1, 5, 6 }, Q[] = {{1, 3}, {4, 6}, {3, 4}}
Output: 46
Explanation:
One possible way is to rearrange the array to arr[] = {2, 6, 10, 6, 5, 1}.
In this arrangement:
The sum of the subarray in the range [1, 3] = 2 + 6 + 10 = 18.
The sum of the subarray in the range [4, 6] = 6 + 5 + 1 = 12.
The sum of the subarray in the range [3, 4] = 10 + 6 = 16.
The total sum of all the subarrays = 46 which is the maximum possible.
Input: arr[] = { 1, 2, 3, 4, 5, 6, 7, 8 }, Q[] = {{1, 4}, {5, 5}, {7, 8}, {8, 8}}
Output: 43
Explanation:
One possible way is to rearrange the array to arr[] = {2, 3, 4, 5, 6, 1, 7, 8}.
In this arrangement:
The sum of the subarray in the range [1, 4] = 2 + 3 + 4 + 5 = 14.
The sum of the subarray in the range [5, 5] = 6 = 6.
The sum of the subarray in the range [7, 8] = 7 + 8 = 15.
The sum of the subarray in the range [8, 8] = 8 = 8.
The total sum of all the subarrays = 43 which is the maximum possible.

Approach: On observing clearly, one conclusion which can be made is that we get the maximum sum when the maximum elements are included in as many subarrays as possible. For this, we need to find the number of times every index is included by iterating all the queries.
For example: Let the array be arr[] = {2, 6, 10, 6, 5, 1} and the queries be Q[][] = {{1, 3}, {4, 6}, {3, 4}}.

1. Step 1: Create a count array C[] of size N. So, initially, the count array C[] = {0, 0, 0, 0, 0, 0}.
2. Step 2: For the query [1, 3], the elements at the index [1, 3] are incremented by 1. The count array after this query becomes {1, 1, 1, 0, 0, 0}.
3. Step 3: Similarly, for the next query, the count array becomes {1, 1, 1, 1, 1, 1} and finally, after the third query, the count array becomes {1, 1, 2, 2, 1, 1}.
4. Step 4: After obtaining the count array, the idea is to use sorting to get the maximum sum.
5. Step 5: After sorting, the array C[] = {1, 1, 1, 1, 2, 2} and arr[] = {1, 2, 5, 6, 6, 10}. The maximum possible sum is the weighted sum of both the arrays, i.e.:

sum = ((1 * 1) + (1 * 2) + (1 * 5) + (1 * 6) + (2 * 6) + (2 * 10)) = 46

Below is the implementation of the above approach:

C++

 // C++ program to find the maximum sum// after rearranging the array for K queries #include using namespace std; // Function to find maximum sum after// rearranging array elementsint maxSumArrangement(int A[], int R[],                      int N, int M){     // Auxiliary array to find the    // count of each selected elements    int count[N];     // Initialize with 0    memset(count, 0, sizeof count);     // Finding count of every element    // to be selected    for (int i = 0; i < M; ++i) {         int l = R[i], r = R[i] + 1;         // Making it to 0-indexing        l--;        r--;         // Prefix sum array concept is used        // to obtain the count array        count[l]++;         if (r < N)            count[r]--;    }     // Iterating over the count array    // to get the final array    for (int i = 1; i < N; ++i) {        count[i] += count[i - 1];    }for (int i = 0; i < N; ++i) {cout<= 0; --i) {        ans += A[i] * count[i];    }     return ans;} // Driver codeint main(){    int A[] = { 2, 6, 10, 1, 5, 6 };    int R[]        = { { 1, 3 }, { 4, 6 }, { 3, 4 } };     int N = sizeof(A) / sizeof(A);    int M = sizeof(R) / sizeof(R);     cout << maxSumArrangement(A, R, N, M);     return 0;}

