Given an array arr[] of size N, the task is to generate and print all possible combinations of R elements in array. Examples:
Input: arr[] = {0, 1, 2, 3}, R = 3 Output: 0 1 2 0 1 3 0 2 3 1 2 3 Input: arr[] = {1, 3, 4, 5, 6, 7}, R = 5 Output: 1 3 4 5 6 1 3 4 5 7 1 3 4 6 7 1 3 5 6 7 1 4 5 6 7 3 4 5 6 7
Approach: Recursive methods are discussed here. In this post, an iterative method to output all combinations for a given array will be discussed. The iterative method acts as a state machine. When the machine is called, it outputs a combination and move to the next one. For a combination of r elements from an array of size n, a given element may be included or excluded from the combination. Let’s have a Boolean array of size n to label whether the corresponding element in data array is included. If the ith element in the data array is included, then the ith element in the boolean array is true or false otherwise. Then, r booleans in the boolean array will be labelled as true. We can initialize the boolean array to have r trues from index 0 to index r – 1. During the iteration, we scan the boolean array from left to right and find the first element which is true and whose previous one is false and the first element which is true and whose next one is false. Then, we have the first continuous tract of trues in the Boolean array. Assume there are m trues in this tract, starting from index Start and ending at index End. The next iteration would be
- Set index End + 1 of the boolean array to true.
- Set index Start to index End – 1 of the boolean array to false.
- Set index 0 to index k – 2 to true.
For example, If the current boolean array is {0, 0, 1, 1, 1, 1, 0, 0, 0, 1, 0, 0}, then k = 4, Start = 2, and End = 5. The next Boolean array would be {1, 1, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0}. In case Start == End where there is only one true in the tract, we simply set index End to false and index End + 1 to true. We also need to record the current Start and End and update Start and End during each iteration. When the last r booleans are set to true, we cannot move to the next combination and we stop. The following image illustrates how the boolean array changes from one iteration to another. 
To output the combination, we just scan the boolean array. If its ith index is true, we print out the ith element of the data array. Below is the implementation of the above approach:
C++
// C++ implementation of the approach#include <iostream>using namespace std;class Combination {private: // Data array for combination int* Indices; // Length of the data array int N; // Number of elements in the combination int R; // The boolean array bool* Flags; // Starting index of the 1st tract of trues int Start; // Ending index of the 1st tract of trues int End;public: // Constructor Combination(int* arr, int n, int r) { this->Indices = arr; this->N = n; this->R = r; this->Flags = nullptr; } ~Combination() { if (this->Flags != nullptr) { delete[] this->Flags; } } // Set the 1st r Booleans to true, // initialize Start and End void GetFirst() { this->Flags = new bool[N]; // Generate the very first combination for (int i = 0; i < this->N; ++i) { if (i < this->R) { Flags[i] = true; } else { Flags[i] = false; } } // Update the starting ending indices // of trues in the boolean array this->Start = 0; this->End = this->R - 1; this->Output(); } // Function that returns true if another // combination can still be generated bool HasNext() { return End < (this->N - 1); } // Function to generate the next combination void Next() { // Only one true in the tract if (this->Start == this->End) { this->Flags[this->End] = false; this->Flags[this->End + 1] = true; this->Start += 1; this->End += 1; while (this->End + 1 < this->N && this->Flags[this->End + 1]) { ++this->End; } } else { // Move the End and reset the End if (this->Start == 0) { Flags[this->End] = false; Flags[this->End + 1] = true; this->End -= 1; } else { Flags[this->End + 1] = true; // Set all the values to false starting from // index Start and ending at index End // in the boolean array for (int i = this->Start; i <= this->End; ++i) { Flags[i] = false; } // Set the beginning elements to true for (int i = 0; i < this->End - this->Start; ++i) { Flags[i] = true; } // Reset the End this->End = this->End - this->Start - 1; this->Start = 0; } } this->Output(); }private: // Function to print the combination generated previouslt void Output() { for (int i = 0, count = 0; i < this->N && count < this->R; ++i) { // If current index is set to true in the boolean array // then element at current index in the original array // is part of the combination generated previously if (Flags[i]) { cout << Indices[i] << " "; ++count; } } cout << endl; }};// Driver codeint main(){ int arr[] = { 0, 1, 2, 3 }; int n = sizeof(arr) / sizeof(int); int r = 3; Combination com(arr, n, r); com.GetFirst(); while (com.HasNext()) { com.Next(); } return 0;} |
Java
// Java implementation of the approachclass Combination { // Data array for combination private int[] Indices; // Number of elements in the combination private int R; // The boolean array private boolean[] Flags; // Starting index of the 1st tract of trues private int Start; // Ending index of the 1st tract of trues private int End; // Constructor public Combination(int[] arr, int r) { this.Indices = arr; this.R = r; } // Set the 1st r Booleans to true, // initialize Start and End public void GetFirst() { Flags = new boolean[this.Indices.length]; // Generate the very first combination for (int i = 0; i < this.R; ++i) { Flags[i] = true; } // Update the starting ending indices // of trues in the boolean array this.Start = 0; this.End = this.R - 1; this.Output(); } // Function that returns true if another // combination can still be generated public boolean HasNext() { return End < (this.Indices.length - 1); } // Function to generate the next combination public void Next() { // Only one true in the tract if (this.Start == this.End) { this.Flags[this.End] = false; this.Flags[this.End + 1] = true; this.Start += 1; this.End += 1; while (this.End + 1 < this.Indices.length && this.Flags[this.End + 1]) { ++this.End; } } else { // Move the End and reset the End if (this.Start == 0) { Flags[this.End] = false; Flags[this.End + 1] = true; this.End -= 1; } else { Flags[this.End + 1] = true; // Set all the values to false starting from // index Start and ending at index End // in the boolean array for (int i = this.Start; i <= this.End; ++i) { Flags[i] = false; } // Set the beginning elements to true for (int i = 0; i < this.End - this.Start; ++i) { Flags[i] = true; } // Reset the End this.End = this.End - this.Start - 1; this.Start = 0; } } this.Output(); } // Function to print the combination generated previouslt private void Output() { for (int i = 0, count = 0; i < Indices.length && count < this.R; ++i) { // If current index is set to true in the boolean array // then element at current index in the original array // is part of the combination generated previously if (Flags[i]) { System.out.print(Indices[i]); System.out.print(" "); ++count; } } System.out.println(); }}// Driver codeclass GFG { public static void main(String[] args) { int[] arr = { 0, 1, 2, 3 }; int r = 3; Combination com = new Combination(arr, r); com.GetFirst(); while (com.HasNext()) { com.Next(); } }}// This code is contributed by Rajput-Ji |
Python3
# Python 3 implementation of the approachclass Combination : # Data array for combination Indices = None # Number of elements in the combination R = 0 # The boolean array Flags = None # Starting index of the 1st tract of trues Start = 0 # Ending index of the 1st tract of trues End = 0 # Constructor def __init__(self, arr, r) : self.Indices = arr self.R = r # Set the 1st r Booleans to true, # initialize Start and End def GetFirst(self) : self.Flags = [False] * (len(self.Indices)) # Generate the very first combination i = 0 while (i < self.R) : self.Flags[i] = True i += 1 # Update the starting ending indices # of trues in the boolean array self.Start = 0 self.End = self.R - 1 self.Output() # Function that returns true if another # combination can still be generated def HasNext(self) : return self.End < (len(self.Indices) - 1) # Function to generate the next combination def Next(self) : # Only one true in the tract if (self.Start == self.End) : self.Flags[self.End] = False self.Flags[self.End + 1] = True self.Start += 1 self.End += 1 