**Quicksort in C++ With Illustration.**

Quicksort is a widely used sorting algorithm which selects a specific element called “pivot” and partitions the array or list to be sorted into two parts based on this pivot s0 that the elements lesser than the pivot are to the left of the list and the elements greater than the pivot are to the right of the list.

Thus the list is partitioned into two sublists. The sublists may not be necessary for the same size. Then Quicksort calls itself recursively to sort these two sublists.

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Table of Contents:

### Introduction

Quicksort works efficiently as well as faster even for larger arrays or lists.

In this tutorial, we will explore more about the working of Quicksort along with some programming examples of the quicksort algorithm.

As a pivot value, we can choose either first, last or the middle value or any random value. The general idea is that ultimately the pivot value is placed at its proper position in the array by moving the other elements in the array to the left or right.

### General Algorithm

**The general algorithm for Quicksort is given below.**

quicksort(A, low, high) begin Declare array A[N] to be sorted low = 1st element; high = last element; pivot if(low < high) begin pivot = partition (A,low,high); quicksort(A,low,pivot-1) quicksort(A,pivot+1,high) End end

Let us now take a look at the pseudocode for Quicksort technique.

### Pseudo Code For Quicksort

//pseudocode for quick sort main algorithm procedure quickSort(arr[], low, high) arr = list to be sorted low – first element of array high – last element of array begin if (low < high) { // pivot – pivot element around which array will be partitioned pivot = partition(arr, low, high); quickSort(arr, low, pivot - 1); // call quicksort recursively to sort sub array before pivot quickSort(arr, pivot + 1, high); // call quicksort recursively to sort sub array after pivot } end procedure //partition procedure selects the last element as pivot. Then places the pivot at the correct position in //the array such that all elements lower than pivot are in the first half of the array and the //elements higher than pivot are at the higher side of the array. procedure partition (arr[], low, high) begin // pivot (Element to be placed at right position) pivot = arr[high]; i = (low - 1) // Index of smaller element for j = low to high { 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) end procedure

**The working of the partitioning algorithm is described below using an example.**

In this illustration, we take the last element as pivot. We can see that the array is successively divided around the pivot element until we have a single element in the array.

Now we present an illustration of the Quicksort below to better understand the concept.

### Illustration

Let us see an illustration of the quicksort algorithm. Consider the following array with the last element as pivot. Also, the first element is labeled low and the last element is high.

From the illustration, we can see that, we move the pointers high and low at both the ends of the array. Whenever low points to the element greater than the pivot and high points to the element lesser than the pivot, then we exchange the positions of these elements and advance the low and high pointers in their respective directions.

This is done until the low and high pointers cross each other. Once they cross each other the pivot element is placed at its proper position and the array is partitioned into two. Then both these sub-arrays are sorted independently using quicksort recursively.

### C++ Example

**Given below is the implementation of the Quicksort algorithm in C++.**

#include <iostream> using namespace std; // Swap two elements - Utility function void swap(int* a, int* b) { int t = *a; *a = *b; *b = t; } // partition the array using last element as pivot int partition (int arr[], int low, int high) { int pivot = arr[high]; // pivot int i = (low - 1); for (int j = low; j <= high- 1; j++) { //if current element is smaller than pivot, increment the low element //swap elements at i and j if (arr[j] <= pivot) { i++; // increment index of smaller element swap(&arr[i], &arr[j]); } } swap(&arr[i + 1], &arr[high]); return (i + 1); } //quicksort algorithm void quickSort(int arr[], int low, int high) { if (low < high) { //partition the array int pivot = partition(arr, low, high); //sort the sub arrays independently quickSort(arr, low, pivot - 1); quickSort(arr, pivot + 1, high); } } void displayArray(int arr[], int size) { int i; for (i=0; i < size; i++) cout<<arr[i]<<"\t"; } int main() { int arr[] = {12,23,3,43,51,35,19,45}; int n = sizeof(arr)/sizeof(arr[0]); cout<<"Input array"<<endl; displayArray(arr,n); cout<<endl; quickSort(arr, 0, n-1); cout<<"Array sorted with quick sort"<<endl; displayArray(arr,n); return 0; }

**Output:**

Input array

12 23 3 43 51 35 19 45

Array sorted with quicksort

3 12 19 23 35 43 45 51

Here we have few routines that are used to partition the array and call quicksort recursively to sort the partition, basic quicksort function, and utility functions to display the array contents and swap the two elements accordingly.

