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Backtracking to find all subsets

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  • Difficulty Level : Medium
  • Last Updated : 28 Jul, 2022
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Given a set of positive integers, find all its subsets. 

Examples: 

Input: array = {1, 2, 3}
Output: // this space denotes null element. 
         1
         1 2
         1 2 3
         1 3
         2
         2 3
         3
Explanation: These are all the subsets that 
can be formed using the array.

Input: 1 2
Output: 
         1 
         2
         1 2
Explanation: These are all the subsets that 
can be formed using the array.
Recommended Practice

The iterative solution is already discussed here: the iterative approach to find all subsets. This article aims to provide a backtracking approach.

Approach: The idea is simple, that if there are n number of elements inside an array, there are n number of choices for the first element in the array. Moving forward to the next recursion call, there will be n-1 number of choices (as we cannot insert the last element again now) for inserting the second element in the array. 
Using the above idea forms a recursive solution to the problem.

Algorithm: 

  1. Create a recursive function that takes the following parameters, input array, the current index, the output array, or current subset, if all the subsets need to be stored then a vector of the array is needed if the subsets need to be printed only then this space can be ignored.
  2. First, print the subset (output array) that has been sent to the function and then run a for loop starting from the ‘index’ to n-1 where n is the size of the input array. We use a loop to demonstrate that we have exactly n number of choices to choose from when adding the first element to our new array. 
  3. Inside the loop, we call the function for the next index after inserting that particular index and then in the next call, we again have (n-1) choices to choose from and so it goes. 
  4. Whenever a call is made for the last index of the array : in that function call, the loop is not run as the condition i<A.size() is not fulfilled and hence, we backtrack to the last recursion call and call the function for the next index after removing the element at that current index.
  1. We finally return when the initial loop ends and we have received all the subsets possible. 
 

Complete Interview Preparation - GFG

Implementation: 

C++




// CPP program to find all subsets by backtracking.
#include <bits/stdc++.h>
using namespace std;
 
// For our vector subset, at every step we have A.size()-i-1
// choices to include. A loop helps us to choose each element
// and then move to the indices further present in the array.
// At the end we backtrack to choose a different index.
void subsetsUtil(vector<int>& A, vector<vector<int> >& res,
                 vector<int>& subset, int index)
{
    res.push_back(subset);
      // Loop to choose from different elements present
      // after the current index 'index'
    for (int i = index; i < A.size(); i++) {
 
        // include the A[i] in subset.
        subset.push_back(A[i]);
 
        // 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;
 
    // keeps track of current element in vector A
      // and the number of elements present in the array subset
    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 total number of subsets possible
    // are O(2^N)
    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;
}


Java




/*package whatever //do not write package name here */
 
import java.io.*;
import java.util.*;
class GFG {
    public static void
    findSubsets(List<List<Integer>> subset, ArrayList<Integer> nums, ArrayList<Integer> output, int index)
    {
      // Base Condition
        if (index == nums.size()) {
            subset.add(output);
            return;
        }
       
        // Not Including Value which is at Index
        findSubsets(subset, nums, new ArrayList<>(output), index + 1);
 
        // Including Value which is at Index
        output.add(nums.get(index));
        findSubsets(subset, nums, new ArrayList<>(output), index + 1);
    }
 
    public static void main(String[] args) {
       
      //Main List for storing all subsets
      List<List<Integer>> subset = new ArrayList<>();
       
      // Input ArrayList
      ArrayList<Integer> input = new ArrayList<>();
      input.add(1);
      input.add(2);
      input.add(3);
       
      findSubsets(subset, input, new ArrayList<>(), 0);
 
      // Comparator is used so that all subset get
      // sorted in ascending order of values
        Collections.sort(subset, (o1, o2) -> {
            int n = Math.min(o1.size(), o2.size());
            for (int i = 0; i < n; i++) {
                if (o1.get(i) == o2.get(i)){
                    continue;
                }else{
                    return o1.get(i) - o2.get(i);
                }
            }
            return 1;
        });
       
