In this article, we will discuss how to implement QuickSort using random pivoting. In QuickSort we first partition the array in place such that all elements to the left of the pivot element are smaller, while all elements to the right of the pivot are greater that the pivot. Then we recursively call the same procedure for left and right subarrays.
Unlike merge sort, we don’t need to merge the two sorted arrays. Thus Quicksort requires lesser auxiliary space than Merge Sort, which is why it is often preferred to Merge Sort. Using a randomly generated pivot we can further improve the time complexity of QuickSort.
We have discussed at two popular methods for partitioning the arrays-Hoare’s vs Lomuto partition scheme
It is advised that the reader has read that article or knows how to implement the QuickSort using either of the two partition schemes.
Algorithm for random pivoting using Lomuto Partitioning
partition(arr[], lo, hi) pivot = arr[hi] i = lo // place for swapping for j := lo to hi – 1 do if arr[j] <= pivot then swap arr[i] with arr[j] i = i + 1 swap arr[i] with arr[hi] return i partition_r(arr[], lo, hi) r = Random Number from lo to hi Swap arr[r] and arr[hi] return partition(arr, lo, hi) quicksort(arr[], lo, hi) if lo < hi p = partition_r(arr, lo, hi) quicksort(arr, lo , p-1) quicksort(arr, p+1, hi)
Implementation using Lomuto Partitioning:
C++
// C++ implementation QuickSort // using Lomuto's partition Scheme. #include <cstdlib> #include <iostream> using namespace std; // This function takes last element // as pivot, places // the pivot element at its correct // position in sorted array, and // places all smaller (smaller than pivot) // to left of pivot and all greater // elements to right of pivot int partition( int arr[], int low, int high) { // pivot int pivot = arr[high]; // Index of smaller element int i = (low - 1); 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 (i + 1); } // Generates Random Pivot, swaps pivot with // end element and calls the partition function int partition_r( int arr[], int low, int high) { // Generate a random number in between // low .. high srand ( time (NULL)); int random = low + rand () % (high - low); // Swap A[random] with A[high] swap(arr[random], arr[high]); return partition(arr, low, high); } /* The main function that implements QuickSort arr[] --> Array to be sorted, low --> Starting index, high --> Ending index */ void quickSort( int arr[], int low, int high) { if (low < high) { /* pi is partitioning index, arr[p] is now at right place */ int pi = partition_r(arr, low, high); // Separately sort elements before // partition and after partition quickSort(arr, low, pi - 1); quickSort(arr, pi + 1, high); } } /* Function to print an array */ void printArray( int arr[], int size) { int i; for (i = 0; i < size; i++) printf ( "%d " , arr[i]); printf ( "\n" ); } // Driver Code int main() { int arr[] = { 10, 7, 8, 9, 1, 5 }; int n = sizeof (arr) / sizeof (arr[0]); quickSort(arr, 0, n - 1); printf ( "Sorted array: \n" ); printArray(arr, n); return 0; } |
Java
// Java program to illustrate // Randomised Quick Sort import java.util.*; class RandomizedQsort { // This Function helps in calculating // random numbers between low(inclusive) // and high(inclusive) static void random( int arr[], int low, int high) { Random rand= new Random(); int pivot = rand.nextInt(high-low)+low; int temp1=arr[pivot]; arr[pivot]=arr[high]; arr[high]=temp1; } /* This function takes last element as pivot, places the pivot element at its correct position in sorted array, and places all smaller (smaller than pivot) to left of pivot and all greater elements to right of pivot */ static int partition( int arr[], int low, int high) { // pivot is choosen randomly random(arr,low,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 if (arr[j] < pivot) { i++; // swap arr[i] and arr[j] 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 ; } /* The main function that implements QuickSort() arr[] --> Array to be sorted, low --> Starting index, high --> Ending index */ static void sort( int arr[], int low, int high) { if (low < high) { /* pi is partitioning index, arr[pi] is now at right place */ int pi = partition(arr, low, high); // Recursively sort elements before // partition and after partition sort(arr, low, pi- 1 ); sort(arr, pi+ 1 , high); } } /* A utility function to print array of size n */ static void printArray( int arr[]) { int n = arr.length; for ( int i = 0 ; i < n; ++i) System.out.print(arr[i]+ " " ); System.out.println(); } // Driver Code public static void main(String args[]) { int arr[] = { 10 , 7 , 8 , 9 , 1 , 5 }; int n = arr.length; sort(arr, 0 , n- 1 ); System.out.println( "Sorted array" ); printArray(arr); } } // This code is contributed by Ritika Gupta. |
Python3
# Python implementation QuickSort using # Lomuto's partition Scheme. import random ''' The function which implements QuickSort. arr :- array to be sorted. start :- starting index of the array. stop :- ending index of the array. ''' def quicksort(arr, start , stop): if (start < stop): # pivotindex is the index where # the pivot lies in the array pivotindex = partitionrand(arr,\ start, stop) # At this stage the array is # partially sorted around the pivot. # Separately sorting the # left half of the array and the # right half of the array. quicksort(arr , start , pivotindex - 1 ) quicksort(arr, pivotindex + 1 , stop) # This function generates random pivot, # swaps the first element with the pivot # and calls the partition function. def partitionrand(arr , start, stop): # Generating a random number between the # starting index of the array and the # ending index of the array. randpivot = random.randrange(start, stop) # Swapping