QuickSort Tail Call Optimization (Reducing worst case space to Log n )

Prerequisite : Tail Call Elimination

In QuickSort, partition function is in-place, but we need extra space for recursive function calls. A simple implementation of QuickSort makes two calls to itself and in worst case requires O(n) space on function call stack. 

The worst case happens when the selected pivot always divides the array such that one part has 0 elements and other part has n-1 elements. For example, in below code, if we choose last element as pivot, we get worst case for sorted arrays (See this for visualization)

C++

#include <iostream>

// Function to partition the array and return the pivot index
int partition(int arr[], int low, int high) {
    int pivot = arr[high]; // Choose the pivot as the last element
    int i = low - 1; // Initialize the index of the smaller element

    for (int j = low; j < high; j++) {
        // If the current element is smaller than or equal to the pivot
        if (arr[j] <= pivot) {
            i++; // Increment the index of the smaller element
            std::swap(arr[i], arr[j]); // Swap arr[i] and arr[j]
        }
    }

    // Swap the pivot element with the element at index (i + 1)
    std::swap(arr[i + 1], arr[high]);

    return i + 1; // Return the pivot index
}

// Function to perform the QuickSort algorithm
void quickSort(int arr[], int low, int high) {
    if (low < high) {
        // Find the pivot index such that elements smaller than the pivot
        // are on the left and elements greater than the pivot are on the right
        int pi = partition(arr, low, high);

        // Recursively sort the elements before and after the pivot
        quickSort(arr, low, pi - 1);
        quickSort(arr, pi + 1, high);
    }
}

int main() {
    int arr[] = {12, 11, 13, 5, 6, 7};
    int n = sizeof(arr) / sizeof(arr[0]);

    std::cout << "Original array: ";
    for (int i = 0; i < n; i++) {
        std::cout << arr[i] << " ";
    }
    std::cout << std::endl;

    quickSort(arr, 0, n - 1);

    std::cout << "Sorted array: ";
    for (int i = 0; i < n; i++) {
        std::cout << arr[i] << " ";
    }
    std::cout << std::endl;

    return 0;
}

C

/* A Simple implementation of QuickSort that makes two
   two recursive calls. */
void quickSort(int arr[], int low, int high)
{
    if (low < high)
    {
        /* pi is partitioning index, arr[p] is now
           at right place */
        int pi = partition(arr, low, high);

        // Separately sort elements before
        // partition and after partition
        quickSort(arr, low, pi - 1);
        quickSort(arr, pi + 1, high);
    }
}
// See below link for complete running code
// https://www.geeksforgeeks.org/quick-sort/

Java

// A Simple implementation of QuickSort that
// makes two recursive calls.
static void quickSort(int arr[], int low, int high)
{
    if (low < high)
    {

        // pi is partitioning index, arr[p] is
        // now at right place
        int 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 rutvik_56

Python3

# Python3 program for the above approach
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 sanjoy_62

C#

// A Simple implementation of QuickSort that
// makes two recursive calls.
static void quickSort(int []arr, int low, int high)
{
    if (low < high)
    {

        // pi is partitioning index, arr[p] is
        // now at right place
        int 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 pratham76.

Javascript

<script>
// A Simple implementation of QuickSort that
// makes two recursive calls.
function quickSort(arr , low , high)
{
    if (low < high)
    {

        // pi is partitioning index, arr[p] is
        // now at right place
        var 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 umadevi9616.
</script>

Can we reduce the auxiliary space for function call stack?
We can limit the auxiliary space to O(Log n). The idea is based on tail call elimination. As seen in the previous post, we can convert the code so that it makes one recursive call. For example, in the below code, we have converted the above code to use a while loop and have reduced the number of recursive calls.