       
      // Printing Subset
      for(int i = 0; i < subset.size(); i++){
          for(int j = 0; j < subset.get(i).size(); j++){
              System.out.print(subset.get(i).get(j) + " ");
          }
          System.out.println();
      }
       
    }
}


Python3




# Python3 program to find all subsets
# by backtracking.
 
# 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
def subsetsUtil(A, subset, index):
    print(*subset)
    for i in range(index, len(A)):
         
        # include the A[i] in subset.
        subset.append(A[i])
         
        # move onto the next element.
        subsetsUtil(A, subset, i + 1)
         
        # exclude the A[i] from subset and
        # triggers backtracking.
        subset.pop(-1)
    return
 
# below function returns the subsets of vector A.
def subsets(A):
    global res
    subset = []
     
    # keeps track of current element in vector A
    index = 0
    subsetsUtil(A, subset, index)
     
# Driver Code
 
# find the subsets of below vector.
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.
subsets(array)
 
# This code is contributed by SHUBHAMSINGH8410


C#




/*package whatever //do not write package name here */
using System;
using System.Collections.Generic;
 
public class GFG {
  public static void
    findSubsets(List<List<int>> subset, List<int> nums, List<int> output, int index)
  {
    // Base Condition
    if (index == nums.Count) {
      subset.Add(output);
      return;
    }
 
    // Not Including Value which is at Index
    findSubsets(subset, nums, new List<int>(output), index + 1);
 
    // Including Value which is at Index
    output.Add(nums[index]);
    findSubsets(subset, nums, new List<int>(output), index + 1);
  }
 
  public static void Main(String[] args) {
 
    // Main List for storing all subsets
    List<List<int>> subset = new List<List<int>>();
 
    // Input List
    List<int> input = new List<int>();
    input.Add(1);
    input.Add(2);
    input.Add(3);
 
    findSubsets(subset, input, new List<int>(), 0);
 
    // Comparator is used so that all subset get
    // sorted in ascending order of values
    subset.Sort((o1, o2) => {
      int n = Math.Min(o1.Count, o2.Count);
      for (int i = 0; i < n; i++) {
        if (o1[i] == o2[i]){
          continue;
        }else{
          return o1[i] - o2[i];
        }
      }
      return 1;
    });
 
    // Printing Subset
    for(int i = 0; i < subset.Count; i++){
      for(int j = 0; j < subset[i].Count; j++){
        Console.Write(subset[i][j] + " ");
      }
      Console.WriteLine();
    }
 
  }
}
 
// This code is contributed by shikhasingrajput


Javascript




<script>
/*package whatever //do not write package name here */
function findSubsets(subset, nums, output, index)
{
 
    // Base Condition
    if (index == nums.length) {
        subset.push(output);
        return;
    }
 
    // Not Including Value which is at Index
    findSubsets(subset, nums, [...output], index + 1);
 
    // Including Value which is at Index
    output.push(nums[index]);
    findSubsets(subset, nums, [...output], index + 1);
}
 
// Main List for storing all subsets
let subset = [];
 
// Input ArrayList
let input = [];
input.push(1);
input.push(2);
input.push(3);
 
findSubsets(subset, input, [], 0);
 
// Comparator is used so that all subset get
// sorted in ascending order of values
subset.sort((o1, o2) => {
    let n = Math.min(o1.length, o2.length);
    for (let i = 0; i < n; i++) {
        if (o1[i] == o2[i]) {
            continue;
        } else {
            return o1[i] - o2[i];
        }
    }
    return 1;
});
 
// Printing Subset
for (let i = 0; i < subset.length; i++) {
    for (let j = 0; j < subset[i].length; j++) {
        document.write(subset[i][j] + " ");
    }
    document.write("<br>");
}
 
// This code is contributed by saurabh_jaiswal.
</script>


Output

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

Complexity Analysis:  

  • Time Complexity: O(n. 2^n). Total number of subsets generated are 2^n, So Time Complexity is O(2^n). If we include the time taken to copy the subset vector into the res vector the time taken will be equal to the size of the subset vector.
  • Auxiliary Space: O(n) There can be at max n recursion calls at a particular time, which would consume O(n) stack space. 

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