# QuickSort Complete Tutorial | Example | Algorithm | Programming | Complexity

In this tutorial, I will explain the QuickSort Algorithm in detail with the help of an example, algorithm and programming. To find out the efficiency of this algorithm as compared to other sorting algorithms, at the end of this article, you will also learn to calculate complexity.

### Introduction to QuickSort:

What is a Quick Sort?

Quick Sort is based on the concept of divide-and-conquer, just the same as merge sort. The basic idea of quicksort is to pick an element called the pivot element and partition the array. The quicksort algorithm is also known as a partition-exchange algorithm.

The partition in quicksort divides the given array into 3 parts:

• Elements less than the pivot element
• Pivot element
• Elements greater than the pivot element

You might be wondering what a pivot element is. A pivot element is that element that is used to partition the array, i.e. it can be the last element, the first element, or any random element.

For example, Consider an array of elements.

`X= {72, 23, 12, 65, 87, 52, 33, 46}`

Here we pick up 46 as the pivot element (i.e. the last element in the array).

Now, after the first pass, the array will look like this-

`X= {23, 12, 33, 46, 87, 52, 72, 65}`

Here, the pivot is set at its position while all the elements that are smaller than the pivot are placed to its left, and the elements greater than the pivot are placed to its right.

So, at the instant {23, 12, 33} and {87, 52, 72, 65} are two sub-arrays and the recursive logic of partitioning elements (dividing the array) with the help of pivot element and keep doing this until and unless the array is sorted completely.

### How does Quick Sort work?

1. Pick an element, called a pivot, or refer to it as the partitioning element, from the array. For instance, let us consider the last element as a pivot.
2. Partition: rearrange the array such that all elements with values less than the pivot come before the pivot (i.e., on the left of the pivot), while all elements with values greater than the pivot come after it (i.e., on the right of pivot). The pivot element will be in its final position after it.
3. Recursively apply the above steps to the sub-array of elements on the left side of the pivot and on the right side of the pivot to get a sorted array.

Now, using pictorial representation, let’s try to sort an array that has initial values as X= {75, 26, 15, 67, 54, 31, 49}.

Here, in the above representation, we select the element at the last index of the array as a pivot, which is 49 and then call partition() to thereby re-arrange the elements of the array in a way that elements less than 49 are before it in and elements greater than 49 are after it.

Then, after the first pass, we consider the subarrays at left and right and select a partition element for them. Here, in the above representation, we select 31 as a pivot from the left array and 67 as a pivot from the right array. And once again call the partition() to split the array.

Thus, we recursively keep partitioning the array with the help of pivot unless we obtain a sorted array.

### How to implement the QuickSort algorithm?

The below function can be used as a recursive approach to sort elements using Quick Sort.

```QuickSort(A,start,end){
if(start >= end){
//calling partition function
pIndex <- Partition(A,start,end);
QuickSort(A, start, pIndex-1);
QuickSort(A,pIndex+1,end);
}
}```

The below function describes how partitioning is performed using the pivot element–

```Partition(A, start, end){
pivot <- A[end];
pIndex <- start;
for i<- start to end
{
if(A[i] <= pivot){
swap(A[i], A[pIndex]); // a function to swap the elements
pIndex <- pIndex + 1;
}
}
swap(A[pIndex],A[end]);
return pIndex;
}```

### QuickSort Programming:

If you understand the workings of the QuickSort algorithm, it is not very difficult to convert the QuickSort algorithm into code for different programming languages.

I have added comments in the code wherever needed. Most of the code is self-explanatory.

