Hoare's vs Lomuto partition scheme in QuickSort
Last Updated :
06 Aug, 2024
We have discussed the implementation of QuickSort using Lomuto partition scheme. Lomuto's partition scheme is easy to implement as compared to Hoare scheme. This has inferior performance to Hoare's QuickSort.
Lomuto's Partition Scheme:
This algorithm works by assuming the pivot element as the last element. If any other element is given as a pivot element then swap it first with the last element. Now initialize two variables i as low and j also low, iterate over the array and increment i when arr[j] <= pivot and swap arr[i] with arr[j] otherwise increment only j. After coming out from the loop swap arr[i+1] with arr[hi]. This i stores the pivot element.
partition(arr[], lo, hi)
pivot = arr[hi]
i = lo-1 // place for swapping
for j := lo to hi – 1 do
if arr[j] <= pivot then
i = i + 1
swap arr[i] with arr[j]
swap arr[i+1] with arr[hi]
return i+1
Refer QuickSort for details of this partitioning scheme.
Below are implementations of this approach:-
C++
#include <bits/stdc++.h>
using namespace std;
/* This function takes the last element as pivot, places
the pivot element at its correct position in sorted
array, and places all smaller (smaller than pivot)
to the left of the pivot and all greater elements
to the right of the pivot */
int partition(vector<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)
{
i++; // increment index of smaller element
swap(arr[i], arr[j]);
}
}
swap(arr[i + 1], arr[high]);
return (i + 1);
}
/* The main function that implements QuickSort
arr[] --> Array to be sorted,
low --> Starting index,
high --> Ending index */
void quickSort(vector<int>& arr, int low, int high)
{
if (low < high)
{
/* pi is partitioning index, arr[p] is now
at right place */
int pi = partition(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(const vector<int>& arr)
{
for (int i : arr)
cout << i << " ";
cout << endl;
}
// Driver program to test above functions
int main()
{
vector<int> arr = {10, 7, 8, 9, 1, 5};
int n = arr.size();
quickSort(arr, 0, n - 1);
cout << "Sorted array: \n";
printArray(arr);
return 0;
}
#include <stdio.h>
#include <stdlib.h>
// Function to swap two elements
void swap(int* a, int* b) {
int t = *a;
*a = *b;
*b = t;
}
/* This function takes the last element as pivot, places
the pivot element at its correct position in sorted
array, and places all smaller (smaller than pivot)
to the left of the pivot and all greater elements to the right
of the pivot */
int 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) {
i++; // increment index of smaller element
swap(&arr[i], &arr[j]);
}
}
swap(&arr[i + 1], &arr[high]);
return (i + 1);
}
/* 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(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) {
for (int i = 0; i < size; i++)
printf("%d ", arr[i]);
printf("\n");
}
// Driver program to test above functions
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 implementation QuickSort
// using Lomuto's partition Scheme
import java.io.*;
class GFG {
static void Swap(int[] array, int position1,
int position2)
{
// Swaps elements in an array
// Copy the first position's element
int temp = array[position1];
// Assign to the second element
array[position1] = array[position2];
// Assign to the first element
array[position2] = temp;
}
/* 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)
{
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) {
i++; // increment index of
// smaller element
Swap(arr, i, j);
}
}
Swap(arr, i + 1, high);
return (i + 1);
}
/* The main function that
implements QuickSort
arr[] --> Array to be sorted,
low --> Starting index,
high --> Ending index */
static 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(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 */
static void printArray(int[] arr, int size)
{
int i;
for (i = 0; i < size; i++)
System.out.print(" " + arr[i]);
System.out.println();
}
// Driver Code
static public void main(String[] args)
{
int[] arr = { 10, 7, 8, 9, 1, 5 };
int n = arr.length;
quickSort(arr, 0, n - 1);
System.out.println("Sorted array: ");
printArray(arr, n);
}
}
// This code is contributed by vt_m.
