Printing all subsets of {1,2,3,…n} without using array or loop

Last Updated : 12 Sep, 2022

Given a natural number n, print all the subsets of the set $\{1, 2, 3, ..., n\}$ without using any array or loop (only the use of recursion is allowed).
Examples:

Input : n = 4
Output : { 1 2 3 4 }
         { 1 2 3 }
         { 1 2 4 }
         { 1 2 }
         { 1 3 4 }
         { 1 3 }
         { 1 4 }
         { 1 }
         { 2 3 4 }
         { 2 3 }
         { 2 4 }
         { 2 }
         { 3 4 }
         { 3 }
         { 4 }
         { }

Input : n = 2
Output : { 1 2 }
         { 1 }
         { 2 }
         { }

Approach:

Start from $num = 2^n - 1$ upto 0.
Consider the binary representation of num with n bits.
Start from the leftmost bit which represents 1, the second bit represents 2, and so on until nth bit which represents n.
Print the number corresponding to the bit if it is set.
Perform the above steps for all values of num until it is equal to 0.

Let’s understand the above approach through an example:
Considering input n = 4, start from $num = 2^n - 1 = 15$ .

and so on … until num = 0.
Below is the implementation of the above approach:

C++

// C++ code to print all subsets
// of {1, 2, 3, n} without using
// array or loop, just recursion.
#include <bits/stdc++.h>
using namespace std;
 
void subset(int, int, int);
 
// This recursive function calls subset
// function to print the subsets one by one.
// numBits --> number of bits needed to
// represent the number (simply input value n).
// num --> Initially equal to 2 ^ n - 1 and
// decreases by 1 every recursion until 0.
void printSubsets(int numOfBits, int num) 
{
    if (num >= 0)
    {
        cout << "{ ";
 
        // Print the subset corresponding to
        // binary representation of num.
        subset(numOfBits - 1, num, numOfBits);
        cout << "}" << endl;
 
        // Call the function recursively to
        // print the next subset.
        printSubsets(numOfBits, num - 1);
    }
    else
        return;
}
 
// This function recursively prints the
// subset corresponding to the binary
// representation of num.
// nthBit --> nth bit from right side
// starting from n and decreases until 0
void subset(int nthBit, int num, int numOfBits)
{
    if (nthBit >= 0) 
    {
        // Print number in given subset only
        // if the bit corresponding to it
        // is set in num.
        if (num & (1 << nthBit)) 
        {
            cout << numOfBits - nthBit << " ";
        }
 
        // Check for the next bit
        subset(nthBit - 1, num, numOfBits);
    } 
    else
        return;
}
 
// Driver Code
int main()
{
    int n = 4;
    printSubsets(n, pow(2, n) - 1);
}
 
// This code is contributed by
// sanjeev2552

Java

// Java code to print all subsets 
// of {1, 2, 3, n} without using
// array or loop, just recursion.
class GfG 
{
 
    // This recursive function calls subset
    // function to print the subsets one by one. 
    // numBits --> number of bits needed to 
    // represent the number (simply input value n).
    // num --> Initially equal to 2 ^ n - 1 and 
    // decreases by 1 every recursion until 0.
    static void printSubSets(int numOfBits, int num) 
    {
        if (num >= 0) 
        {
            System.out.print("{ ");
             
            // Print the subset corresponding to 
            // binary representation of num.
            subset(numOfBits - 1, num, numOfBits);
            System.out.println("}");
             
            // Call the function recursively to 
            // print the next subset.
            printSubSets(numOfBits, num - 1);
 
        } else
            return;
    }
 
    // This function recursively prints the 
    // subset corresponding to the binary 
    // representation of num.
    // nthBit --> nth bit from right side 
    // starting from n and decreases until 0.
    static void subset(int nthBit, int num, int numOfBits) 
    {
        if (nthBit >= 0) 
        {
            // Print number in given subset only
            // if the bit corresponding to it 
            // is set in num.
            if ((num & (1 << nthBit)) != 0)
            {
                System.out.print(numOfBits - nthBit + " ");
 
            }
             
            // Check for the next bit 
            subset(nthBit - 1, num, numOfBits);
        } else
            return;
    }
     
    // Driver code
    public static void main(String[] args) 
    {
        int n = 4;
        printSubSets(n, (int) (Math.pow(2, n)) -1);
    }
}
 
// This code is contributed by laststringx

Python3

# Python3 code to print all subsets 
# of {1, 2, 3, …n} without using
# array or loop, just recursion.
 
# This recursive function calls subset
# function to print the subsets one by one. 
# numBits --> number of bits needed to 
# represent the number (simply input value n).
# num --> Initially equal to 2 ^ n - 1 and 
# decreases by 1 every recursion until 0.
def printSubsets(numOfBits, num):
     
    if num >= 0:
        print("{", end = " ")
 