#### C/C++ Programming:

 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 `/* C program for QuickSort implementation*/` `#include` `//swapElement is utility function to swap two elements` `void` `swapElements(``int``* a, ``int``* b)` `{` `    ``int` `t = *a;` `    ``*a = *b;` `    ``*b = t;` `}` `//This fucntuin puts all the smaller elements at left ` `//and bigger elements at right of the pivot element ` `//in the array` `int` `partition (``int` `nArray[], ``int` `nStart, ``int` `nEnd)` `{` `        ``//Choosing last element as pivot` `    ``int` `pivot = nArray[nEnd];` `        ``//Index of smaller element ` `    ``int` `i = (nStart - 1); ` `    ``for` `(``int` `j = nStart; j <= nEnd- 1; j++)` `    ``{` `        ``// If current element is smaller than or` `        ``// equal to pivot` `        ``if` `(nArray[j] <= pivot)` `        ``{` `            ``i++; ``// increment index of smaller element` `            ``swapElements(&nArray[i], &nArray[j]);` `        ``}` `    ``}` `    ``swapElements(&nArray[i + 1], &nArray[nEnd]);` `    ``return` `(i + 1);` `}` `/* Function to QuickSort the array` `nArray[] - Input array to sort,` `nStart - Starting index of nArray,` `nEnd - Ending index of nArray */` `void` `QSort(``int` `nArray[], ``int` `nStart, ``int` `nEnd)` `{` `    ``if` `(nStart < nEnd)` `    ``{` `        ``//Here, pivot is partitioning index, ` `        ``//nArray[pivot] is now at right place` `        ``int` `pivot = partition(nArray, nStart, nEnd);` `        ``//Recursive call to QuickSort the elements ` `        ``//in the array after partition` `        ``QSort(nArray, nStart, pivot - 1);` `        ``QSort(nArray, pivot + 1, nEnd);` `    ``}` `}` `//Simple function to print ` `//the elements of an array` `void` `printArray(``int` `nArray[], ``int` `size)` `{` `    ``int` `i;` `    ``for` `(i=0; i < size; i++)` `        ``printf``(``"%d "``, nArray[i]);` `}` `// Main function for QuickSort` `int` `main()` `{` `    ``int` `arrSample[] = {75,26,15,67,85,54,31,49};` `    ``int` `nSize = ``sizeof``(arrSample)/``sizeof``(arrSample[0]);` `    ` `    ``printf``(``"\nInitial Array to Sort: "``);` `    ``printArray(arrSample, nSize);` `    ` `    ``QSort(arrSample, 0, nSize-1);` `    ` `    ``printf``(``"\nArray after QuickSort: "``);` `    ``printArray(arrSample, nSize);` `    ``return` `0;` `}`

Output of C/C++ QuickSort Program

```Initial Array to Sort: 75 26 15 67 85 54 31 49
Array after QuickSort: 15 26 31 49 54 67 75 85```

#### Python Programming:

 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 `# Python program for Quicksort` `#This fucntuin puts all the smaller elements at left ` `#and bigger elements at right of the pivot element ` `#in the array` `def` `partition(nArray,nStart,nEnd):` `    ``# index of smaller element` `    ``i ``=` `( nStart``-``1` `)         ` `    ``# pivot` `    ``pivot ``=` `nArray[nEnd]     ` `    ``for` `j ``in` `range``(nStart , nEnd):` `        ``# If current element is smaller than or` `        ``# equal to pivot` `        ``if` `nArray[j] <``=` `pivot:` `        ` `            ``# increment index of smaller element` `            ``i ``=` `i``+``1` `            ``nArray[i],nArray[j] ``=` `nArray[j],nArray[i]` `    ``nArray[i``+``1``],nArray[nEnd] ``=` `nArray[nEnd],nArray[i``+``1``]` `    ``return` `( i``+``1` `)` `#Function to QuickSort the array` `#nArray[] - Input array to sort,` `#nStart - Starting index of nArray,` `#nEnd - Ending index of nArray */` `def` `QSort(nArray,nStart,nEnd):` `    ``if` `nStart < nEnd:` `        ``# pivot is partitioning index, ` `        ``#nArray[pivot] is now at right place` `        ``pivot ``=` `partition(nArray,nStart,nEnd)` `        ``#Recursive call to QuickSort the elements ` `        ``#in the array after partition` `        ``QSort(nArray, nStart, pivot``-``1``)` `        ``QSort(nArray, pivot``+``1``, nEnd)` `#QuickSort program runs from here` `nArray ``=` `[``75``, ``26``, ``15``, ``67``, ``85``, ``54``, ``31``, ``49``]` `nSize ``=` `len``(nArray)` `print``(``"Initial Array to Sort:"``);` `for` `i ``in` `range``(nSize):` `    ``print` `(``"%d"` `%``nArray[i]),` `    ` `QSort(nArray,``0``,nSize``-``1``)` `print``(``"\nArray after QuickSort:"``)` `for` `i ``in` `range``(nSize):` `    ``print` `(``"%d"` `%``nArray[i]),`

Note: The `nArray` mentioned in the above Python program is Python list. So, don’t get confused with the name.

The output of the Python QuickSort Program

