Quicksort

1. Overview

Like Merge Sort, QuickSort is a Divide and Conquer algorithm. It picks an element as pivot and partitions the given array around the picked pivot. There are many different versions of quickSort that pick pivot in different ways.

• Always pick the first element as a pivot.
• Always pick the last element as the pivot (implemented below)
• Pick a random element as pivot.
• Pick the median as a pivot.

2. Description

2.1 Procedure

/* low  --> Starting index,  high  --> Ending index */
quickSort(arr[], low, high)
{
    if (low < high)
    {
        /* pi is partitioning index, arr[pi] is now
           at right place */
        pi = partition(arr, low, high);

        quickSort(arr, low, pi - 1);  // Before pi
        quickSort(arr, pi + 1, high); // After pi
    }
}

/* 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 */
partition (arr[], low, high)
{
    // pivot (Element to be placed at right position)
    pivot = arr[high];  
 
    i = (low - 1)  // Index of smaller element

    for (j = low; j <= high- 1; j++)
    {
        // If current element is smaller than the pivot
        if (arr[j] < pivot)
        {
            i++;    // increment index of smaller element
            swap arr[i] and arr[j]
        }
    }
    swap arr[i + 1] and arr[high])
    return (i + 1)
}

2.1.1 Illustration of partition():

arr[] = {10, 80, 30, 90, 40, 50, 70}
Indexes:  0   1   2   3   4   5   6 

low = 0, high =  6, pivot = arr[h] = 70
Initialize index of smaller element, i = -1

Traverse elements from j = low to high-1
j = 0 : Since arr[j] <= pivot, do i++ and swap(arr[i], arr[j])
i = 0 
arr[] = {10, 80, 30, 90, 40, 50, 70} // No change as i and j 
                                     // are same

j = 1 : Since arr[j] > pivot, do nothing
// No change in i and arr[]

j = 2 : Since arr[j] <= pivot, do i++ and swap(arr[i], arr[j])
i = 1
arr[] = {10, 30, 80, 90, 40, 50, 70} // We swap 80 and 30 

j = 3 : Since arr[j] > pivot, do nothing
// No change in i and arr[]

j = 4 : Since arr[j] <= pivot, do i++ and swap(arr[i], arr[j])
i = 2
arr[] = {10, 30, 40, 90, 80, 50, 70} // 80 and 40 Swapped
j = 5 : Since arr[j] <= pivot, do i++ and swap arr[i] with arr[j] 
i = 3 
arr[] = {10, 30, 40, 50, 80, 90, 70} // 90 and 50 Swapped 

We come out of loop because j is now equal to high-1.
Finally we place pivot at correct position by swapping
arr[i+1] and arr[high] (or pivot) 
arr[] = {10, 30, 40, 50, 70, 90, 80} // 80 and 70 Swapped 

Now 70 is at its correct place. All elements smaller than
70 are before it and all elements greater than 70 are after
it.

2.1.2 Implementation

# Python program for implementation of Quicksort 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 
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 the 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) 
  
# Driver code to test above 
arr = [10, 7, 8, 9, 1, 5] 
n = len(arr) 
quickSort(arr,0,n-1) 
print ("Sorted array is:") 
for i in range(n): 
    print ("%d" %arr[i]), 
  
# This code is contributed by Mohit Kumra 

Other implementations using Linked list

• Singly Linked List
• Doubly Linked List

2.2 Time Complexity

Particular	Worst
Worst-case performance	O($n^{2}$)
Best-case performance	O(n log n)
3-Way quicksort Best-case performance	O(n)
Average performance	O(n log n)
Worst-case space complexity	O(n)
Worst-case space complexity (Sedgewick 1978)	O(log n)

2.2.1 Analysis of complexity

 T(n) = T(k) + T(n-k-1) + O(n)

This notation can be solved using Mater Theorem.

• Worst Case

 T(n) = T(0) + T(n-1) + O(n)
which is equivalent to  
# O(n^2)
 T(n) = T(n-1) + O(n)

• Best Case

# O(N log N)
T(n) = 2T(n/2) + O(n)

 

 

• Average Case

# O(N log N)
T(n) = T(n/9) + T(9n/10) + O(n)

 

2.3 3-Way Quicksort

In 3 Way QuickSort, an array arr[l..r] is divided into 3 parts:

• a) arr[l..i] elements less than pivot.
  
• b) arr[i+1..j-1] elements equal to pivot.
  
• c) arr[j..r] elements greater than pivot.

3. Reference

https://en.wikipedia.org/wiki/Quicksort

https://www.geeksforgeeks.org/quick-sort/

https://www.youtube.com/watch?v=PgBzjlCcFvc&feature=emb_title

https://en.wikipedia.org/wiki/Master_theorem

https://www.geeksforgeeks.org/3-way-quicksort-dutch-national-flag/

https://www.geeksforgeeks.org/quicksort-on-singly-linked-list/

https://www.geeksforgeeks.org/quicksort-for-linked-list/

https://www.geeksforgeeks.org/quicksort-tail-call-optimization-reducing-worst-case-space-log-n/

https://www.geeksforgeeks.org/analysis-of-algorithms-set-2-asymptotic-analysis/