Power Set in Lexicographic order

This article is about generating Power set in lexicographical order. 

Examples :

Input : abc
Output : a ab abc ac b bc c

The idea is to sort array first. After sorting, one by one fix characters and recursively generates all subsets starting from them. After every recursive call, we remove last character so that next permutation can be generated. 

Implementation:

C++

// CPP program to generate power set in
// lexicographic order.
#include <bits/stdc++.h>
using namespace std;

// str : Stores input string
// n : Length of str.
void func(string s, vector<string>& str, int n, int pow_set)
{
    int i, j;
    for (i = 0; i < pow_set; i++) {
        string x;
        for (j = 0; j < n; j++) {
            if (i & 1 << j) {
                x = x + s[j];
            }
        }
        if (i != 0)
            str.push_back(x);
    }
}
int main()
{
    int n;
    string s;
    vector<string> str;
    s = "cab";
    n = s.length();
    int pow_set = pow(2, n);
    func(s, str, n, pow_set);
    sort(str.begin(), str.end());
    for (int i = 0; i < str.size(); i++)
        cout << str[i] << " ";
    cout << endl;

    return 0;
}

Java

// Java program to generate power set in
// lexicographic order.
import java.util.*;

class GFG {

    // str : Stores input string
    // n : Length of str.
    // curr : Stores current permutation
    // index : Index in current permutation, curr
    static void permuteRec(String str, int n,
                           int index, String curr)
    {
        // base case
        if (index == n) {
            return;
        }
        System.out.println(curr);
        for (int i = index + 1; i < n; i++) {

            curr += str.charAt(i);
            permuteRec(str, n, i, curr);

            // backtracking
            curr = curr.substring(0, curr.length() - 1);
        }
        return;
    }

    // Generates power set in lexicographic
    // order.
    static void powerSet(String str)
    {
        char[] arr = str.toCharArray();
        Arrays.sort(arr);
        permuteRec(new String(arr), str.length(), -1, "");
    }

    // Driver code
    public static void main(String[] args)
    {
        String str = "cab";
        powerSet(str);
    }
}

/* This code contributed by PrinciRaj1992 */

Python3

# Python3 program to generate power
# set in lexicographic order.

# str : Stores input string
# n : Length of str.
# curr : Stores current permutation
# index : Index in current permutation, curr
def permuteRec(string, n, index = -1, curr = ""):

    # base case
    if index == n:
        return

    if len(curr) > 0:
        print(curr)

    for i in range(index + 1, n):
        curr += string[i]
        permuteRec(string, n, i, curr)

        # backtracking
        curr = curr[:len(curr) - 1]

# Generates power set in lexicographic order
def powerSet(string):
    string = ''.join(sorted(string))
    permuteRec(string, len(string))

# Driver Code
if __name__ == "__main__":
    string = "cab"
    powerSet(string)

# This code is contributed by vibhu4agarwal

C#

// C# program to generate power set in
// lexicographic order.
using System;

class GFG {

    // str : Stores input string
    // n : Length of str.
    // curr : Stores current permutation
    // index : Index in current permutation, curr
    static void permuteRec(String str, int n,
                           int index, String curr)
    {
        // base case
        if (index == n) {
            return;
        }
        Console.WriteLine(curr);
        for (int i = index + 1; i < n; i++) {

            curr += str[i];
            permuteRec(str, n, i, curr);

            // backtracking
            curr = curr.Substring(0, curr.Length - 1);
        }
        return;
    }

    // Generates power set in lexicographic
    // order.
    static void powerSet(String str)
    {
        char[] arr = str.ToCharArray();
        Array.Sort(arr);
        permuteRec(new String(arr), str.Length, -1, "");
    }

    // Driver code
    public static void Main(String[] args)
    {
        String str = "cab";
        powerSet(str);
    }
}

// This code contributed by Rajput-Ji

PHP

<?php
// PHP program to generate power
// set in lexicographic order.

// str : Stores input string
// n : Length of str.
// curr : Stores current permutation
// index : Index in current permutation, curr
function permuteRec($str, $n, $index = -1,
                              $curr = "")
{
    // base case
    if ($index == $n)
        return;

    echo $curr."\n";
    for ($i = $index + 1; $i < $n; $i++)
    {

        $curr=$curr.$str[$i];
        permuteRec($str, $n, $i, $curr);

        // backtracking
        $curr ="";
    }
    return;
}

// Generates power set in lexicographic
// order.
function powerSet($str)
{

    $str = str_split($str);
    sort($str);
    permuteRec($str, sizeof($str));
}

// Driver code
$str = "cab";
powerSet($str);

// This code is contributed by Mithun Kumar
?>

Javascript

<script>
// javascript program to generate power set in
// lexicographic order.

    // str : Stores input string
    // n : Length of str.
    // curr : Stores current permutation
    // index : Index in current permutation, curr
    function permuteRec( str , n , index,  curr) {
        // base case
        if (index == n) {
            return;
        }
        document.write(curr+" ");
        for (var i = index + 1; i < n; i++) {

            curr += str[i];
            permuteRec(str, n, i, curr);

            // backtracking
            curr = curr.substring(0, curr.length - 1);
        }
        return;
    }

    // Generates power set in lexicographic
    // order.
    function powerSet(str) {
        var arr = str.split("");
        arr.sort();
        permuteRec(arr, str.length, -1, "");
    }

    // Driver code

        var str = "cab";
        powerSet(str);

// This code contributed by umadevi9616
</script>

Output

a ab b c ca cab cb 

Time Complexity: O(n*2ⁿ) 
Auxiliary Space: O(1)

Method (binary numbers)

The idea is to use binary numbers  to generate the power set of a given set of elements in lexicographical order

• Sort the given set in lexicographical order.
• Define a variable “n” to represent the size of the set.
• Use a loop to generate all possible binary numbers of length “n”.
• For each binary number, convert it to a string of 0s and 1s, 
• Add the current subset to the output list.
• Sort the output list in lexicographical order.
• Print the sorted list of subsets.

Python3

def generate_power_set(s):
    # Sort the set in lexicographical order
    s = ''.join(sorted(s))

    n = len(s)
    subsets = []
    # Generate all possible binary strings of length n
    for i in range(2**n):
        # Convert the integer i to a binary string of length n
        binary = bin(i)[2:].zfill(n)
        # Generate the subset based on the binary string
        subset = ''.join([s[j] for j in range(n) if binary[j] == '1'])
        subsets.append(subset)
    # Sort the subsets in lexicographically order
    subsets.sort()
    # Print the subsets in sorted order
    for subset in subsets:
        print(subset)

# Example usage
s = 'abc'
generate_power_set(s)

Output

a
ab
abc
ac
b
bc
c

Time complexity :O(2^n * n), where n is the length of the input set.
Space complexity  :O(2^n * n), since the output list of subsets can potentially contain 2^n elements