Skip to content
Related Articles
Sum of minimum element of all sub-sequences of a sorted array
• Difficulty Level : Medium
• Last Updated : 29 Apr, 2021

Given a sorted array A of n integers. The task is to find the sum of the minimum of all possible subsequences of A.
Note: Considering there will be no overflow of numbers.

Examples:

Input: A = [1, 2, 4, 5]
Output: 29
Subsequences are [1], [2], [4], [5], [1, 2], [1, 4], [1, 5], [2, 4], [2, 5], [4, 5] [1, 2, 4], [1, 2, 5], [1, 4, 5], [2, 4, 5], [1, 2, 4, 5]
Minimums are 1, 2, 4, 5, 1, 1, 1, 2, 2, 4, 1, 1, 1, 2, 1.
Sum is 29
Input: A = [1, 2, 3]
Output: 11

Approach: The Naive approach is to generate all possible subsequences, find their minimum and add them to the result.
Efficient Approach: It is given that the array is sorted, so observe that the minimum element occurs 2n-1 times, the second minimum occurs 2n-2 times, and so on… Let’s take an example:

arr[] = {1, 2, 3}
Subsequences are {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}
Minimum of each subsequence: {1}, {2}, {3}, {1}, {1}, {2}, {1}.
where
1 occurs 4 times i.e. 2 n-1 where n = 3.
2 occurs 2 times i.e. 2n-2 where n = 3.
3 occurs 1 times i.e. 2n-3 where n = 3.

So, traverse the array and add current element i.e. arr[i]* pow(2, n-1-i) to the sum.
Below is the implementation of the above approach:

## C++

 `// C++ implementation of the above approach``#include ``using` `namespace` `std;` `// Function to find the sum``// of minimum of all subsequence``int` `findMinSum(``int` `arr[], ``int` `n)``{` `    ``int` `occ = n - 1, sum = 0;``    ``for` `(``int` `i = 0; i < n; i++) {``        ``sum += arr[i] * ``pow``(2, occ);``        ``occ--;``    ``}` `    ``return` `sum;``}` `// Driver code``int` `main()``{``    ``int` `arr[] = { 1, 2, 4, 5 };``    ``int` `n = ``sizeof``(arr) / ``sizeof``(arr[0]);` `    ``cout << findMinSum(arr, n);` `    ``return` `0;``}`

## Java

 `// Java implementation of the above approach``class` `GfG``{` `// Function to find the sum``// of minimum of all subsequence``static` `int` `findMinSum(``int` `arr[], ``int` `n)``{` `    ``int` `occ = n - ``1``, sum = ``0``;``    ``for` `(``int` `i = ``0``; i < n; i++)``    ``{``        ``sum += arr[i] * (``int``)Math.pow(``2``, occ);``        ``occ--;``    ``}` `    ``return` `sum;``}` `// Driver code``public` `static` `void` `main(String[] args)``{``    ``int` `arr[] = { ``1``, ``2``, ``4``, ``5` `};``    ``int` `n = arr.length;` `    ``System.out.println(findMinSum(arr, n));``}``}` `// This code is contributed by Prerna Saini`

## Python3

 `# Python3 implementation of the``# above approach` `# Function to find the sum``# of minimum of all subsequence``def` `findMinSum(arr, n):` `    ``occ ``=` `n ``-` `1``    ``Sum` `=` `0``    ``for` `i ``in` `range``(n):``        ``Sum` `+``=` `arr[i] ``*` `pow``(``2``, occ)``        ``occ ``-``=` `1``    ` `    ``return` `Sum` `# Driver code``arr ``=` `[``1``, ``2``, ``4``, ``5``]``n ``=` `len``(arr)` `print``(findMinSum(arr, n))` `# This code is contributed``# by mohit kumar`

## C#

 `// C# implementation of the above approach``using` `System;` `class` `GFG``{``    ` `// Function to find the sum``// of minimum of all subsequence``static` `int` `findMinSum(``int` `[]arr, ``int` `n)``{` `    ``int` `occ = n - 1, sum = 0;``    ``for` `(``int` `i = 0; i < n; i++)``    ``{``        ``sum += arr[i] *(``int``) Math.Pow(2, occ);``        ``occ--;``    ``}` `    ``return` `sum;``}` `// Driver code``public` `static` `void` `Main(String []args)``{``    ``int` `[]arr = { 1, 2, 4, 5 };``    ``int` `n = arr.Length;` `    ``Console.WriteLine( findMinSum(arr, n));``}``}``// This code is contributed by Arnab Kundu`

## PHP

 ``

## Javascript

 ``
Output:
`29`

Note: To find the Sum of maximum element of all subsequences 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 mathematical concepts for competitive programming with the Essential Maths for CP Course at a student-friendly price. To complete your preparation from learning a language to DS Algo and many more,  please refer Complete Interview Preparation Course.

My Personal Notes arrow_drop_up