C++

/* QuickSort after tail call elimination */
#include <iostream>

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 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]; // pivot
    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++; // increment index of smaller element
            swap(&arr[i], &arr[j]);
        }
    }
    swap(&arr[i + 1], &arr[high]);
    return (i + 1);
}

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

        // Separately sort elements before
        // partition and after partition
        quickSort(arr, low, pi - 1);
        quickSort(arr, pi + 1, high);
    }
}
/* Function to print an array */
void printArray(int arr[], int size)
{
    for (int i = 0; i < size; i++)
        cout << arr[i] << " ";
    cout << endl;
}
// Driver program to test above functions
int main()
{
    int arr[] = {10, 7, 8, 9, 1, 5};
    int n = sizeof(arr)/sizeof(arr[0]);
    quickSort(arr, 0, n - 1);
    cout << "Sorted array: \n";
    printArray(arr, n);
    return 0;
}
// This code code is contributed by shivhack999

C

/* QuickSort after tail call elimination using while loop */
void quickSort(int arr[], int low, int high)
{
    while (low < high)
    {
        /* pi is partitioning index, arr[p] is now
           at right place */
        int pi = partition(arr, low, high);

        // Separately sort elements before
        // partition and after partition
        quickSort(arr, low, pi - 1);

        low = pi+1;
    }
}
// See below link for complete running code
// https://ide.geeksforgeeks.org/qrlM31

Java

/* QuickSort after tail call elimination using while loop */
static void quickSort(int arr[], int low, int high)
{
    while (low < high)
    {
        /* pi is partitioning index, arr[p] is now
           at right place */
        int pi = partition(arr, low, high);

        // Separately sort elements before
        // partition and after partition
        quickSort(arr, low, pi - 1);

        low = pi+1;
    }
}
// See below link for complete running code
// https://ide.geeksforgeeks.org/qrlM31

// This code is contributed by gauravrajput1

Python3

# QuickSort after tail call elimination using while loop '''
def quickSort(arr, low, high):
    while (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)

        low = pi+1

# See below link for complete running code
# https:#ide.geeksforgeeks.org/qrlM31

# This code is contributed by gauravrajput1

C#

/* QuickSort after tail call elimination using while loop */
static void quickSort(int []arr, int low, int high)
{
    while (low < high)
    {
        /* pi is partitioning index, arr[p] is now
           at right place */
        int pi = partition(arr, low, high);

        // Separately sort elements before
        // partition and after partition
        quickSort(arr, low, pi - 1);

        low = pi+1;
    }
}
// See below link for complete running code
// https://ide.geeksforgeeks.org/qrlM31


// This code contributed by gauravrajput1

Javascript

<script>
/* QuickSort after tail call elimination using while loop */
function quickSort(arr , low , high)
{
    while (low < high)
    {
        /* pi is partitioning index, arr[p] is now
           at right place */
        var pi = partition(arr, low, high);

        // Separately sort elements before
        // partition and after partition
        quickSort(arr, low, pi - 1);

        low = pi+1;
    }
}
// See below link for complete running code
// https://ide.geeksforgeeks.org/qrlM31

// This code is contributed by gauravrajput1
</script>

Although we have reduced number of recursive calls, the above code can still use O(n) auxiliary space in worst case. In worst case, it is possible that array is divided in a way that the first part always has n-1 elements. For example, this may happen when last element is choses as pivot and array is sorted in increasing order. 

We can optimize the above code to make a recursive call only for the smaller part after partition. Below is implementation of this idea.

Further Optimization : 

C++

// C++ program of the above approach
#include <bits/stdc++.h>
using namespace std;

void quickSort(int arr[], int low, int high)
{
  while (low < high)
  {
    /* pi is partitioning index, arr[p] is now
           at right place */
    int pi = partition(arr, low, high);

    // If left part is smaller, then recur for left
    // part and handle right part iteratively
    if (pi - low < high - pi)
    {
      quickSort(arr, low, pi - 1);
      low = pi + 1;
    }

    // Else recur for right part
    else
    {
      quickSort(arr, pi + 1, high);
      high = pi - 1;
    }
  }
}

// This code is contributed by code_hunt.

