Iterative Quick Sort

Following is a typical recursive implementation of Quick Sort that uses last element as pivot.

C++

```/* A typical recursive C/C++  implementation of QuickSort */

/* This function takes last element as pivot, places
the pivot element at its correct position in sorted
array, and places all smaller (smaller than pivot)
to left of pivot and all greater elements to right
of pivot */
int partition (int arr[], int l, int h)
{
int x = arr[h];
int i = (l - 1);

for (int j = l; j <= h- 1; j++)
{
if (arr[j] <= x)
{
i++;
swap (&arr[i], &arr[j]);
}
}
swap (&arr[i + 1], &arr[h]);
return (i + 1);
}

/* A[] --> Array to be sorted,
l  --> Starting index,
h  --> Ending index */
void quickSort(int A[], int l, int h)
{
if (l < h)
{
/* Partitioning index */
int p = partition(A, l, h);
quickSort(A, l, p - 1);
quickSort(A, p + 1, h);
}
}
```

Python

```# A typical recursive Python  implementation of QuickSort */

# This function takes last element as pivot, places
# the pivot element at its correct position in sorted
# array, and places all smaller (smaller than pivot)
# to left of pivot and all greater elements to right
# of pivot
def partition(arr,low,high):
i = ( low-1 )         # index of smaller element
pivot = arr[high]     # pivot

for j in range(low , high):

# If current element is smaller than or
# equal to pivot
if   arr[j] <= pivot:

# increment index of smaller element
i = i+1
arr[i],arr[j] = arr[j],arr[i]

arr[i+1],arr[high] = arr[high],arr[i+1]
return ( i+1 )

# The main function that implements QuickSort
# arr[] --> Array to be sorted,
# low  --> Starting index,
# high  --> Ending index

# Function to do Quick sort
def quickSort(arr,low,high):
if low < high:

# pi is partitioning index, arr[p] is now
# at right place
pi = partition(arr,low,high)

# Separately sort elements before
# partition and after partition
quickSort(arr, low, pi-1)
quickSort(arr, pi+1, high)

# This code is contributed by Mohit Kumra
```

Java

```// Java program for implementation of QuickSort
class QuickSort
{
/* This function takes last element as pivot,
places the pivot element at its correct
position in sorted array, and places all
smaller (smaller than pivot) to left of
pivot and all greater elements to right
of pivot */
int partition(int arr[], int low, int high)
{
int pivot = arr[high];
int i = (low-1); // index of smaller element
for (int j=low; j<=high-1; j++)
{
// If current element is smaller than or
// equal to pivot
if (arr[j] <= pivot)
{
i++;

// swap arr[i] and arr[j]
int temp = arr[i];
arr[i] = arr[j];
arr[j] = temp;
}
}

// swap arr[i+1] and arr[high] (or pivot)
int temp = arr[i+1];
arr[i+1] = arr[high];
arr[high] = temp;

return i+1;
}

/* The main function that implements QuickSort()
arr[] --> Array to be sorted,
low  --> Starting index,
high  --> Ending index */
void qSort(int arr[], int low, int high)
{
if (low < high)
{
/* pi is partitioning index, arr[pi] is
now at right place */
int pi = partition(arr, low, high);

// Recursively sort elements before
// partition and after partition
qSort(arr, low, pi-1);
qSort(arr, pi+1, high);
}
}
}
```

The above implementation can be optimized in many ways

1) The above implementation uses last index as pivot. This causes worst-case behavior on already sorted arrays, which is a commonly occurring case. The problem can be solved by choosing either a random index for the pivot, or choosing the middle index of the partition or choosing the median of the first, middle and last element of the partition for the pivot. (See this for details)

2) To reduce the recursion depth, recur first for the smaller half of the array, and use a tail call to recurse into the other.

3) Insertion sort works better for small subarrays. Insertion sort can be used for invocations on such small arrays (i.e. where the length is less than a threshold t determined experimentally). For example, this library implementation of qsort uses insertion sort below size 7.

Despite above optimizations, the function remains recursive and uses function call stack to store intermediate values of l and h. The function call stack stores other bookkeeping information together with parameters. Also, function calls involve overheads like storing activation record of the caller function and then resuming execution.

The above function can be easily converted to iterative version with the help of an auxiliary stack. Following is an iterative implementation of the above recursive code.