while (self.End + 1 < len(self.Indices) and self.Flags[self.End + 1]) : self.End += 1 else : # Move the End and reset the End if (self.Start == 0) : self.Flags[self.End] = False self.Flags[self.End + 1] = True self.End -= 1 else : self.Flags[self.End + 1] = True # Set all the values to false starting from # index Start and ending at index End # in the boolean array i = self.Start while (i <= self.End) : self.Flags[i] = False i += 1 # Set the beginning elements to true i = 0 while (i < self.End - self.Start) : self.Flags[i] = True i += 1 # Reset the End self.End = self.End - self.Start - 1 self.Start = 0 self.Output() # Function to print the combination generated previouslt def Output(self) : i = 0 count = 0 while (i < len(self.Indices) and count < self.R) : # If current index is set to true in the boolean array # then element at current index in the original array # is part of the combination generated previously if (self.Flags[i]) : print(self.Indices[i], end ="") print(" ", end ="") count += 1 i += 1 print() # Driver codeclass GFG : @staticmethod def main( args) : arr = [0, 1, 2, 3] r = 3 com = Combination(arr, r) com.GetFirst() while (com.HasNext()) : com.Next() if __name__=="__main__": GFG.main([]) # This code is contributed by aadityaburujwale. |
C#
// C# implementation of the approachusing System;namespace IterativeCombination {class Combination { // Data array for combination private int[] Indices; // Number of elements in the combination private int R; // The boolean array private bool[] Flags; // Starting index of the 1st tract of trues private int Start; // Ending index of the 1st tract of trues private int End; // Constructor public Combination(int[] arr, int r) { this.Indices = arr; this.R = r; } // Set the 1st r Booleans to true, // initialize Start and End public void GetFirst() { Flags = new bool[this.Indices.Length]; // Generate the very first combination for (int i = 0; i < this.R; ++i) { Flags[i] = true; } // Update the starting ending indices // of trues in the boolean array this.Start = 0; this.End = this.R - 1; this.Output(); } // Function that returns true if another // combination can still be generated public bool HasNext() { return End < (this.Indices.Length - 1); } // Function to generate the next combination public void Next() { // Only one true in the tract if (this.Start == this.End) { this.Flags[this.End] = false; this.Flags[this.End + 1] = true; this.Start += 1; this.End += 1; while (this.End + 1 < this.Indices.Length && this.Flags[this.End + 1]) { ++this.End; } } else { // Move the End and reset the End if (this.Start == 0) { Flags[this.End] = false; Flags[this.End + 1] = true; this.End -= 1; } else { Flags[this.End + 1] = true; // Set all the values to false starting from // index Start and ending at index End // in the boolean array for (int i = this.Start; i <= this.End; ++i) { Flags[i] = false; } // Set the beginning elements to true for (int i = 0; i < this.End - this.Start; ++i) { Flags[i] = true; } // Reset the End this.End = this.End - this.Start - 1; this.Start = 0; } } this.Output(); } // Function to print the combination generated previouslt private void Output() { for (int i = 0, count = 0; i < Indices.Length && count < this.R; ++i) { // If current index is set to true in the boolean array // then element at current index in the original array // is part of the combination generated previously if (Flags[i]) { Console.Write(Indices[i]); Console.Write(" "); ++count; } } Console.WriteLine(); }}// Driver codeclass AppDriver { static void Main() { int[] arr = { 0, 1, 2, 3 }; int r = 3; Combination com = new Combination(arr, r); com.GetFirst(); while (com.HasNext()) { com.Next(); } }}} |
Javascript
