Following is a typical recursive implementation of Quick Sort that uses last element as pivot.
C++
// CPP code for recursive function of Quicksort #include <bits/stdc++.h> using namespace std; // Function to swap numbers void swap(int* a, int* b) { int temp = *a; *a = *b; *b = temp; } /* 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); } } // Driver code int main() { int n = 5; int arr[n] = { 4, 2, 6, 9, 2 }; quickSort(arr, 0, n - 1); for (int i = 0; i < n; i++) { cout << arr[i] << " "; } return 0; } |
Java
// Java program for implementation of QuickSort import java.util.*; 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 */ static 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 */ static 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); } } // Driver code public static void main(String args[]) { int n = 5; int arr[] = { 4, 2, 6, 9, 2 }; qSort(arr, 0, n - 1); for (int i = 0; i < n; i++) { System.out.print(arr[i] + " "); } } } |
Python3
# A typical recursive Python # implementation of QuickSort # 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 += 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) # Driver Code if __name__ == '__main__' : arr = [4, 2, 6, 9, 2] n = len(arr) # Calling quickSort function quickSort(arr, 0, n - 1) for i in range(n): print(arr[i], end = " ") |
C#
// C# program for implementation of // QuickSort using System; class GFG { /* 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 */ static int partition(int[] arr, int low, int high) { int temp; int pivot = arr[high]; // index of smaller element int i = (low - 1); 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] temp = arr[i]; arr[i] = arr[j]; arr[j] = temp; } } // swap arr[i+1] and arr[high] // (or pivot) 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 */ static 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); } } // Driver code public static void Main() { int n = 5; int[] arr = { 4, 2, 6, 9, 2 }; qSort(arr, 0, n - 1); for (int i = 0; i < n; i++) Console.Write(arr[i] + " "); } } // This code is contributed by nitin mittal. |
PHP
<?php // PHP code for recursive function // of Quicksort // Function to swap numbers function swap(&$a, &$b) { $temp = $a; $a = $b; $b = $temp; } /* This function takes last element as pivot, places the pivot element at its correct position in sorted array, and places all smaller (smaller than pivot) to left of pivot and all greater elements to right of pivot */function partition (&$arr, $l, $h) { $x = $arr[$h]; $i = ($l - 1); for ($j = $l; $j <= $h - 1; $j++) { if ($arr[$j] <= $x) { $i++; swap ($arr[$i], $arr[$j]); } } swap ($arr[$i + 1], $arr[$h]); return ($i + 1); } /* A[] --> Array to be sorted, l --> Starting index, h --> Ending index */function quickSort(&$A, $l, $h) { if ($l < $h) { /* Partitioning index */ $p = partition($A, $l, $h); quickSort($A, $l, $p - 1); quickSort($A, $p + 1, $h); } } // Driver code $n = 5; $arr = array(4, 2, 6, 9, 2); quickSort($arr, 0, $n - 1); for($i = 0; $i < $n; $i++) { echo $arr[$i] . " "; } // This code is contributed by // rathbhupendra ?> |
Javascript
<script> // JavaScript program for 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 */ function partition(arr, low, high) { let temp; let pivot = arr[high]; // index of smaller element let i = (low - 1); for (let 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] temp = arr[i]; arr[i] = arr[j]; arr[j] = temp; } } // swap arr[i+1] and arr[high] // (or pivot) 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 */ function qSort(arr, low, high) { if (low < high) { /* pi is partitioning index, arr[pi] is now at right place */ let pi = partition(arr, low, high); // Recursively sort elements // before partition and after // partition qSort(arr, low, pi - 1); qSort(arr, pi + 1, high); } } let n = 5; let arr = [ 4, 2, 6, 9, 2 ]; qSort(arr, 0, n - 1); for (let i = 0; i < n; i++) document.write(arr[i] + " "); </script> |
Output:
2 2 4 6 9
The above implementation can be optimized in many ways
1) The above implementation uses the last index as a 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 Quicksort uses insertion sort below size 7.
Despite the 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 records of the caller function and then resuming execution.
The above function can be easily converted to an iterative version with the help of an auxiliary stack. Following is an iterative implementation of the above recursive code.
