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Backtracking to find all subsets

Backtracking to find all subsets: Here, we are going to learn to find out the subsets of a given set of numbers using backtracking.
Submitted by Souvik Saha, on February 03, 2020

Description:

This is a standard interview problem to find out the subsets of a given set of numbers using backtracking.

Problem statement:

There is a set contains N number of elements. You have to find out all the possible subsets of the given set and print them.

    Input:
    Test case T
    //T no. of lines with the value of N and corresponding values.
    E.g.
    2
    
    4
    1 2 3 4
	
    2 
    7 8

    1<=T<=100
    1<=N<=1000 

    Output:
    Print the subsets of the set.

Example

    T=2

    Input:
    N=4
    1 2 3 4
    
    Output: 
    1 2
    1 2 3
    1 2 3 4
    1 2 4
    1 3
    1 3 4
    1 4
    2
    2 3
    2 3 4
    2 4
    3
    3 4
    4
    
    Input:
    N=2 
    7 8 
    
    Output:
    7
    7 8
    8

Explanation with example

Let, there is a Set S having N number of elements,

    S = {X1, X2, X3, ..., Xn}

The process to print the subsets of the set is a problem of combination and permutation. To get the result we use the backtracking process.

    Let,
    f(i) = function to insert the ith number into a subset.

Here, we take a subset of that set in our consideration and consider two things,

1. An element is a part of that subset ( f(i) ).
2. An element is not a part of that subset ( not f(i) ).

    For: N=3
    S = { 1 2 3 }

Algorithm:

Here we use the vector STL to store the subsets.

    traverse(arr, n, current_pos,set,subset){
        if(Current_pos is greater or equals to the n)Then
            return
        end if
        for i = current_pos to the end of the set
            insert the element of arr[i] into subset
            insert the subset into the set
            traverse(arr,n,i+1,set,subset)
            pop the element from subset
        end for
    }

C++ implementation:

#include <bits/stdc++.h>
using namespace std;

void traverse(int* arr, int n, int pos, vector<vector<int> >& v, vector<int> vi)
{
    //if the current position is greater than or equals to n
    if (pos >= n)
        return;
    for (int i = pos; i < n; i++) {
        vi.push_back(arr[i]);
        v.push_back(vi);
        traverse(arr, n, i + 1, v, vi);
        vi.pop_back();
    }
}

vector<vector<int> > find_subset(int* arr, int n)
{
    vector<vector<int> > v;
    vector<int> vi;
    int pos = 0;
    traverse(arr, n, pos, v, vi);
    return v;
}

void print(vector<vector<int> > v)
{
    for (int i = 0; i < v.size(); i++) {
        for (int j = 0; j < v[i].size(); j++) {
            cout << v[i][j] << " ";
        }
        cout << endl;
    }
}

int main()
{
    int t;

    cout << "Test Case : ";
    cin >> t;

    while (t--) {
        int n;

        cout << "Enter the value of n : ";
        cin >> n;

        int arr[n];

        //taking the set elements
        cout << "Enter the values : ";
        for (int i = 0; i < n; i++) {
            cin >> arr[i];
        }

        vector<vector<int> > v = find_subset(arr, n);

        //print the vector
        print(v);
    }
    return 0;
}

Output

Test Case : 2
Enter the value of n : 4
Enter the values : 1 2 3 4
1
1 2
1 2 3
1 2 3 4
1 2 4
1 3
1 3 4
1 4
2
2 3
2 3 4
2 4
3
3 4
4
Enter the value of n : 2
Enter the values : 7 8
7
7 8
8