C

/* This QuickSort requires O(Log n) auxiliary space in
   worst case. */
void quickSort(int arr[], int low, int high)
{
    while (low < high)
    {
        /* pi is partitioning index, arr[p] is now
           at right place */
        int pi = partition(arr, low, high);

        // If left part is smaller, then recur for left
        // part and handle right part iteratively
        if (pi - low < high - pi)
        {
            quickSort(arr, low, pi - 1);
            low = pi + 1;
        }

        // Else recur for right part
        else
        {
            quickSort(arr, pi + 1, high);
            high = pi - 1;
        }
    }
}
// See below link for complete running code
// https://ide.geeksforgeeks.org/LHxwPk

Java

/* This QuickSort requires O(Log n) auxiliary space in
   worst case. */
static void quickSort(int arr[], int low, int high)
{
    while (low < high)
    {
        /* pi is partitioning index, arr[p] is now
           at right place */
        int pi = partition(arr, low, high);

        // If left part is smaller, then recur for left
        // part and handle right part iteratively
        if (pi - low < high - pi)
        {
            quickSort(arr, low, pi - 1);
            low = pi + 1;
        }

        // Else recur for right part
        else
        {
            quickSort(arr, pi + 1, high);
            high = pi - 1;
        }
    }
}
// See below link for complete running code
// https://ide.geeksforgeeks.org/LHxwPk

// This code is contributed by gauravrajput1

Python3

''' This QuickSort requires O(Log n) auxiliary space in
   worst case. '''
def quickSort(arr, low, high)
{
    while (low < high):
        ''' pi is partitioning index, arr[p] is now
           at right place '''
        pi = partition(arr, low, high);

        # If left part is smaller, then recur for left
        # part and handle right part iteratively
        if (pi - low < high - pi):
            quickSort(arr, low, pi - 1);
            low = pi + 1;

        # Else recur for right part
        else:
            quickSort(arr, pi + 1, high);
            high = pi - 1;

# See below link for complete running code
# https:#ide.geeksforgeeks.org/LHxwPk

# This code is contributed by gauravrajput1

C#

/* This QuickSort requires O(Log n) auxiliary space in
   worst case. */
static void quickSort(int []arr, int low, int high)
{
    while (low < high)
    {
        /* pi is partitioning index, arr[p] is now
           at right place */
        int pi = partition(arr, low, high);

        // If left part is smaller, then recur for left
        // part and handle right part iteratively
        if (pi - low < high - pi)
        {
            quickSort(arr, low, pi - 1);
            low = pi + 1;
        }

        // Else recur for right part
        else
        {
            quickSort(arr, pi + 1, high);
            high = pi - 1;
        }
    }
}
// See below link for complete running code
// https://ide.geeksforgeeks.org/LHxwPk


// This code is contributed by gauravrajput1

Javascript

<script>
/* This QuickSort requires O(Log n) auxiliary space in
   worst case. */
function quickSort(arr , low , high)
{
    while (low < high)
    {
        /* pi is partitioning index, arr[p] is now
           at right place */
        var pi = partition(arr, low, high);

        // If left part is smaller, then recur for left
        // part and handle right part iteratively
        if (pi - low < high - pi)
        {
            quickSort(arr, low, pi - 1);
            low = pi + 1;
        }

        // Else recur for right part
        else
        {
            quickSort(arr, pi + 1, high);
            high = pi - 1;
        }
    }
}
// See below link for complete running code
// https://ide.geeksforgeeks.org/LHxwPk


// This code contributed by gauravrajput1
</script>

In the above code, if left part becomes smaller, then we make recursive call for left part. Else for the right part. In worst case (for space), when both parts are of equal sizes in all recursive calls, we use O(Log n) extra space. 

Reference:
http://www.cs.nthu.edu.tw/~wkhon/algo08-tutorials/tutorial2b.pdf

This article is contributed by Dheeraj Jain. Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above