        # Print the subset corresponding to 
        # binary representation of num.
        subset(numOfBits-1, num, numOfBits)
        print("}")
 
        # Call the function recursively to 
        # print the next subset.
        printSubsets(numOfBits, num-1)
         
    else:
        return
 
# This function recursively prints the 
# subset corresponding to the binary 
# representation of num.
# nthBit --> nth bit from right side 
# starting from n and decreases until 0.
def subset(nthBit, num, numOfBits):
     
    if nthBit >= 0:
         
        # Print number in given subset only
        # if the bit corresponding to it 
        # is set in num.
        if num & (1 << nthBit) != 0:
            print(numOfBits - nthBit, end = " ")
         
        # Check for the next bit 
        subset(nthBit-1, num, numOfBits)
         
    else:
        return
 
# Driver Code    
n = 4
printSubsets(n, 2**n - 1)

C#

// C# code to print all subsets 
// of {1, 2, 3, n} without using 
// array or loop, just recursion.
using System;
 
class GfG 
{ 
 
    // This recursive function calls subset 
    // function to print the subsets one by one. 
    // numBits --> number of bits needed to 
    // represent the number (simply input value n). 
    // num --> Initially equal to 2 ^ n - 1 and 
    // decreases by 1 every recursion until 0. 
    static void printSubSets(int numOfBits, int num) 
    { 
        if (num >= 0) 
        { 
            Console.Write("{ "); 
             
            // Print the subset corresponding to 
            // binary representation of num. 
            subset(numOfBits - 1, num, numOfBits); 
            Console.WriteLine("}"); 
             
            // Call the function recursively to 
            // print the next subset. 
            printSubSets(numOfBits, num - 1); 
 
        } else
            return; 
    } 
 
    // This function recursively prints the 
    // subset corresponding to the binary 
    // representation of num. 
    // nthBit --> nth bit from right side 
    // starting from n and decreases until 0. 
    static void subset(int nthBit, int num, int numOfBits) 
    { 
        if (nthBit >= 0) 
        { 
            // Print number in given subset only 
            // if the bit corresponding to it 
            // is set in num. 
            if ((num & (1 << nthBit)) != 0) 
            { 
                Console.Write(numOfBits - nthBit + " "); 
 
            } 
             
            // Check for the next bit 
            subset(nthBit - 1, num, numOfBits); 
        } else
            return; 
    } 
     
    // Driver codeM
    public static void Main(String[] args) 
    { 
        int n = 4; 
        printSubSets(n, (int) (Math.Pow(2, n)) -1); 
    } 
}
 
// This code is contributed by Srathore

Javascript

<script>
 
// Javascript code to print all subsets
// of {1, 2, 3, n} without using
// array or loop, just recursion.
 
// This recursive function calls subset
// function to print the subsets one by one.
// numBits --> number of bits needed to
// represent the number (simply input value n).
// num --> Initially equal to 2 ^ n - 1 and
// decreases by 1 every recursion until 0.
function printSubsets(numOfBits, num) 
{
    if (num >= 0)
    {
        document.write( "{ ");
 
        // Print the subset corresponding to
        // binary representation of num.
        subset(numOfBits - 1, num, numOfBits);
        document.write( "}<br>" );
 
        // Call the function recursively to
        // print the next subset.
        printSubsets(numOfBits, num - 1);
    }
    else
        return;
}
 
// This function recursively prints the
// subset corresponding to the binary
// representation of num.
// nthBit --> nth bit from right side
// starting from n and decreases until 0
function subset(nthBit, num, numOfBits)
{
    if (nthBit >= 0) 
    {
        // Print number in given subset only
        // if the bit corresponding to it
        // is set in num.
        if (num & (1 << nthBit)) 
        {
            document.write( numOfBits - nthBit + " ");
        }
 
        // Check for the next bit
        subset(nthBit - 1, num, numOfBits);
    } 
    else
        return;
}
 
// Driver Code
var n = 4;
printSubsets(n, Math.pow(2, n) - 1);
 
</script>

Output:

{ 1 2 3 4 }
{ 1 2 3 }
{ 1 2 4 }
{ 1 2 }
{ 1 3 4 }
{ 1 3 }
{ 1 4 }
{ 1 }
{ 2 3 4 }
{ 2 3 }
{ 2 4 }
{ 2 }
{ 3 4 }
{ 3 }
{ 4 }
{ }

Time Complexity: $O(n*2^n)$

Auxiliary Space: O(n) for call stack

Calculate Stirling numbers which represents the number of ways to arrange r objects around n different circles

rituraj_jain

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Practice Tags :

Printing all subsets of {1,2,3,…n} without using array or loop

C++

Java

Python3

C#

Javascript

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