Quick Sort

Published on Thursday, 14 September 2017

Based on the idea that we can partition an array around a pivot (chosen element of the array), we can put the smaller (than pivot) elements on the left side and place greater elements on the right side of the pivot, then, we keep doing the same thing for the left side and the right side. More details of how this algorithm works is on "How it works" section.

quick sort with pivot figure

Fig: fundamental concept of Quick-Sort, partitioning

Quick sort, one of the fundamental sorting algorithms of computer science sorts an array efficiently on average.

A few important things about Quick Sort,

  • expected running time θ(n lg n)
  • sorts in place (does not use additional memory)
  • works well even in virtual environment
  • worst case does not depend on any particular input

Description of the Algorithm

Steps,

  • Divide
  • Conquer
  • Combine

Divide

Partition of the array A[pr..r] into 2 parts, A[p .. q-1] and A[q-1 .. r]. Index q comes from the partitioning procedure.

Pivot can be chosen in a number of ways,

Conquer

We start with partitioning the array. We sort 2 sub-arrays recursively by calling quick sort on partitions.

Combine

Each recursive call to 'quick-sort' sorts the particular segment of the array. After recursive calls return the array is already sorted. Therefore, no action is required to be taken on combine step.

How it works

The idea is that we choose a pivot element from the array. We put all elements less than the pivot to the left side and we put all elements greater than the pivot item to the right side. We consider the left side and the right side as 2 sub-arrays where we apply the same procedure. After recursively applying the mechanism in the end, the array is sorted.

Worst Case of Quick Sort

Worst case happens if every call of partition the chosen pivot creates a partition with (n-1) element and 1 element. This can be the case when an array is sorted in reverse order.

For example an array with 5 elements,
 5 4 3 2 1

Steps can be visualized below,
1. QuickSort(A, 0, 4)

Partition procedure chooses A[4] = 1 as pivot. Nothing is smaller than 1 so the
iteration/traversing element does nothing. In the end, it swaps A[4] and A[0]. 
At this stage, the array looks like,
 1 4 3 2 5

and the procedure returns 0. After partition function returns two calls are spawned,
 QuickSort(A, 0, -1)
 QuickSort(A, 1, 4)

2. QuickSort(A, 0, -1) does nothing
3. QuickSort(A, 1, 4) has following input,
 4 3 2 5

its partition procedure chooses A[4] as pivot and returns 4. All elements swap
with themselves; array stays unchanged.

It spawns two quick sorts.
 QuickSort(A, 1, 3)
 QuickSort(A, 5, 4)

4. QuickSort(A, 1, 3) has input as,
 4 3 2

Partition does a swap in the end and leaves 2 in index 1. It returns 1.
it looks like following then,
 2 3 4
 
It spawns two quick sort calls.
 QuickSort(A, 1, 0)
 QuickSort(A, 2, 3)
 
5. QuickSort(A, 5, 4) does nothing and the array is sorted.
6. QuickSort(A, 1, 0) takes no action.
7. QuickSort(A, 2, 3) Partition takes index 3 as pivot. Array stays
unchanged. 3 is returned by Partition. 2 calls are spawned,
 QuickSort(A, 2, 2)
 QuickSort(A, 4, 3)


Worst case of quick sort depends on the partitioning. As we can see index
returned by partition function are,
 0, 4, 1, 3, 2

It means instead pivot dividing the array into two parts it has traversed through every index of the array. Therefore, we get O(N) for each call of quick sort. pivot is selected n times. This gives overall O(n^2) time for quick sort.

The Generic Algorithm written in book, Intro to Algo by C.L.R.S can be applied to sort a part of an array specifying starting index and the ending index. Usually when I write a sorting algorithm we apply to the entire array calling like this,

QuickSort(A);

Usually in C# or Java languages there is a Length attribute of the array, C has sizeof. Another example input,

5
1 4 2 4 3

Primary References

  1. From the book, Introduction to Algorithm - C.L.R.S
  • Performance analysis on Section# 7.2
  • Random Sampling on Section# 7.3
  • Worst case analysis on Section# 7.4
  1. CS Stony Brook Dr Rezaul's Lecture
  2. Cracking the Coding Interview, 3rd ed. - Gayle Mcdowell