First, we call the quicksort function with the input array. Inside the quicksort function, we call the partition function. In the partition function, we use the last element as the pivot element for the array. Once the pivot is decided, the array is partitioned into two parts and then the quicksort function is called to independently sort both the sub arrays.

When the quicksort function returns, the array is sorted such that the pivot element is at its correct location and the elements lesser than the pivot is at the left of the pivot and the elements greater than the pivot is at the right of the pivot.

Next, we will implement the quicksort algorithm in Java.

### Java Example

// Quicksort implementation in Java class QuickSort { //partition the array with last element as pivot int partition(int arr[], int low, int high) { int pivot = arr[high]; int i = (low-1); // index of smaller element for (int j=low; j<high; j++) { // If current element is smaller than or // equal to pivot,then swap the elements if (arr[j] <= pivot) { i++; int temp = arr[i]; arr[i] = arr[j]; arr[j] = temp; } } // swap arr[i+1] and arr[high] (or pivot) int temp = arr[i+1]; arr[i+1] = arr[high]; arr[high] = temp; return i+1; } //quicksort function void quick_sort(int arr[], int low, int high) { if (low < high) { //partition the array around pivot int pivot = partition(arr, low, high); //call quick_sort recursively to sort sub arrays quick_sort(arr, low, pivot-1); quick_sort(arr, pivot+1, high); } } //display the contents of array static void displayArray(int arr[]) { int n = arr.length; for (int i=0; i<n; ++i) System.out.print(arr[i]+" "); System.out.println(); } } class Main{ public static void main(String args[]) { int arr[] = {12,23,3,43,51,35,19,45}; int n = arr.length; QuickSort obj = new QuickSort(); System.out.println("Input array"); obj.displayArray(arr); obj.quick_sort(arr, 0, n-1); System.out.println("Array after sorting with quick sort"); obj.displayArray(arr); } }

**Output:**

Input array

12 23 3 43 51 35 19 45

Array after sorting with quicksort

3 12 19 23 35 43 45 51

In the Java implementation as well, we have used the same logic that we used in C++ implementation. We have used the last element in the array as the pivot and quicksort is performed on the array in order to place the pivot element at its proper position.

### Complexity Analysis Of The Quicksort Algorithm

The time taken by quicksort to sort an array depends on the input array and partition strategy or method.

If k is the number of elements less than the pivot and n is the total number of elements, then the general time taken by quicksort can be expressed as follows:

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

Here, T(k) and T(n-k-1) are the time taken by recursive calls and O(n) is the time taken by partitioning call.

Let us analyze this time complexity for quicksort in detail.

**#1) Worst case**: Worst case in quicksort technique occurs mostly when we select the lowest or highest element in the array as a pivot. (In the above illustration we have selected the highest element as the pivot). In such a situation worst case occurs when the array to be sorted is already sorted in ascending or descending order.

**Hence the above expression for total time taken changes as:**

** T(n) = T(0) + T(n-1) + O(n) **that resolves to** O(n ^{2})**

**#2) Best case: **The best case for quicksort always occurs when the pivot element selected is the middle of the array.

**Thus the recurrence for the best case is:**

** T(n) = 2T(n/2) + O(n) = O(nlogn)**

**#3) Average case: **To analyze the average case for quicksort, we should consider all the array permutations and then calculate the time taken by each of these permutations. In a nutshell, the average time for quicksort also becomes O(nlogn).

**Given below are the various complexities for Quicksort technique:**

Worst case time complexity | O(n 2 ) |

Best case time complexity | O(n*log n) |

Average time complexity | O(n*log n) |

Space complexity | O(n*log n) |

We can implement quicksort in many different ways just by changing the choice of the pivot element (middle, first or last), however, the worst-case rarely occurs for quicksort.

