Quicksort is an algorithm based on divide and conquer approach in which the array is split into subarrays and these sub-arrays are recursively called to sort the elements.

How QuickSort Works?

1. A pivot element is chosen from the array. You can choose any element from the array as the pivot element.
   Here, we have taken the rightmost (ie. the last element) of the array as the pivot element.

   

2. The elements smaller than the pivot element are put on the left and the elements greater than the pivot element are put on the right.
   

   

   The above arrangement is achieved by the following steps.

   1. A pointer is fixed at the pivot element. The pivot element is compared with the elements beginning from the first index. If the element greater than the pivot element is reached, a second pointer is set for that element.
   2. Now, the pivot element is compared with the other elements (a third pointer). If an element smaller than the pivot element is reached, the smaller element is swapped with the greater element found earlier.
      

      

   3. The process goes on until the second last element is reached.
      Finally, the pivot element is swapped with the second pointer.

      

   4. Now the left and right subparts of this pivot element are taken for further processing in the steps below.
3. Pivot elements are again chosen for the left and the right sub-parts separately. Within these sub-parts, the pivot elements are placed at their right position. Then, step 2 is repeated.
   

   

4. The sub-parts are again divided into smaller sub-parts until each subpart is formed of a single element.
5. At this point, the array is already sorted.

Quicksort uses recursion for sorting the sub-parts.

On the basis of Divide and conquer approach, quicksort algorithm can be explained as:

• Divide
  The array is divided into subparts taking pivot as the partitioning point. The elements smaller than the pivot are placed to the left of the pivot and the elements greater than the pivot are placed to the right.
• Conquer
  The left and the right subparts are again partitioned using the by selecting pivot elements for them. This can be achieved by recursively passing the subparts into the algorithm.
• Combine
  This step does not play a significant role in quicksort. The array is already sorted at the end of the conquer step.

You can understand the working of quicksort with the help of the illustrations below.

Quick Sort Algorithm

quickSort(array, leftmostIndex, rightmostIndex)
  if (leftmostIndex < rightmostIndex)
    pivotIndex <- partition(array,leftmostIndex, rightmostIndex)
    quickSort(array, leftmostIndex, pivotIndex)
    quickSort(array, pivotIndex + 1, rightmostIndex)

partition(array, leftmostIndex, rightmostIndex)
  set rightmostIndex as pivotIndex
  storeIndex <- leftmostIndex - 1
  for i <- leftmostIndex + 1 to rightmostIndex
  if element[i] < pivotElement
    swap element[i] and element[storeIndex]
    storeIndex++
  swap pivotElement and element[storeIndex+1]
return storeIndex + 1

Python Examples

/* Quick sort in Python */

/* Function to partition the array on the basis of pivot element */
def partition(array, low, high):

    /* Select the pivot element */
    pivot = array[high]
    i = low - 1

    /* Put the elements smaller than pivot on the left and greater 
       than pivot on the right of pivot */
    for j in range(low, high):
        if array[j] <= pivot:
            i = i + 1
            (array[i], array[j]) = (array[j], array[i])

    (array[i + 1], array[high]) = (array[high], array[i + 1])

    return i + 1


def quickSort(array, low, high):
    if low < high:

        /* Select pivot position and put all the elements smaller 
           than pivot on left and greater than pivot on right */
        pi = partition(array, low, high)

        /* Sort the elements on the left of pivot */
        quickSort(array, low, pi - 1)

        /* Sort the elements on the right of pivot */
        quickSort(array, pi + 1, high)


data = [8, 7, 2, 1, 0, 9, 6]
size = len(data)
quickSort(data, 0, size - 1)
print('Sorted Array in Ascending Order:')
print(data)

Quicksort Complexity

Time Complexities

• Worst Case Complexity [Big-O]: O(n2)
  It occurs when the pivot element picked is either the greatest or the smallest element.
  This condition leads to the case in which the pivot element lies in an extreme end of the sorted array. One sub-array is always empty and another sub-array contains n - 1 elements. Thus, quicksort is called only on this sub-array.
  However, the quick sort algorithm has better performance for scattered pivots.
• Best Case Complexity [Big-omega]: O(n*log n)
  It occurs when the pivot element is always the middle element or near to the middle element.
• Average Case Complexity [Big-theta]: O(n*log n)
  It occurs when the above conditions do not occur.

Space Complexity

The space complexity for quicksort is O(log n).

Quicksort Applications

Quicksort is implemented when

• the programming language is good for recursion
• time complexity matters
• space complexity matters