''' Python3 implementation QuickSort using Lomuto's partition
Scheme.'''
''' 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 '''
def partition(arr, low, high):
# pivot
pivot = arr[high]
# Index of smaller element
i = (low - 1)
for j in range(low, high):
# If current element is smaller than or
# equal to pivot
if (arr[j] <= pivot):
# increment index of smaller element
i += 1
arr[i], arr[j] = arr[j], arr[i]
arr[i + 1], arr[high] = arr[high], arr[i + 1]
return (i + 1)
''' The main function that implements QuickSort
arr --> Array to be sorted,
low --> Starting index,
high --> Ending index '''
def quickSort(arr, low, high):
if (low < high):
''' pi is partitioning index, arr[p] is now
at right place '''
pi = partition(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 '''
def printArray(arr, size):
for i in range(size):
print(arr[i], end = " ")
print()
# Driver code
arr = [10, 7, 8, 9, 1, 5]
n = len(arr)
quickSort(arr, 0, n - 1)
print("Sorted array:")
printArray(arr, n)
# This code is contributed by SHUBHAMSINGH10
// C# implementation QuickSort
// using Lomuto's partition Scheme
using System;
class GFG {
static void Swap(int[] array, int position1,
int position2)
{
// Swaps elements in an array
// Copy the first position's element
int temp = array[position1];
// Assign to the second element
array[position1] = array[position2];
// Assign to the first element
array[position2] = temp;
}
/* 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)
{
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) {
i++; // increment index of
// smaller element
Swap(arr, i, j);
}
}
Swap(arr, i + 1, high);
return (i + 1);
}
/* The main function that
implements QuickSort
arr[] --> Array to be sorted,
low --> Starting index,
high --> Ending index */
static 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(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 */
static void printArray(int[] arr, int size)
{
int i;
for (i = 0; i < size; i++)
Console.Write(" " + arr[i]);
Console.WriteLine();
}
// Driver Code
static public void Main()
{
int[] arr = { 10, 7, 8, 9, 1, 5 };
int n = arr.Length;
quickSort(arr, 0, n - 1);
Console.WriteLine("Sorted array: ");
printArray(arr, n);
}
}
// This code is contributed by vt_m.
/* This function takes the last element as pivot, places
the pivot element at its correct position in sorted
array, and places all smaller (smaller than pivot)
to the left of the pivot and all greater elements
to the right of the pivot */
function partition(arr, low, high) {
let pivot = arr[high]; // pivot
let i = low - 1; // Index of smaller element
for (let j = low; j <= high - 1; j++) {
// If current element is smaller than or
// equal to pivot
if (arr[j] <= pivot) {
i++; // increment index of smaller element
// Swap arr[i] and arr[j]
[arr[i], arr[j]] = [arr[j], arr[i]];
}
}
// Swap arr[i + 1] and arr[high]
[arr[i + 1], arr[high]] = [arr[high], arr[i + 1]];
return i + 1;
}
/* The main function that implements QuickSort
arr[] --> Array to be sorted,
low --> Starting index,
high --> Ending index */
function quickSort(arr, low, high) {
if (low < high) {
/* pi is partitioning index, arr[pi] is now
at right place */
let pi = partition(arr, low, high);
// Separately sort elements before
// partition and after partition
quickSort(arr, low, pi - 1);
quickSort(arr, pi + 1, high);
}
}
// Driver code to test above functions
let arr = [10, 7, 8, 9, 1, 5];
let n = arr.length;
quickSort(arr, 0, n - 1);
console.log("Sorted array:");
console.log(arr.join(' '));
OutputSorted array:
1 5 7 8 9 10
Time Complexity: O(N2)
Auxiliary Space: O(1)
Hoare's Partition Scheme:
Hoare's Partition Scheme works by initializing two indexes that start at two ends, the two indexes move toward each other until an inversion is (A smaller value on the left side and greater value on the right side) found. When an inversion is found, two values are swapped and the process is repeated.