Java

 // Java program to find the maximum sum// after rearranging the array for K queriesimport java.util.*; class GFG{          // Function to find maximum sum after    // rearranging array elements    static int maxSumArrangement(int A[], int R[][],                          int N, int M)    {              // Auxiliary array to find the        // count of each selected elements        int count[] = new int[N];        int i;              // Finding count of every element        // to be selected        for ( i = 0; i < M; ++i) {                  int l = R[i], r = R[i] + 1;                  // Making it to 0-indexing            l--;            r--;                  // Prefix sum array concept is used            // to obtain the count array            count[l]++;                  if (r < N)                count[r]--;        }              // Iterating over the count array        // to get the final array        for (i = 1; i < N; ++i) {            count[i] += count[i - 1];        }              // Variable to store the maximum sum        int ans = 0;              // Sorting both the arrays        Arrays.sort( count);        Arrays.sort(A);              // Loop to find the maximum sum        for (i = N - 1; i >= 0; --i) {            ans += A[i] * count[i];        }              return ans;    }          // Driver code    public static void main(String []args)    {        int A[] = { 2, 6, 10, 1, 5, 6 };        int R[][]            = { { 1, 3 }, { 4, 6 }, { 3, 4 } };              int N = A.length;        int M = R.length;              System.out.print(maxSumArrangement(A, R, N, M));          }} // This code is contributed by chitranayal

Python3

 # Python3 program to find the maximum sum# after rearranging the array for K queries # Function to find maximum sum after# rearranging array elementsdef maxSumArrangement( A,  R,  N,  M):     # Auxiliary array to find the    # count of each selected elements    # Initialize with 0    count = [0 for i in range(N)]     # Finding count of every element    # to be selected    for i in range(M):         l = R[i]        r = R[i] + 1         # Making it to 0-indexing        l = l - 1        r = r - 1         # Prefix sum array concept is used        # to obtain the count array        count[l] = count[l] + 1         if (r < N):            count[r] = count[r] - 1     # Iterating over the count array    # to get the final array    for i in range(1, N):        count[i] = count[i] + count[i - 1]     # Variable to store the maximum sum    ans = 0     # Sorting both the arrays    count.sort()    A.sort()     # Loop to find the maximum sum    for i in range(N - 1, -1, -1):        ans = ans + A[i] * count[i]     return ans # Driver codeA = [ 2, 6, 10, 1, 5, 6 ]R = [ [ 1, 3 ], [ 4, 6 ], [ 3, 4 ] ] N = len(A)M = len(R) print(maxSumArrangement(A, R, N, M)) # This code is contributed by Sanjit_Prasad

C#

 // C# program to find the maximum sum// after rearranging the array for K queriesusing System; class GFG{           // Function to find maximum sum after    // rearranging array elements    static int maxSumArrangement(int []A, int [,]R,                          int N, int M)    {               // Auxiliary array to find the        // count of each selected elements        int []count = new int[N];        int i;               // Finding count of every element        // to be selected        for ( i = 0; i < M; ++i) {                   int l = R[i, 0], r = R[i, 1] + 1;                   // Making it to 0-indexing            l--;            r--;                   // Prefix sum array concept is used            // to obtain the count array            count[l]++;                   if (r < N)                count[r]--;        }               // Iterating over the count array        // to get the readonly array        for (i = 1; i < N; ++i) {            count[i] += count[i - 1];        }               // Variable to store the maximum sum        int ans = 0;               // Sorting both the arrays        Array.Sort( count);        Array.Sort(A);               // Loop to find the maximum sum        for (i = N - 1; i >= 0; --i) {            ans += A[i] * count[i];        }               return ans;    }           // Driver code    public static void Main(String []args)    {        int []A = { 2, 6, 10, 1, 5, 6 };        int [,]R            = { { 1, 3 }, { 4, 6 }, { 3, 4 } };               int N = A.Length;        int M = R.GetLength(0);               Console.Write(maxSumArrangement(A, R, N, M));         }} // This code is contributed by Princi Singh

Javascript


Output:
46

Time Complexity: O(N* log(N))

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