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:

filter_none

edit
close

play_arrow

link
brightness_4
code

// 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;
}
chevron_right

filter_none

edit
close

play_arrow

link
brightness_4
code

// 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
chevron_right

filter_none

edit
close

play_arrow

link
brightness_4
code

# 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
chevron_right

filter_none

edit
close

play_arrow

link
brightness_4
code

// 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
chevron_right

filter_none

edit
close

play_arrow

link
brightness_4
code

<?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
?>
chevron_right

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.

Attention reader! Don’t stop learning now. Get hold of all the important DSA concepts with the DSA Self Paced Course at a student-friendly price and become industry ready.





Check out this Author's contributed articles.

If you like GeeksforGeeks and would like to contribute, you can also write an article using contribute.geeksforgeeks.org or mail your article to contribute@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.

Please Improve this article if you find anything incorrect by clicking on the "Improve Article" button below.



Article Tags :
Practice Tags :