Algorithm:
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
// Find a value in right side smaller
// than pivot
do
j--;
while (arr[j] > pivot);
if i >= j then
return j
swap arr[i] with arr[j]
Below are implementations of this approach:-
C++
#include <bits/stdc++.h>
using namespace std;
/* This function takes the first element as pivot, and places
all the elements smaller than the pivot on the left side
and all the elements greater than the pivot on
the right side. It returns the index of the last element
on the smaller side */
int partition(vector<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]);
}
}
/* The main function that implements QuickSort
arr[] --> Array to be sorted,
low --> Starting index,
high --> Ending index */
void quickSort(vector<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);
// Separately sort elements before
// partition and after partition
quickSort(arr, low, pi);
quickSort(arr, pi + 1, high);
}
}
/* Function to print an array */
void printArray(const vector<int>& arr) {
for (int i : arr)
cout << i << " ";
cout << endl;
}
// Driver Code
int main() {
vector<int> arr = {10, 7, 8, 9, 1, 5};
quickSort(arr, 0, arr.size() - 1);
cout << "Sorted array: \n";
printArray(arr);
return 0;
}
#include <stdio.h>
/* This function takes the first element as pivot, and places
all the elements smaller than the pivot on the left side
and all the elements greater than the pivot on
the right side. It returns the index of the last element
on the smaller side */
int partition(int arr[], int low, int high) {
int pivot = arr[low];
int i = low - 1, j = high + 1;
while (1) {
// 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] and arr[j]
int temp = arr[i];
arr[i] = arr[j];
arr[j] = temp;
}
}
/* 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[pi]
is now at right place */
int pi = partition(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 size) {
for (int i = 0; i < size; i++)
printf("%d ", arr[i]);
printf("\n");
}
// Driver code to test above functions
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 implementation of QuickSort
// using Hoare's partition scheme
import java.io.*;
class GFG {
/* This function takes first element as pivot, and
places all the elements smaller than the pivot on the
left side and all the elements greater than the pivot
on the right side. It returns the index of the last
element on the smaller side*/
static 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;
int temp = arr[i];
arr[i] = arr[j];
arr[j] = temp;
// swap(arr[i], arr[j]);
}
}
/* The main function that
implements QuickSort
arr[] --> Array to be sorted,
low --> Starting index,
high --> Ending index */
static 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(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 */
static void printArray(int[] arr, int n)
{
for (int i = 0; i < n; i++)
System.out.print(" " + arr[i]);
System.out.println();
}
// Driver Code
static public void main(String[] args)
{
int[] arr = { 10, 7, 8, 9, 1, 5 };
int n = arr.length;
quickSort(arr, 0, n - 1);
System.out.println("Sorted array: ");
printArray(arr, n);
}
}
// This code is contributed by vt_m.
''' Python implementation of QuickSort using Hoare's
partition scheme. '''
''' This function takes first element as pivot, and places
all the elements smaller than the pivot on the left side
and all the elements greater than the pivot on
the right side. It returns the index of the last element
on the smaller side '''
def partition(arr, low, high):
pivot = arr[low]
i = low - 1
j = high + 1
while (True):
# Find leftmost element greater than
# or equal to pivot
i += 1
while (arr[i] < pivot):
i += 1
# Find rightmost element smaller than
# or equal to pivot
j -= 1
while (arr[j] > pivot):