the starting element of # the array and the pivot arr[start], arr[randpivot] = \ arr[randpivot], arr[start] return partition(arr, start, stop) ''' This function takes the first element as pivot, places the pivot element at the correct position in the sorted array. All the elements are re-arranged according to the pivot, the elements smaller than the pivot is places on the left and the elements greater than the pivot is placed to the right of pivot. ''' def partition(arr,start,stop): pivot = start # pivot # a variable to memorize where the i = start + 1 # partition in the array starts from. for j in range (start + 1 , stop + 1 ): # if the current element is smaller # or equal to pivot, shift it to the # left side of the partition. if arr[j] < = arr[pivot]: arr[i] , arr[j] = arr[j] , arr[i] i = i + 1 arr[pivot] , arr[i - 1 ] = \ arr[i - 1 ] , arr[pivot] pivot = i - 1 return (pivot) # Driver Code if __name__ = = "__main__" : array = [ 10 , 7 , 8 , 9 , 1 , 5 ] quicksort(array, 0 , len (array) - 1 ) print (array) # This code is contributed by soumyasaurav |
Sorted array: 1 5 7 8 9 10
Algorithm for random pivoting using Hoare Partitioning
partition(arr[], lo, hi) pivot = arr[lo] i = lo - 1 // Initialize left index j = hi + 1 // Initialize right index // Find a value in left side greater // than pivot do i = i + 1 while arr[i] pivot if i >= j then return j swap arr[i] with arr[j] partition_r(arr[], lo, hi) r = Random number from lo to hi Swap arr[r] and arr[lo] return partition(arr, lo, hi) quicksort(arr[], lo, hi) if lo < hi p = partition_r(arr, lo, hi) quicksort(arr, lo, p) quicksort(arr, p+1, hi)
Implementation using Hoare’s Partitioning:
C++
// C++ implementation of QuickSort // using Hoare's partition scheme #include <cstdlib> #include <iostream> using namespace std; // This function takes last element as // pivot, places the pivot element at // its correct position in sorted // array, and places all smaller // (smaller than pivot) to left of pivot // and all greater elements to right int partition( int arr[], int low, int high) { int pivot = arr[low]; int i = low - 1, j = high + 1; while ( true ) { // Find leftmost element greater than // or equal to pivot do { i++; } while (arr[i] < pivot); // Find rightmost element smaller than // or equal to pivot do { j--; } while (arr[j] > pivot); // If two pointers met if (i >= j) return j; swap(arr[i], arr[j]); } } // Generates Random Pivot, swaps pivot with // end element and calls the partition function // In Hoare partition the low element is selected // as first pivot int partition_r( int arr[], int low, int high) { // Generate a random number in between // low .. high srand ( time (NULL)); int random = low + rand () % (high - low); // Swap A[random] with A[high] swap(arr[random], arr[low]); return partition(arr, low, high); } // The main function that implements QuickSort // arr[] --> Array to be sorted, // low --> Starting index, // high --> Ending index void quickSort( int arr[], int low, int high) { if (low < high) { // pi is partitioning index, // arr[p] is now at right place int pi = partition_r(arr, low, high); // Separately sort elements before // partition and after partition quickSort(arr, low, pi); quickSort(arr, pi + 1, high); } } // Function to print an array void printArray( int arr[], int n) { for ( int i = 0; i < n; i++) printf ( "%d " , arr[i]); printf ( "\n" ); } // Driver Code int main() { int arr[] = { 10, 7, 8, 9, 1, 5 }; int n = sizeof (arr) / sizeof (arr[0]); quickSort(arr, 0, n - 1); printf ( "Sorted array: \n" ); printArray(arr, n); return 0; } |
Python3
# Python implementation QuickSort using # Hoare's partition Scheme. import random ''' The function which implements randomised QuickSort, using Haore's partition scheme. arr :- array to be sorted. start :- starting index of the array. stop :- ending index of the array. ''' def quicksort(arr, start, stop): if (start < stop): # pivotindex is the index where # the pivot lies in the array pivotindex = partitionrand(arr,\ start, stop) # At this stage the array is # partially sorted around the pivot. # separately sorting the left half of # the array and the right # half of the array. quicksort(arr , start , pivotindex) quicksort(arr, pivotindex + 1 , stop) # This function generates random pivot, # swaps the first element with the pivot # and calls the partition function. def partitionrand(arr , start, stop): # Generating a random number between # the starting index of the array and # the ending index of the array. randpivot = random.randrange(start, stop) # Swapping the starting element of # the array and the pivot arr[start], arr[randpivot] = \ arr[randpivot], arr[start] return partition(arr, start, stop) ''' This function takes the first element as pivot, places the pivot element at the correct position in the sorted array. All the elements are re-arranged according to the pivot, the elements smaller than the pivot is places on the left and the elements greater than the pivot is placed to the right of pivot. ''' def partition(arr,start,stop): pivot = start # pivot i = start - 1 j = stop + 1 while True : while True : i = i + 1 if arr[i] > = arr[pivot]: break while True : j = j - 1 if arr[j] < = arr[pivot]: break if i > = j: return j arr[i] , arr[j] = arr[j] , arr[i] # Driver Code if __name__ = = "__main__" : array = [ 10 , 7 , 8 , 9 , 1 , 5 ] quicksort(array, 0 , len (array) - 1 ) print (array) # This code is contributed by soumyasaurav |
Sorted array: 1 5 7 8 9 10
Analysis of Randomized Quick Sort
Notes
- Using random pivoting we improve the expected or average time complexity to O (N log N). The Worst-Case complexity is still O ( N^2 ).
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