C++
// An iterative implementation of quick sort #include <bits/stdc++.h> using namespace std; // 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) cout << arr[i] << " "; } // Driver code 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; } // This is code is contributed by rathbhupendra |
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; } |
Java
// Java program for implementation of QuickSort import java.util.*; 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 */ static int partition(int arr[], int low, int high) { int pivot = arr[high]; // index of smaller element int i = (low - 1); 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; } /* A[] --> Array to be sorted, l --> Starting index, h --> Ending index */ static void quickSortIterative(int arr[], int l, int h) { // Create an auxiliary stack int[] stack = new int[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; } } } // Driver code public static void main(String args[]) { int arr[] = { 4, 3, 5, 2, 1, 3, 2, 3 }; int n = 8; // Function calling quickSortIterative(arr, 0, n - 1); for (int i = 0; i < n; i++) { System.out.print(arr[i] + " "); } } } |
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 |
C#
// C# program for implementation of QuickSort using System; class GFG { /* 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 */ static int partition(int[] arr, int low, int high) { int temp; int pivot = arr[high]; // index of smaller element int i = (low - 1); 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] temp = arr[i]; arr[i] = arr[j]; arr[j] = temp; } } // swap arr[i+1] and arr[high] // (or pivot) temp = arr[i + 1]; arr[i + 1] = arr[high]; arr[high] = temp; return i + 1; } /* A[] --> Array to be sorted, l --> Starting index, h --> Ending index */ static void quickSortIterative(int[] arr, int l, int h) { // Create an auxiliary stack int[] stack = new int[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; } } } // Driver code public static void Main() { int[] arr = { 4, 3, 5, 2, 1, 3, 2, 3 }; int n = 8; // Function calling quickSortIterative(arr, 0, n - 1); for (int i = 0; i < n; i++) Console.Write(arr[i] + " "); } } // This code is contributed by anuj_67. |
PHP
<?php // An iterative implementation of quick sort // A utility function to swap two elements function swap ( &$a, &$b ) { $t = $a; $a = $b; $b = $t; } /* This function is same in both iterative and recursive*/function partition (&$arr, $l, $h) { $x = $arr[$h]; $i = ($l - 1); for ($j = $l; $j <= $h- 1; $j++) { if ($arr[$j] <= $x) { $i++; swap ($arr[$i], $arr[$j]); } } swap ($arr[$i + 1], $arr[$h]); return ($i + 1); } /* A[] --> Array to be sorted, l --> Starting index, h --> Ending index */function quickSortIterative (&$arr, $l, $h) { // Create an auxiliary stack $stack=array_fill(0, $h - $l + 1, 0); // initialize top of stack $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 $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 function printArr( $arr, $n ) { for ( $i = 0; $i < $n; ++$i ) echo $arr[$i]." "; } // Driver code $arr = array(4, 3, 5, 2, 1, 3, 2, 3); $n = count($arr); quickSortIterative($arr, 0, $n - 1 ); printArr($arr, $n ); // This is code is contributed by chandan_jnu ?> |
Javascript
<script> // Javascript program for 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 */ function partition(arr, low, high) { let temp; let pivot = arr[high]; // index of smaller element let i = (low - 1); for (let 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] temp = arr[i]; arr[i] = arr[j]; arr[j] = temp; } } // swap arr[i+1] and arr[high] // (or pivot) temp = arr[i + 1]; arr[i + 1] = arr[high]; arr[high] = temp; return i + 1; } /* A[] --> Array to be sorted, l --> Starting index, h --> Ending index */ function quickSortIterative(arr, l, h) { // Create an auxiliary stack let stack = new Array(h - l + 1); stack.fill(0); // initialize top of stack let 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 let 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; } } } let arr = [ 4, 3, 5, 2, 1, 3, 2, 3 ]; let n = 8; // Function calling quickSortIterative(arr, 0, n - 1); for (let i = 0; i < n; i++) document.write(arr[i] + " "); // This code is contributed by mukesh07. </script> |
Output:
1 2 2 3 3 3 4 5
Time Complexity: O(n*log(n))
Auxiliary Space: O(n)
The above-mentioned optimizations for recursive quicksort can also be applied to the iterative version.
1) Partition process is the same in both recursive and iterative. The same techniques to choose optimal pivot can also be applied to the iterative version.
2) To reduce the stack size, first push the indexes of smaller half.
3) Use insertion sort when the size reduces below an experimentally calculated threshold.
References:
http://en.wikipedia.org/wiki/Quicksort
This article is compiled by Aashish Barnwal and reviewed by neveropen team. Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above.
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