Quicksort is a sorting algorithm based on the divide and conquer approach where An array is divided into subarrays by selecting a pivot element (element selected from the array). There are many different versions of quickSort that pick pivot in different ways.

While dividing the array, the pivot element should be positioned in such a way that elements less than pivot are kept on the left side and elements greater than pivot are on the right side of the pivot. The left and right subarrays are also divided using the same approach. This process continues until each subarray contains a single element. At this point, elements are already sorted. Finally, elements are combined to form a sorted array.

Gayle Laakmann McDowell's Explanation on QuickSort

The pivot value divides the list into two parts. And recursively, we find the pivot for each sub-lists until all lists contains only one element.

Quick Sort Pivot Algorithm

Based on our understanding of partitioning in quick sort, we will now try to write an algorithm for it, which is as follows.

Step 1 − Choose the highest index value has pivot

Step 2 − Take two variables to point left and right of the list excluding pivot

Step 3 − left points to the low index

Step 4 − right points to the high

Step 5 − while value at left is less than pivot move right

Step 6 − while value at right is greater than pivot move left

Step 7 − if both step 5 and step 6 does not match swap left and right

Step 8 − if left ≥ right, the point where they met is new pivot

Quick Sort Algorithm

Using pivot algorithm recursively, we end up with smaller possible partitions. Each partition is then processed for quick sort. We define recursive algorithm for quicksort as follows −

Step 1 − Make the right-most index value pivot

Step 2 − partition the array using pivot value

Step 3 − quicksort left partition recursively

Step 4 − quicksort right partition recursively

Implementations

Quick Sort - Java Implementation

Quick Sort - Python Implementation

Analysis of QuickSort

Time taken by QuickSort, in general, can be written as follows.

T(n) = T(k) + T(n-k-1) + \theta (n)

The first two terms are for two recursive calls, the last term is for the partition process. k is the number of elements that are smaller than the pivot. The time taken by QuickSort depends upon the input array and partition strategy. Following are three cases.

Worst Case:

The worst case occurs when the partition process always picks the greatest or smallest element as the pivot. If we consider the above partition strategy where the last element is always picked as a pivot, the worst case would occur when the array is already sorted in increasing or decreasing order. Following is recurrence for the worst case.

T(n) = T(0) + T(n-1) + \theta (n)which is equivalent to T(n) = T(n-1) + \theta (n)

The solution to the above recurrence is (n2).

Best Case:

The best case occurs when the partition process always picks the middle element as the pivot. The following is recurrence for the best case.

T(n) = 2T(n/2) + \theta (n)

The solution for the above recurrence is (nLogn). It can be solved using case 2 of Master Theorem.

Average Case:

To do average case analysis, we need to consider all possible permutation of array and calculate time taken by every permutation which doesn’t look easy. We can get an idea of average case by considering the case when partition puts O(n/9) elements in one set and O(9n/10) elements in other set. Following is recurrence for this case.

T(n) = T(n/9) + T(9n/10) + \theta (n)

The solution of above recurrence is also O(nLogn):

Although the worst case time complexity of QuickSort is O(n2) which is more than many other sorting algorithms like Merge Sort and Heap Sort, QuickSort is faster in practice, because its inner loop can be efficiently implemented on most architectures, and in most real-world data. QuickSort can be implemented in different ways by changing the choice of pivot, so that the worst case rarely occurs for a given type of data. However, merge sort is generally considered better when data is huge and stored in external storage.

Is QuickSort stable?

The default implementation is not stable. However any sorting algorithm can be made stable by considering indexes as comparison parameter.

Is QuickSort In-place?

As per the broad definition of in-place algorithm it qualifies as an in-place sorting algorithm as it uses extra space only for storing recursive function calls but not for manipulating the input.

What is 3-Way QuickSort?

In simple QuickSort algorithm, we select an element as pivot, partition the array around pivot and recur for subarrays on left and right of pivot. Consider an array which has many redundant elements. For example, {1, 4, 2, 4, 2, 4, 1, 2, 4, 1, 2, 2, 2, 2, 4, 1, 4, 4, 4}. If 4 is picked as pivot in Simple QuickSort, we fix only one 4 and recursively process remaining occurrences. In 3 Way QuickSort, an array arr[l..r] is divided in 3 parts:

arr[l..i] elements less than pivot. arr[i+1..j-1] elements equal to pivot. arr[j..r] elements greater than pivot. See this for implementation.

How to implement QuickSort for Linked Lists?

QuickSort on Singly Linked List QuickSort on Doubly Linked List

Can we implement QuickSort Iteratively?

Yes, please refer Iterative Quick Sort.

Why Quick Sort is preferred over MergeSort for sorting Arrays ?

Quick Sort in its general form is an in-place sort (i.e. it doesn’t require any extra storage) whereas merge sort requires O(N) extra storage, N denoting the array size which may be quite expensive. Allocating and de-allocating the extra space used for merge sort increases the running time of the algorithm. Comparing average complexity we find that both type of sorts have O(NlogN) average complexity but the constants differ. For arrays, merge sort loses due to the use of extra O(N) storage space. Most practical implementations of Quick Sort use randomized version. The randomized version has expected time complexity of O(nLogn). The worst case is possible in randomized version also, but worst case doesn’t occur for a particular pattern (like sorted array) and randomized Quick Sort works well in practice. Quick Sort is also a cache friendly sorting algorithm as it has good locality of reference when used for arrays. Quick Sort is also tail recursive, therefore tail call optimizations is done.

Why MergeSort is preferred over QuickSort for Linked Lists ?

In case of linked lists the case is different mainly due to difference in memory allocation of arrays and linked lists. Unlike arrays, linked list nodes may not be adjacent in memory. Unlike array, in linked list, we can insert items in the middle in O(1) extra space and O(1) time. Therefore merge operation of merge sort can be implemented without extra space for linked lists. In arrays, we can do random access as elements are continuous in memory. Let us say we have an integer (4-byte) array A and let the address of A[0] be x then to access A[i], we can directly access the memory at (x + i*4). Unlike arrays, we can not do random access in linked list. Quick Sort requires a lot of this kind of access. In linked list to access i’th index, we have to travel each and every node from the head to i’th node as we don’t have continuous block of memory. Therefore, the overhead increases for quick sort. Merge sort accesses data sequentially and the need of random access is low.

How to optimize QuickSort so that it takes O(Log n) extra space in worst case?

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

Reference:

https://www.programiz.com/dsa/quick-sort https://www.geeksforgeeks.org/quick-sort/?ref=lbp https://www.tutorialspoint.com/data_structures_algorithms/quick_sort_algorithm.htm