# Power Set in Lexicographic order

Last Updated : 23 May, 2024

Examples :

`Input : abcOutput : 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 */ ``` Python ```# 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 ``` Javascript ```// 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; } console.log(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 ``` 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 ?> ```

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.
C++ ```#include <algorithm> #include <iostream> #include <string> #include <vector> using namespace std; void generate_power_set(string s) { // Sort the set in lexicographical order sort(s.begin(), s.end()); int n = s.length(); vector<string> subsets; // Generate all possible binary strings of length n for (int i = 0; i < (1 << n); i++) { string binary = ""; int num = i; // Convert the integer i to a binary string of // length n for (int j = 0; j < n; j++) { binary = char('0' + num % 2) + binary; num /= 2; } // Generate the subset based on the binary string string subset = ""; for (int j = 0; j < n; j++) { if (binary[j] == '1') { subset += s[j]; } } subsets.push_back(subset); } // Sort the subsets in lexicographically order sort(subsets.begin(), subsets.end()); // Print the subsets in sorted order for (auto& subset : subsets) { cout << subset << " "; } cout << endl; } int main() { string s = "abc"; generate_power_set(s); return 0; } // This code is contributed by abhinav_m22 ``` Java ```import java.util.ArrayList; import java.util.Arrays; import java.util.List; class Program { // Function to generate the power set of a string static void generatePowerSet(String s) { // Convert the string to a character array char[] charArray = s.toCharArray(); // Sort the character array Arrays.sort(charArray); String sortedString = new String(charArray); int n = sortedString.length(); List<String> subsets = new ArrayList<>(); // Generate all possible binary strings of length n for (int i = 0; i < (1 << n); i++) { StringBuilder binary = new StringBuilder(); int num = i; // Convert the integer i to a binary string of length n for (int j = 0; j < n; j++) { binary.insert(0, num % 2); num /= 2; } // Generate the subset based on the binary string StringBuilder subset = new StringBuilder(); for (int j = 0; j < n; j++) { if (binary.charAt(j) == '1') { subset.append(sortedString.charAt(j)); } } subsets.add(subset.toString()); } // Sort the subsets in lexicographically order subsets.sort(null); // Print the subsets in sorted order for (String subset : subsets) { System.out.print(subset + " "); } System.out.println(); } public static void main(String[] args) { String s = "abc"; generatePowerSet(s); } } ``` Python ```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) ``` C# ```using System; using System.Collections.Generic; using System.Linq; class Program { // Function to generate the power set of a string static void GeneratePowerSet(string s) { // Sort the set in lexicographical order char[] charArray = s.ToCharArray(); Array.Sort(charArray); string sortedString = new string(charArray); int n = sortedString.Length; List<string> subsets = new List<string>(); // Generate all possible binary strings of length n for (int i = 0; i < (1 << n); i++) { string binary = ""; int num = i; // Convert the integer i to a binary string of // length n for (int j = 0; j < n; j++) { binary = (num % 2) + binary; num /= 2; } // Generate the subset based on the binary // string string subset = ""; for (int j = 0; j < n; j++) { if (binary[j] == '1') { subset += sortedString[j]; } } subsets.Add(subset); } // Sort the subsets in lexicographically order subsets.Sort(); // Print the subsets in sorted order foreach(var subset in subsets) { Console.Write(subset + " "); } Console.WriteLine(); } static void Main() { string s = "abc"; GeneratePowerSet(s); } } ``` Javascript ```function generatePowerSet(s) { // Convert the string to an array of characters let charArray = Array.from(s); // Sort the character array charArray.sort(); let sortedString = charArray.join(''); let n = sortedString.length; let subsets = []; // Generate all possible binary strings of length n for (let i = 0; i < (1 << n); i++) { let binary = ''; let num = i; // Convert the integer i to a binary string of length n for (let j = 0; j < n; j++) { binary = (num % 2) + binary; num = Math.floor(num / 2); } // Generate the subset based on the binary string let subset = ''; for (let j = 0; j < n; j++) { if (binary.charAt(j) === '1') { subset += sortedString.charAt(j); } } subsets.push(subset); } // Sort the subsets in lexicographically order subsets.sort(); // Print the subsets in sorted order console.log(subsets.join(' ')); } let s = 'abc'; generatePowerSet(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

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