//Javascript code for the above approachclass Combination {// Data array for combinationIndices = null;// Number of elements in the combinationR = 0;// The boolean arrayFlags = null;// Starting index of the 1st tract of truesStart = 0;// Ending index of the 1st tract of truesEnd = 0;// Constructorconstructor(arr, r) { this.Indices = arr; this.R = r;}// Set the 1st r Booleans to true,// initialize Start and EndGetFirst() { this.Flags = Array(this.Indices.length).fill(false); // Generate the very first combination let i = 0; while (i < this.R) { this.Flags[i] = true; i += 1; } // Update the starting ending indices // of trues in the boolean array this.Start = 0; this.End = this.R - 1; this.Output();}// Function that returns true if another// combination can still be generatedHasNext() { return this.End < (this.Indices.length - 1);}// Function to generate the next combinationNext() { // Only one true in the tract if (this.Start === this.End) { this.Flags[this.End] = false; this.Flags[this.End + 1] = true; this.Start += 1; this.End += 1; while (this.End + 1 < this.Indices.length && this.Flags[this.End + 1]) { this.End += 1; } } else { // Move the End and reset the End if (this.Start === 0) { this.Flags[this.End] = false; this.Flags[this.End + 1] = true; this.End -= 1; } else { this.Flags[this.End + 1] = true; // Set all the values to false starting from // index Start and ending at index End // in the boolean array let i = this.Start; while (i <= this.End) { this.Flags[i] = false; i += 1; } // Set the beginning elements to true i = 0; while (i < this.End - this.Start) { this.Flags[i] = true; i += 1; } // Reset the End this.End = this.End - this.Start - 1; this.Start = 0; } this.Output(); }}// Function to print the combination generated previousltOutput() { let i = 0; let count = 0; while (i < this.Indices.length && count < this.R) { // If current index is set to true in the boolean array // then element at current index in the original array is part of the combination generated previouslyif (this.Flags[i]) {document.write(this.Indices[i], " ");count += 1;}i += 1;}document.write("<br>");}}// Driver codeclass GFG {static main() {let arr = [0, 1, 2, 3];let r = 3;let com = new Combination(arr, r);com.GetFirst();while (com.HasNext()) {com.Next();}}}if (require.main === module) {GFG.main();} |
Javascript
//JS code for the above approachclass Combination {// Data array for combinationIndices = null;// Number of elements in the combinationR = 0;// The boolean arrayFlags = null;// Starting index of the 1st tract of truesStart = 0;// Ending index of the 1st tract of truesEnd = 0;// Constructorconstructor(arr, r) { this.Indices = arr; this.R = r;}// Set the 1st r Booleans to true,// initialize Start and EndGetFirst() { this.Flags = Array(this.Indices.length).fill(false); // Generate the very first combination for (let i = 0; i < this.R; i++) { this.Flags[i] = true; } // Update the starting ending indices // of trues in the boolean array this.Start = 0; this.End = this.R - 1; this.Output();}// Function that returns true if another// combination can still be generatedHasNext() { return this.End < (this.Indices.length - 1);}// Function to generate the next combinationNext(){ // Only one true in the tract if (this.Start === this.End) { this.Flags[this.End] = false; this.Flags[this.End + 1] = true; this.Start += 1; this.End += 1; while (this.End + 1 < this.Indices.length && this.Flags[this.End + 1]) { this.End += 1;}} else{// Move the End and reset the Endif (this.Start === 0) {this.Flags[this.End] = false;this.Flags[this.End + 1] = true;this.End -= 1;} else {this.Flags[this.End + 1] = true; // Set all the values to false starting from // index Start and ending at index End // in the boolean array for (let i = this.Start; i <= this.End; i++) { this.Flags[i] = false; } // Set the beginning elements to true for (let i = 0; i < this.End - this.Start; i++) { this.Flags[i] = true; } // Reset the End this.End = this.End - this.Start - 1; this.Start = 0; } } this.Output();}// Function to print the combination generated previouslyOutput() { for (let i = 0, count = 0; i < this.Indices.length && count < this.R; i++) { // If current index is set to true in the boolean array // then element at current index in the original array // is part of the combination generated previously if (this.Flags[i]) { console.log(this.Indices[i], " "); count += 1; } } console.log("<br>");}}// Driver codeclass GFG {static main() {let arr = [0, 1, 2, 3];let r = 3;let com = new Combination(arr, r);com.GetFirst();while (com.HasNext()) {com.Next();}}}if (require.main === module) {GFG.main();}// This code is contributed by lokeshpotta20. |
Output:
0 1 2 0 1 3 0 2 3 1 2 3
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