Iterative Quick Sort

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

/* A typical recursive implementation of quick sort */   /* 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)     {                int p = partition(A, l, h); /* Partitioning index */         quickSort(A, l, p - 1);          quickSort(A, p + 1, h);     } }

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.

// 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; }

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.