```Initial Array to Sort:
75 26 15 67 85 54 31 49
Array after QuickSort:
15 26 31 49 54 67 75 85
```

#### Java Programming:

Now, have a look at how quicksort is implemented using a Java program.

 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 `public` `class` `QSort {` `    ``void` `swap(``int` `a1,``int` `a2){` `        ``int` `temp = a1;` `        ``a1=a2;` `        ``a2=temp;` `     ``}` `     ``void` `QuickSort(``int` `A[],``int` `start,``int` `end){` `         ``if``(start <= end){` `             ``//calling partition function` `             ``int` `pIndex =Partition(A,start,end);` `              ``QuickSort(A, start, pIndex-``1``);` `              ``QuickSort(A,pIndex+``1``,end);` `         ``}` `    ``}` `    ``int` `Partition(``int` `A[],``int` `start,``int` `end){` `        ``int` `pivot= A[end];` `        ``// set the partition index as start initially` `        ``int` `pIndex = start;        ` `         ``for` `(``int` `i=start;i

Output of Java QuickSort Program:

`15 26 31 49 54 67 75 85`

#### Time Complexity of QuickSort:

The equation to calculate the time taken by the Quicksort to sort all the elements in the array can be formulated based on the size of the array.

In every partition, the array is divided into two subarrays.

Best Case Time Complexity:

Let’s take the best case:

The best case occurs when the middle element gets selected for every partition.  The array is get divided into two equal-sized subarrays.

So the time complexity can be formulated as,

`T(n) = T(n/2) + T(n/2) + Cn`

Where T(n) is the time taken by QuickSort to sort the array of size n elements.
C is some other constant.

So the equation becomes

`T(n) = 2T(n/2) + Cn`

The above recursive formula can be resolved in time `O(nlog(n)`.

Worst Case Time Complexity:

Worst-case complexity occurs when the pivot elements get selected at the end of the array every time while partitioning.

So the array will get partitioned into two subarrays having elements (n-1) and 1.

So the time complexity equation in the worst case is

`T(n) = T(n-1) + T(1) + C`

After solving the above recursive function, it is equivalent to

`T(n) = O(n^2)`

Note: Unlike all the major sorting algorithms, QuickSort takes more time if the input array is already sorted.

Average Case Time Complexity:

In the average case, you have to consider all the possible positions for choosing a pivot element. It is not easy to calculate.

If we calculate the recursive solution for all the possible pivots, it is equivalent to `O(nLog(n))`,So the complexity.

#### Other Sorting Algorithms you should try to improve your coding skills.

• Write a program to check if the array is sorted. (C/C++)
• Selection Sort (PythonC/C++)
• Bubble Sort (C/C++)

I would recommend trying to implement the quicksort algorithm yourself. You can always refer to the code mentioned in this tutorial. Let us know if you have any queries.

Happy Learning!

1. David G. Pickett says:

In a C++/OO world, one would want a quicksort equipped sorting container object.

Downsides to quicksort are the lack of parallel processing of partial input at input time. Tree containers sort as items are added, for instance, so when the last item is added, the result is immediately available.

There are processing situations where you want to start draining sorted data before all input is written, which tree containers support, although if you inserted an item before the point in the tree where output is occurring, it is not included. This is only possible if the data is somewhat sorted already, like trades coming from threads running trade engines for many different symbols sorting by time or trade order, where one engine/thread might be slightly ahead of another. You could pull the data that was 15 second old only, so the sort has complete control of the data being sorted.

1. Interesting! Thanks, David for putting your thoughts. You said it right.

What I think, we can enforce paralleling processing as the subarrays you get after partition are independent. But, in the case of partial input with QuickSort, we can not ensure sorting with minimal effort. We have to deal with the array complete data array. In fact, QuicSort takes more time when input elements are already sorted.

There is always ups and downsides to each sorting algorithm. Choosing a better one is solely depends on the input or situation you are working on.

2. Glen Wood says:

2022 Thank you very much for what you do. I’m a stundent from Chile. 🙂