### 3-way Quicksort

In original quicksort technique, we usually select a pivot element and then divide the array into sub-arrays around this pivot so that one sub-array consists of elements less than the pivot and another consists of elements greater than the pivot.

But what if we select a pivot element and there is more than one element in the array that is equal to pivot?

**For Example,** consider the following array {5,76,23,65,4,4,5,4,1,1,2,2,2,2}. If we perform a simple quicksort on this array and select 4 as a pivot element, then we will fix only one occurrence of element 4 and the rest will be partitioned along with the other elements.

**Instead, if we use 3-way quicksort, then we will divide the array [l…r] into three sub-arrays as follows:**

- Array[l…i] – Here, i is the pivot and this array contains elements less than the pivot.
- Array[i+1…j-1] – Contains the elements that are equal to the pivot.
- Array[j…r] – Contains elements greater than the pivot.

Thus 3-way quicksort can be used when we have more than one redundant element in the array.

### Randomized Quicksort

The quicksort technique is called randomized quicksort technique when we use random numbers to select the pivot element. In randomized quicksort, it is called “central pivot” and it divides the array in such a way that each side has at-least ¼ elements.

**The pseudo-code for randomized quicksort is given below:**

// Sorts an array arr[low..high] using randomized quick sort randomQuickSort(array[], low, high) array – array to be sorted low – lowest element in array high – highest element in array begin 1. If low >= high, then EXIT. //select central pivot 2. While pivot 'pi' is not a Central Pivot. (i) Choose uniformly at random a number from [low..high]. Let pi be the randomly picked number. (ii) Count elements in array[low..high] that are smaller than array[pi]. Let this count be a_low. (iii) Count elements in array[low..high] that are greater than array[pi]. Let this count be a_high. (iv) Let n = (high-low+1). If a_low >= n/4 and a_high >= n/4, then pi is a central pivot. //partition the array 3. Partition array[low..high] around the pivot pi. 4. // sort first half randomQuickSort(array, low, a_low-1) 5. // sort second half randomQuickSort(array, high-a_high+1, high) end procedure

In the above code on “randomQuickSort”, in step # 2 we select the central pivot. In step 2, the probability of the selected element being the central pivot is ½. Hence the loop in step 2 is expected to run 2 times. Thus the time complexity for step 2 in randomized quicksort is O(n).

Using a loop to select the central pivot is not the ideal way to implement randomized quicksort. Instead, we can randomly select an element and call it central pivot or reshuffle the array elements. The expected worst-case time complexity for randomized quicksort algorithm is O(nlogn).

### Quicksort vs. Merge Sort

In this section, we will discuss the main differences between Quick sort and Merge sort.

Comparison Parameter | Quick sort | Merge sort |
---|---|---|

partitioning | The array is partitioned around a pivot element and is not necessarily always into two halves. It can be partitioned in any ratio. | The array is partitioned into two halves(n/2). |

Worst case complexity | O(n 2 ) – a lot of comparisons are required in the worst case. | O(nlogn) – same as the average case |

Data sets usage | Cannot work well with larger data sets. | Works well with all the datasets irrespective of size. |

Additional space | In-place – doesn’t need additional space. | Not in- place – needs additional space to store auxiliary array. |

Sorting method | Internal – data is sorted in the main memory. | External – uses external memory for storing data arrays. |

Efficiency | Faster and efficient for small size lists. | Fast and efficient for larger lists. |

stability | Not stable as two elements with the same values will not be placed in the same order. | Stable – two elements with the same values will appear in the same order in the sorted output. |

Arrays/linked lists | More preferred for arrays. | Works well for linked lists. |

### Conclusion

As the name itself suggests, quicksort is the algorithm that sorts the list quickly than any other sorting algorithms. Just like merge sort, quick sort also adopts a divide and conquer strategy.

As we have already seen, using quick sort we divide the list into sub-arrays using the pivot element. Then these sub-arrays are independently sorted. At the end of the algorithm, the entire array is completely sorted.

Quicksort is faster and works efficiently for sorting data structures. Quicksort is a popular sorting algorithm and sometimes is even preferred over merge sort algorithm.

**In our next tutorial, we will discuss more on Shell sort in detail.**

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