Sum of maximum elements of all possible sub-arrays of an array

Given an array arr[], the task is to find the sum of the maximum elements of every possible sub-array of the array.

Examples:

Input: arr[] = {1, 3, 2}
Output: 15
All possible sub-arrays are {1}, {2}, {3}, {1, 3}, {3, 2} and {1, 3, 2}
And, the sum of all the maximum elements is 1 + 2 + 3 + 3 + 3 + 3 = 15

Input: arr[] = {3, 1}
Output: 7

A simple method is to generate all the sub-arrays and then sum the maximum elements in all of them. The time complexity of this solution will be O(n3).

Better method:
The key for optimization is the question-

In how many segments, the value at an index will be maximum?

The next idea that might come into our mind will be for every index i in the array arr, we try to find:
Left count: We iterate towards left of the index i till we don’t encounter an element strictly greater than arr[i] or we don’t reach the left end of the array. Let us call this count for the index i of the given array as CLi.
Right count: We iterate towards right of the index till we don’t encounter an element greater than or equal to the value at the index or we don’t reach the right end. Let us call this count for the index i of the given array as CRi.

(CLi + 1) * (CRi + 1) will be the number of sub-arrays for the current index i in which its value will be maximum because there are CLi + 1 ways to choose elements from left side (including choosing no element) and CRi + 1 ways to choose elements from the right side.

The time complexity of this approach will be O(n2)

Best method:
This problem can be solved using stack data structure in O(n) time. The idea remains the same as is in the previous approach. For the sake of saving some time, we will use stack from the Standard Template Library of C++.

Left count: Let CLi represent the left count for an index i. CLi for an index i can be defined as the number of elements between the index i and the right most element whose value is strictly greater than arr[i] having index less than i. If, there is no such element, then CLi for an element will be equal to the number of elements to the left of the index i.

To achieve this, we will insert only the index of the elements from left to right into the stack. Let us suppose, we are inserting an index i in the stack and j be the index of the topmost element currently present in the stack. While the value arr[i] is greater than or equal to the value at the top most index in the stack and the stack is not empty, keep popping the elements in the stack. Whenever, an element is popped, the left count(CLi) of the current index(i) is updated as CLi = CLi + CLj + 1.

Right count: We calculate the right count for all the indexes in the similar way. The only difference is we push the elements in stack while traversing right to left in the array. While arr[i] is strictly greater than the value at the top most index in the stack and the stack is not empty, keep popping the elements. Whenever, an element is popped, the right count of the current index(i) is updated as CRi = CRi + CRj + 1.

Final step: Let ans be the variable containing the final answer. We will initialize it with 0. Then, we will iterate through all the indexes from 1 to n of the array and update the ans as ans = ans + (CLi + 1) * (CRi + 1) * arr[i] for all possible values of i from 1 to n.

Below is the implementation of the above approach:

C++

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// C++ implementation of the approach
#include <iostream>
#include <stack>
using namespace std;
  
// Function to return the required sum
int findMaxSum(int arr[], int n)
{
    // Arrays for maintaining left and right count
    int CL[n] = { 0 }, CR[n] = { 0 };
  
    // Stack for storing the indexes
    stack<int> q;
  
    // Calculate left count for every index
    for (int i = 0; i < n; i++) {
        while (q.size() != 0 && arr[q.top()] <= arr[i]) {
            CL[i] += CL[q.top()] + 1;
            q.pop();
        }
        q.push(i);
    }
  
    // Clear the stack
    while (q.size() != 0)
        q.pop();
  
    // Calculate right count for every index
    for (int i = n - 1; i >= 0; i--) {
        while (q.size() != 0 && arr[q.top()] < arr[i]) {
            CR[i] += CR[q.top()] + 1;
            q.pop();
        }
        q.push(i);
    }
  
    // Clear the stack
    while (q.size() != 0)
        q.pop();
  
    // To store the required sum
    int ans = 0;
  
    // Calculate the final sum
    for (int i = 0; i < n; i++)
        ans += (CL[i] + 1) * (CR[i] + 1) * arr[i];
  
    return ans;
}
  
// Driver code
int main()
{
    int arr[] = { 1, 3, 2 };
    int n = sizeof(arr) / sizeof(arr[0]);
    cout << findMaxSum(arr, n);
}

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Python3

# Python3 implementation of the approach

# Function to return the required sum
def findMinSum(arr, n):

# Arrays for maintaining left
# and right count
CL = [0] * n
CR = [0] * n

# Stack for storing the indexes
q = []

# Calculate left count for every index
for i in range(0, n):
while (len(q) != 0 and
arr[q[-1]] <= arr[i]): CL[i] += CL[q[-1]] + 1 q.pop() q.append(i) # Clear the stack while len(q) != 0: q.pop() # Calculate right count for every index for i in range(n - 1, -1, -1): while (len(q) != 0 and arr[q[-1]] < arr[i]): CR[i] += CR[q[-1]] + 1 q.pop() q.append(i) # Clear the stack while len(q) != 0: q.pop() # To store the required sum ans = 0 # Calculate the final sum for i in range(0, n): ans += (CL[i] + 1) * (CR[i] + 1) * arr[i] return ans # Driver code if __name__ == "__main__": arr = [1, 3, 2] n = len(arr) print(findMinSum(arr, n)) # This code is contributed by Rituraj Jain [tabbyending]

Output:

15

Time complexity: O(n)



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