Know Your Sorting Algorithm | Set 2 (Introsort- C++’s Sorting Weapon)

12 Apr, 2022
We have discussed sorting weapons used by different languages in previous article. In this article, C++’s Sorting Weapon, Introsort is discussed. What is Introsort? 
Simply putting, it is the best sorting algorithm around. It is a hybrid sorting algorithm, which means that it uses more than one sorting algorithms as a routine. 
Which standard sorting algorithms are used in Introsort 
Introsort being a hybrid sorting algorithm uses three sorting algorithm to minimise the running time, Quicksort, Heapsort and Insertion Sort 
How does it work? 
Introsort begins with quicksort and if the recursion depth goes more than a particular limit it switches to Heapsort to avoid Quicksort’s worse case O(N2) time complexity. It also uses insertion sort when the number of elements to sort is quite less. So first it creates a partition. Three cases arises from here.
If the partition size is such that there is a possibility to exceed the maximum depth limit then the Introsort switches to Heapsort. We define the maximum depth limit as 2*log(N)If the partition size is too small then Quicksort decays to Insertion Sort. We define this cutoff as 16 (due to research). So if the partition size is less than 16 then we will do insertion sort.If the partition size is under the limit and not too small (i.e- between 16 and 2*log(N)), then it performs a simple quicksort.
If the partition size is such that there is a possibility to exceed the maximum depth limit then the Introsort switches to Heapsort. We define the maximum depth limit as 2*log(N)
If the partition size is too small then Quicksort decays to Insertion Sort. We define this cutoff as 16 (due to research). So if the partition size is less than 16 then we will do insertion sort.
If the partition size is under the limit and not too small (i.e- between 16 and 2*log(N)), then it performs a simple quicksort.
Why is it better than simple Quicksort or Why the need of Introsort? 
Since Quicksort can have a worse case O(N2) time complexity and it also increases the recursion stack space (O(log N) if tail recursion applied), so to avoid all these, we need to switch the algorithm from Quicksort to another if there is a chance of worse case. So Introsort solves this problem by switching to Heapsort. Also due to larger constant factor, quicksort can perform even worse than O(N2) sorting algorithm when N is small enough. So it switches to insertion sort to decrease the running time of sorting. Also if a bad pivot-selection is done then the quicksort does no better than the bubble-sort. 
Why is Insertion Sort used (and not Bubble Sort, etc)?
 Insertion sort offers following advantages.
It is a known and established fact that insertion sort is the most optimal comparison-based sorting algorithm for small arrays.It has a good locality of referenceIt is an adaptive sorting algorithm, i.e- it outperforms all the other algorithms if the array elements are partially sorted.
It is a known and established fact that insertion sort is the most optimal comparison-based sorting algorithm for small arrays.
It has a good locality of reference
It is an adaptive sorting algorithm, i.e- it outperforms all the other algorithms if the array elements are partially sorted.
Why is Heapsort used (and not Mergesort etc)? 
This is solely because of memory requirements. Merge sort requires O(N) space whereas Heapsort is an in-place O(1) space algorithm. 
Why is Heapsort not used in place of Quicksort when the partition size is under the limit ? 
This question is same as why Quicksort generally outperforms Heapsort ? The answer is, although Heapsort also being O(N log N) in average as well as worse case and O(1) space also, we still don’t use it when the partition size is under the limit because the extra hidden constant factor in Heapsort is quite larger than that of Quicksort. 
Why is cut-off 16 for switching from quick sort to insertion sort, and 2*logN for switching from quick sort to heap sort ? 
These values are chosen empirically as an approximate because of various tests and researches conducted. 
