Sum of minimum element of all subarrays of a sorted array

Given a sorted array A of n integers. The task is to find the sum of minimum of all possible subarrays of A.

Examples:

Input: A = [ 1, 2, 4, 5]
Output: 23
Subsequences are [1], [2], [4], [5], [1, 2], [2, 4], [4, 5] [1, 2, 4], [2, 4, 5], [1, 2, 4, 5]
Minimums are 1, 2, 4, 5, 1, 2, 4, 1, 2, 1.
Sum is 23



Input: A = [1, 2, 3]
Output: 10

Approach: The Naive approach is to generate all possible subarrays, find their minimum and add them to result.

Efficient Approach: It is given that the array is sorted, so observe that minimum element occurs N times, the second minimum occurs N-1 times and so on… Let’s take an example:

arr[] = {1, 2, 3}
Subarrays are {1}, {2}, {3}, {1, 2}, {2, 3}, {1, 2, 3}
Minimum of each subarray: {1}, {2}, {3}, {1}, {2}, {1}.
where
1 occurs 3 times i.e. n times where n = 3.
2 occurs 2 times i.e. n-1 times where n = 3.
3 occurs 1 times i.e. n-2 times where n = 3.

So, traverse the array and add current element i.e. (arr[i]* n-i) to the sum.

Below is the implementation of the above approach:

C++

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// C++ implementation of the above approach
#include <bits/stdc++.h>
using namespace std;
  
// Function to find the sum
// of minimum of all subarrays
int findMinSum(int arr[], int n)
{
  
    int sum = 0;
    for (int i = 0; i < n; i++)
        sum += arr[i] * (n - i);
  
    return sum;
}
  
// Driver code
int main()
{
    int arr[] = { 3, 5, 7, 8 };
    int n = sizeof(arr) / sizeof(arr[0]);
  
    cout << findMinSum(arr, n);
  
    return 0;
}

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Java

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// Java implementation of the above approach 
class GfG 
{
  
// Function to find the sum 
// of minimum of all subarrays 
static int findMinSum(int arr[], int n) 
  
    int sum = 0
    for (int i = 0; i < n; i++) 
        sum += arr[i] * (n - i); 
  
    return sum; 
  
// Driver code 
public static void main(String[] args) 
    int arr[] = { 3, 5, 7, 8 }; 
    int n = arr.length; 
  
    System.out.println(findMinSum(arr, n)); 
}
  
// This code is contributed by Prerna Saini

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Python3

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# Python3 implementation of the 
# above approach 
  
# Function to find the sum 
# of minimum of all subarrays 
def findMinSum(arr, n):
    sum = 0
    for i in range(0, n): 
        sum += arr[i] * (n - i) 
    return sum
  
# Driver code 
arr = [3, 5, 7, 8
n = len(arr)
  
print(findMinSum(arr, n)) 
  
# This code has been contributed 
# by 29AjayKumar

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C#

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// C# implementation of the above approach 
using System;
  
class GfG 
  
// Function to find the sum 
// of minimum of all subarrays 
static int findMinSum(int []arr, int n) 
  
    int sum = 0; 
    for (int i = 0; i < n; i++) 
        sum += arr[i] * (n - i); 
  
    return sum; 
  
// Driver code 
public static void Main(String []args) 
    int []arr = { 3, 5, 7, 8 }; 
    int n = arr.Length; 
  
    Console.WriteLine(findMinSum(arr, n)); 
  
// This code is contributed by Arnab Kundu

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PHP

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<?php
  
// PHP implementation of the above approach 
// Function to find the sum 
// of minimum of all subarrays 
function findMinSum($arr,$n
  
    $sum = 0; 
    for ($i = 0; $i < $n; $i++) 
        $sum += $arr[$i] * ($n - $i); 
  
    return $sum
  
// Driver code 
$arr = array( 3, 5, 7, 8 ); 
$n = count($arr); 
  
echo findMinSum($arr, $n); 
      
// This code is contributed by Arnab Kundu
?>

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Output:

49

Note: To find the Sum of maximum element of all subarrays in a sorted array, just traverse the array in reverse order and apply the same formula for Sum.



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