 Open in App
Not now

# Sum of subsets of all the subsets of an array | O(2^N)

• Difficulty Level : Medium
• Last Updated : 27 Jan, 2023

Given an array arr[] of length N, the task is to find the overall sum of subsets of all the subsets of the array.
Examples:

Input: arr[] = {1, 1}
Output:
All possible subsets:
a) {} : 0
All the possible subsets of this subset
will be {}, Sum = 0
b) {1} : 1
All the possible subsets of this subset
will be {} and {1}, Sum = 0 + 1 = 1
c) {1} : 1
All the possible subsets of this subset
will be {} and {1}, Sum = 0 + 1 = 1
d) {1, 1} : 4
All the possible subsets of this subset
will be {}, {1}, {1} and {1, 1}, Sum = 0 + 1 + 1 + 2 = 4
Thus, ans = 0 + 1 + 1 + 4 = 6
Input: arr[] = {1, 4, 2, 12}
Output: 513

Approach: In this article, an approach with O(N * 2N) time complexity to solve the given problem will be discussed.
First, generate all the possible subsets of the array. There will be 2N subsets in total. Then for each subset, find the sum of all of its subsets.
For, that it can be observed that in an array of length L, every element will come exactly 2(L – 1) times in the sum of subsets. So, the contribution of each element will be 2(L – 1) times its values.
Below is the implementation of the above approach:

## C++

 `// C++ implementation of the approach``#include ``using` `namespace` `std;` `// Function to sum of all subsets of a``// given array``void` `subsetSum(vector<``int``>& c, ``int``& ans)``{``    ``int` `L = c.size();``    ``int` `mul = (``int``)``pow``(2, L - 1);``    ``for` `(``int` `i = 0; i < c.size(); i++)``        ``ans += c[i] * mul;``}` `// Function to generate the subsets``void` `subsetGen(``int``* arr, ``int` `i, ``int` `n,``               ``int``& ans, vector<``int``>& c)``{``    ``// Base-case``    ``if` `(i == n) {` `        ``// Finding the sum of all the subsets``        ``// of the generated subset``        ``subsetSum(c, ans);``        ``return``;``    ``}` `    ``// Recursively accepting and rejecting``    ``// the current number``    ``subsetGen(arr, i + 1, n, ans, c);``    ``c.push_back(arr[i]);``    ``subsetGen(arr, i + 1, n, ans, c);``    ``c.pop_back();``}` `// Driver code``int` `main()``{``    ``int` `arr[] = { 1, 1 };``    ``int` `n = ``sizeof``(arr) / ``sizeof``(``int``);` `    ``// To store the final ans``    ``int` `ans = 0;``    ``vector<``int``> c;` `    ``subsetGen(arr, 0, n, ans, c);``    ``cout << ans;` `    ``return` `0;``}`

## Java

 `// Java implementation of the approach``import` `java.util.*;` `class` `GFG``{` `// To store the final ans``static` `int` `ans;` `// Function to sum of all subsets of a``// given array``static` `void` `subsetSum(Vector c)``{``    ``int` `L = c.size();``    ``int` `mul = (``int``)Math.pow(``2``, L - ``1``);``    ``for` `(``int` `i = ``0``; i < c.size(); i++)``        ``ans += c.get(i) * mul;``}` `// Function to generate the subsets``static` `void` `subsetGen(``int` `[]arr, ``int` `i,``                      ``int` `n, Vector c)``{``    ``// Base-case``    ``if` `(i == n)``    ``{` `        ``// Finding the sum of all the subsets``        ``// of the generated subset``        ``subsetSum(c);``        ``return``;``    ``}` `    ``// Recursively accepting and rejecting``    ``// the current number``    ``subsetGen(arr, i + ``1``, n, c);``    ``c.add(arr[i]);``    ``subsetGen(arr, i + ``1``, n, c);``    ``c.remove(``0``);``}` `// Driver code``public` `static` `void` `main(String []args)``{``    ``int` `arr[] = { ``1``, ``1` `};``    ``int` `n = arr.length;` `    ``Vector c = ``new` `Vector();` `    ``subsetGen(arr, ``0``, n, c);``    ``System.out.println(ans);``}``}` `// This code is contributed by 29AjayKumar`

## Python3

 `# Python3 implementation of the approach` `# store the answer``c ``=` `[]``ans ``=` `0` `# Function to sum of all subsets of a``# given array``def` `subsetSum():``    ``global` `ans``    ``L ``=` `len``(c)``    ``mul ``=` `pow``(``2``, L ``-` `1``)``    ``i ``=` `0``    ``while` `( i < ``len``(c)):``        ``ans ``+``=` `c[i] ``*` `mul``        ``i ``+``=` `1``        ` `# Function to generate the subsets``def` `subsetGen(arr, i, n):` `    ``# Base-case``    ``if` `(i ``=``=` `n) :` `        ``# Finding the sum of all the subsets``        ``# of the generated subset``        ``subsetSum()``        ``return``    ` `    ``# Recursively accepting and rejecting``    ``# the current number``    ``subsetGen(arr, i ``+` `1``, n)``    ``c.append(arr[i])``    ``subsetGen(arr, i ``+` `1``, n)``    ``c.pop()` `# Driver code``if` `__name__ ``=``=` `"__main__"` `:` `    ``arr ``=` `[ ``1``, ``1` `]``    ``n ``=` `len``(arr)` `    ``subsetGen(arr, ``0``, n)``    ``print` `(ans)``    ` `# This code is contributed by Arnab Kundu`

## C#

 `// C# implementation of the approach``using` `System;``using` `System.Collections.Generic;` `class` `GFG``{` `// To store the final ans``static` `int` `ans;` `// Function to sum of all subsets of a``// given array``static` `void` `subsetSum(List<``int``> c)``{``    ``int` `L = c.Count;``    ``int` `mul = (``int``)Math.Pow(2, L - 1);``    ``for` `(``int` `i = 0; i < c.Count; i++)``        ``ans += c[i] * mul;``}` `// Function to generate the subsets``static` `void` `subsetGen(``int` `[]arr, ``int` `i,``                      ``int` `n, List<``int``> c)``{``    ``// Base-case``    ``if` `(i == n)``    ``{` `        ``// Finding the sum of all the subsets``        ``// of the generated subset``        ``subsetSum(c);``        ``return``;``    ``}` `    ``// Recursively accepting and rejecting``    ``// the current number``    ``subsetGen(arr, i + 1, n, c);``    ``c.Add(arr[i]);``    ``subsetGen(arr, i + 1, n, c);``    ``c.RemoveAt(0);``}` `// Driver code``public` `static` `void` `Main(String []args)``{``    ``int` `[]arr = { 1, 1 };``    ``int` `n = arr.Length;` `    ``List<``int``> c = ``new` `List<``int``>();` `    ``subsetGen(arr, 0, n, c);``    ``Console.WriteLine(ans);``}``}` `// This code is contributed by Rajput-Ji`

## Javascript

 ``

Output:

`6`

Time Complexity: O(2^n), where n is the size of the given array.

Subset generation takes O(2^n) time as there are 2^n subsets of a given set.

Space Complexity: O(n).

Recursion stack will be used which will take O(n) space.

My Personal Notes arrow_drop_up