C/C++

```// An iterative implementation of quick sort
#include <stdio.h>

// A utility function to swap two elements
void swap ( int* a, int* b )
{
int t = *a;
*a = *b;
*b = t;
}

/* This function is same in both iterative and recursive*/
int partition (int arr[], int l, int h)
{
int x = arr[h];
int i = (l - 1);

for (int j = l; j <= h- 1; j++)
{
if (arr[j] <= x)
{
i++;
swap (&arr[i], &arr[j]);
}
}
swap (&arr[i + 1], &arr[h]);
return (i + 1);
}

/* A[] --> Array to be sorted,
l  --> Starting index,
h  --> Ending index */
void quickSortIterative (int arr[], int l, int h)
{
// Create an auxiliary stack
int stack[ h - l + 1 ];

// initialize top of stack
int top = -1;

// push initial values of l and h to stack
stack[ ++top ] = l;
stack[ ++top ] = h;

// Keep popping from stack while is not empty
while ( top >= 0 )
{
// Pop h and l
h = stack[ top-- ];
l = stack[ top-- ];

// Set pivot element at its correct position
// in sorted array
int p = partition( arr, l, h );

// If there are elements on left side of pivot,
// then push left side to stack
if ( p-1 > l )
{
stack[ ++top ] = l;
stack[ ++top ] = p - 1;
}

// If there are elements on right side of pivot,
// then push right side to stack
if ( p+1 < h )
{
stack[ ++top ] = p + 1;
stack[ ++top ] = h;
}
}
}

// A utility function to print contents of arr
void printArr( int arr[], int n )
{
int i;
for ( i = 0; i < n; ++i )
printf( "%d ", arr[i] );
}

// Driver program to test above functions
int main()
{
int arr[] = {4, 3, 5, 2, 1, 3, 2, 3};
int n = sizeof( arr ) / sizeof( *arr );
quickSortIterative( arr, 0, n - 1 );
printArr( arr, n );
return 0;
}
```

Python

```# Python program for implementation of Quicksort

# This function is same in both iterative and recursive
def partition(arr,l,h):
i = ( l - 1 )
x = arr[h]

for j in range(l , h):
if   arr[j] <= x:

# increment index of smaller element
i = i+1
arr[i],arr[j] = arr[j],arr[i]

arr[i+1],arr[h] = arr[h],arr[i+1]
return (i+1)

# Function to do Quick sort
# arr[] --> Array to be sorted,
# l  --> Starting index,
# h  --> Ending index
def quickSortIterative(arr,l,h):

# Create an auxiliary stack
size = h - l + 1
stack = [0] * (size)

# initialize top of stack
top = -1

# push initial values of l and h to stack
top = top + 1
stack[top] = l
top = top + 1
stack[top] = h

# Keep popping from stack while is not empty
while top >= 0:

# Pop h and l
h = stack[top]
top = top - 1
l = stack[top]
top = top - 1

# Set pivot element at its correct position in
# sorted array
p = partition( arr, l, h )

# If there are elements on left side of pivot,
# then push left side to stack
if p-1 > l:
top = top + 1
stack[top] = l
top = top + 1
stack[top] = p - 1

# If there are elements on right side of pivot,
# then push right side to stack
if p+1 < h:
top = top + 1
stack[top] = p + 1
top = top + 1
stack[top] = h

# Driver code to test above
arr = [4, 3, 5, 2, 1, 3, 2, 3]
n = len(arr)
quickSortIterative(arr, 0, n-1)
print ("Sorted array is:")
for i in range(n):
print ("%d" %arr[i]),

# This code is contributed by Mohit Kumra
```

Java

```// Java implementation of iterative quick sort
class IterativeQuickSort
{
void swap(int arr[],int i,int j)
{
int t = arr[i];
arr[i] = arr[j];
arr[j] = t;
}

/* This function is same in both iterative and
recursive*/
int partition (int arr[], int l, int h)
{
int x = arr[h];
int i = (l - 1);

for (int j = l; j <= h- 1; j++)
{
if (arr[j] <= x)
{
i++;
// swap arr[i] and arr[j]
swap(arr,i,j);
}
}
// swap arr[i+1] and arr[h]
swap(arr,i+1,h);
return (i + 1);
}

// Sorts arr[l..h] using iterative QuickSort
void QuickSort(int arr[], int l, int h)
{
// create auxiliary stack
int stack[] = new int[h-l+1];

// initialize top of stack
int top = -1;

// push initial values in the stack
stack[++top] = l;
stack[++top] = h;

// keep popping elements until stack is not empty
while (top >= 0)
{
// pop h and l
h = stack[top--];
l = stack[top--];

// set pivot element at it's proper position
int p = partition(arr, l, h);

// If there are elements on left side of pivot,
// then push left side to stack
if ( p-1 > l )
{
stack[ ++top ] = l;
stack[ ++top ] = p - 1;
}

// If there are elements on right side of pivot,
// then push right side to stack
if ( p+1 < h )
{
stack[ ++top ] = p + 1;
stack[ ++top ] = h;
}
}
}

// A utility function to print contents of arr
void printArr( int arr[], int n )
{
int i;
for ( i = 0; i < n; ++i )
System.out.print(arr[i]+" ");
}

// Driver code to test above
public static void main(String args[])
{
IterativeQuickSort ob = new IterativeQuickSort();
int arr[] = {4, 3, 5, 2, 1, 3, 2, 3};
ob.QuickSort(arr, 0, arr.length-1);
ob.printArr(arr, arr.length);
}
}
/*This code is contributed by Rajat Mishra */
```

Output:
`1 2 2 3 3 3 4 5`

The above mentioned optimizations for recursive quick sort can also be applied to iterative version.

1) Partition process is same in both recursive and iterative. The same techniques to choose optimal pivot can also be applied to iterative version.

2) To reduce the stack size, first push the indexes of smaller half.

3) Use insertion sort when the size reduces below a experimentally calculated threshold.

References:
http://en.wikipedia.org/wiki/Quicksort

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