Backtracking to find all subsets

Given a set of positive integers, find all its subsets.
Examples:

Input : 1 2 3
Output :     // this space denotes null element. 
         1
         1 2
         1 2 3
         1 3
         2
         2 3
         3

Input : 1 2
Output : 
         1 
         2
         1 2

We have already discussed iterative approach to find all subsets. This article aims to provide a backtracking approach.

Idea is that if we have n number of elements inside an array, we have exactly two choices for each of the elements. Either we include that element in our subset or we do not include it.

[sourcecode language=”CPP” highlight=””]
// CPP program to find all subsets by backtracking.
#include <bits/stdc++.h>
using namespace std;

// In the array A at every step we have two
// choices for each element either we can
// ignore the element or we can include the
// element in our subset
void subsetsUtil(vector<int>& A, vector<vector<int> >& res,
vector<int>& subset, int index)
{
for (int i = index; i < A.size(); i++) {

// include the A[i] in subset.
subset.push_back(A[i]);
res.push_back(subset);

// move onto the next element.
subsetsUtil(A, res, subset, i + 1);

// exclude the A[i] from subset and triggers
// backtracking.
subset.pop_back();
}

return;
}

// below function returns the subsets of vector A.
vector<vector<int> > subsets(vector<int>& A)
{
vector<int> subset;
vector<vector<int> > res;

// include the null element in the set.
res.push_back(subset);

// keeps track of current element in vector A;
int index = 0;
subsetsUtil(A, res, subset, index);

return res;
}

// Driver Code.
int main()
{
// find the subsets of below vector.
vector<int> array = { 1, 2, 3 };

// res will store all subsets.
// O(2 ^ (number of elements inside array))
// because at every step we have two choices
// either include or ignore.
vector<vector<int> > res = subsets(array);

// Print result
for (int i = 0; i < res.size(); i++) {
for (int j = 0; j < res[i].size(); j++)
cout << res[i][j] << " ";
cout << endl;
}

return 0;
}
[/sourcecode]
Output:


1 
1 2 
1 2 3 
1 3 
2 
2 3 
3 

Time Complexity : O(2 ^ n)



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