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.
// 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 code int 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 implementation of the approach class 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 code class 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 |
# Python 3 implementation of the approach class 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 code class 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# implementation of the approach using 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 code class 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 code for the above approach class Combination { // Data array for combination Indices = null ;
// Number of elements in the combination R = 0; // The boolean array Flags = null ;
// Starting index of the 1st tract of trues Start = 0; // Ending index of the 1st tract of trues End = 0; // Constructor constructor(arr, r) { this .Indices = arr;
this .R = r;
} // Set the 1st r Booleans to true, // initialize Start and End GetFirst() { 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 generated HasNext() { return this .End < ( this .Indices.length - 1);
} // Function to generate the next combination 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 += 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 previouslt Output() { 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 previously
if ( this .Flags[i]) {
document.write( this .Indices[i], " " );
count += 1; } i += 1; } document.write( "<br>" );
} } // Driver code class 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(); } |
//JS code for the above approach class Combination { // Data array for combination Indices = null ;
// Number of elements in the combination R = 0; // The boolean array Flags = null ;
// Starting index of the 1st tract of trues Start = 0; // Ending index of the 1st tract of trues End = 0; // Constructor constructor(arr, r) { this .Indices = arr;
this .R = r;
} // Set the 1st r Booleans to true, // initialize Start and End GetFirst() { 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 generated HasNext() { return this .End < ( this .Indices.length - 1);
} // Function to generate the next combination 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 += 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
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 previously Output() { 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 code class 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. |
0 1 2 0 1 3 0 2 3 1 2 3