CPP
/* A Program to sort the array using Introsort.  The most popular C++ STL Algorithm- sort()  uses Introsort. */ #include<bits/stdc++.h>using namespace std; // A utility function to swap the values pointed by// the two pointersvoid swapValue(int *a, int *b){    int *temp = a;    a = b;    b = temp;    return;} /* Function to sort an array using insertion sort*/void InsertionSort(int arr[], int *begin, int *end){    // Get the left and the right index of the subarray    // to be sorted    int left = begin - arr;    int right = end - arr;     for (int i = left+1; i <= right; i++)    {        int key = arr[i];        int j = i-1;        /* Move elements of arr[0..i-1], that are          greater than key, to one position ahead          of their current position */        while (j >= left && arr[j] > key)        {            arr[j+1] = arr[j];            j = j-1;        }        arr[j+1] = key;   }    return;} // A function to partition the array and return// the partition pointint* Partition(int arr[], int low, int high){    int pivot = arr[high];    // pivot    int i = (low - 1);  // Index of smaller element     for (int j = low; j <= high- 1; j++)    {        // If current element is smaller than or        // equal to pivot        if (arr[j] <= pivot)        {            // increment index of smaller element            i++;             swap(arr[i], arr[j]);        }    }    swap(arr[i + 1], arr[high]);    return (arr + i + 1);}  // A function that find the middle of the// values pointed by the pointers a, b, c// and return that pointerint *MedianOfThree(int * a, int * b, int * c){    if (*a < *b && *b < *c)        return (b);     if (*a < *c && *c <= *b)        return (c);     if (*b <= *a && *a < *c)        return (a);     if (*b < *c && *c <= *a)        return (c);     if (*c <= *a && *a < *b)        return (a);     if (*c <= *b && *b <= *a)        return (b);} // A Utility function to perform intro sortvoid IntrosortUtil(int arr[], int * begin,                  int * end, int depthLimit){    // Count the number of elements    int size = end - begin;       // If partition size is low then do insertion sort    if (size < 16)    {        InsertionSort(arr, begin, end);        return;    }     // If the depth is zero use heapsort    if (depthLimit == 0)    {        make_heap(begin, end+1);        sort_heap(begin, end+1);        return;    }     // Else use a median-of-three concept to    // find a good pivot    int * pivot = MedianOfThree(begin, begin+size/2, end);     // Swap the values pointed by the two pointers    swapValue(pivot, end);    // Perform Quick Sort    int * partitionPoint = Partition(arr, begin-arr, end-arr);    IntrosortUtil(arr, begin, partitionPoint-1, depthLimit - 1);    IntrosortUtil(arr, partitionPoint + 1, end, depthLimit - 1);     return;} /* Implementation of introsort*/void Introsort(int arr[], int *begin, int *end){    int depthLimit = 2 * log(end-begin);     // Perform a recursive Introsort    IntrosortUtil(arr, begin, end, depthLimit);       return;} // A utility function to print an array of size nvoid printArray(int arr[], int n){   for (int i=0; i < n; i++)       printf("%d ", arr[i]);   printf("\n");} // Driver program to test Introsortint main(){    int arr[] = {3, 1, 23, -9, 233, 23, -313, 32, -9};    int n = sizeof(arr) / sizeof(arr[0]);     // Pass the array, the pointer to the first element and    // the pointer to the last element    Introsort(arr, arr, arr+n-1);    printArray(arr, n);     return(0);}
-313 -9 -9 1 3 23 23 32 233
Is Introsort stable ? 
Since Quicksort is also not stable so Introsort is also not stable. 
Time Complexity Best Case – O(N log N) Average Case- O(N log N) Worse Case- O(N log N) where, N = number of elements to be sorted. Auxiliary Space Just like quicksort, it may use O(log N) auxiliary recursion stack space. Know Your Sorting Algorithm | Set 2 (Introsort- C++’s Sorting Weapon) 
References https://en.wikipedia.org/wiki/Introsort 
This article is contributed by Rachit Belwariar. If you like GeeksforGeeks and would like to contribute, you can also write an article and mail your article to review-team@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks. Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above
callmeHK
dford
simmytarika5
Insertion Sort
Merge Sort
Quick Sort
Sorting
Sorting
Merge Sort
Writing code in comment?
Please use ide.geeksforgeeks.org,
generate link and share the link here.
Merge two sorted arrays
Count Inversions in an array | Set 1 (Using Merge Sort)
Sort an array of 0s, 1s and 2s | Dutch National Flag problem
Chocolate Distribution Problem
Find a triplet that sum to a given value
k largest(or smallest) elements in an array
sort() in Python
Merge Sort for Linked Lists
Longest Consecutive Subsequence
Python Program for Bubble Sort