j -= 1
# If two pointers met.
if (i >= j):
return j
arr[i], arr[j] = arr[j], arr[i]
''' The main function that implements QuickSort
arr --> Array to be sorted,
low --> Starting index,
high --> Ending index '''
def quickSort(arr, low, high):
''' pi is partitioning index, arr[p] is now
at right place '''
if (low < high):
pi = partition(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 '''
def printArray(arr, n):
for i in range(n):
print(arr[i], end=" ")
print()
# Driver code
arr = [10, 7, 8, 9, 1, 5]
n = len(arr)
quickSort(arr, 0, n - 1)
print("Sorted array:")
printArray(arr, n)
# This code is contributed by shubhamsingh10
// C# implementation of QuickSort
// using Hoare's partition scheme
using System;
class GFG {
/* This function takes first element as pivot, and
places all the elements smaller than the pivot on the
left side and all the elements greater than the pivot
on the right side. It returns the index of the last
element on the smaller side*/
static 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;
int temp = arr[i];
arr[i] = arr[j];
arr[j] = temp;
// swap(arr[i], arr[j]);
}
}
/* The main function that
implements QuickSort
arr[] --> Array to be sorted,
low --> Starting index,
high --> Ending index */
static 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(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 */
static void printArray(int[] arr, int n)
{
for (int i = 0; i < n; i++)
Console.Write(" " + arr[i]);
Console.WriteLine();
}
// Driver Code
static public void Main()
{
int[] arr = { 10, 7, 8, 9, 1, 5 };
int n = arr.Length;
quickSort(arr, 0, n - 1);
Console.WriteLine("Sorted array: ");
printArray(arr, n);
}
}
// This code is contributed by vt_m.
/* This function takes the first element as pivot, and places
all the elements smaller than the pivot on the left side
and all the elements greater than the pivot on
the right side. It returns the index of the last element
on the smaller side */
function partition(arr, low, high) {
let pivot = arr[low];
let 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] and arr[j]
[arr[i], arr[j]] = [arr[j], arr[i]];
}
}
/* The main function that implements QuickSort
arr[] --> Array to be sorted,
low --> Starting index,
high --> Ending index */
function quickSort(arr, low, high) {
if (low < high) {
/* pi is partitioning index, arr[pi] is now
at right place */
let pi = partition(arr, low, high);
// Separately sort elements before
// partition and after partition
quickSort(arr, low, pi);
quickSort(arr, pi + 1, high);
}
}
// Driver code to test above functions
let arr = [10, 7, 8, 9, 1, 5];
let n = arr.length;
quickSort(arr, 0, n - 1);
console.log("Sorted array:");
console.log(arr.join(' '));
OutputSorted array:
1 5 7 8 9 10
Time Complexity: O(N)
Auxiliary Space: O(1)
Note : If we change Hoare's partition to pick the last element as pivot, then the Hoare's partition may cause QuickSort to go into an infinite recursion. For example, {10, 5, 6, 20} and pivot is arr[high], then returned index will always be high and call to same QuickSort will be made. To handle a random pivot, we can always swap that random element with the first element and simply follow the above algorithm.
Comparison:
- Hoare's scheme is more efficient than Lomuto's partition scheme because it does three times fewer swaps on average, and it creates efficient partitions even when all values are equal.
- Like Lomuto's partition scheme, Hoare partitioning also causes Quick sort to degrade to O(n^2) when the input array is already sorted, it also doesn't produce a stable sort.
- Note that in this scheme, the pivot's final location is not necessarily at the index that was returned, and the next two segments that the main algorithm recurs on are (lo..p) and (p+1..hi) as opposed to (lo..p-1) and (p+1..hi) as in Lomuto's scheme.
- Both Hoare's Partition, as well as Lomuto's partition, are unstable.
| Hoare partition algorithm | Lomuto partition algorithm |
|---|
| Generally, the first item or the element is assumed to be the initial pivot element. Some choose the middle element and even the last element. | Generally, a random element of the array is located and picked and then exchanged with the first or the last element to give initial pivot values. In the aforementioned algorithm, the last element of the list is considered as the initial pivot element. |
| It is a linear algorithm. | It is also a linear algorithm. |
| It is relatively faster. | It is slower. |
| It is slightly difficult to understand and to implement. | It is easy to understand and easy to implement. |
| It doesn't fix the pivot element in the correct position. | It fixes the pivot element in the correct position. |
Source : https://en.wikipedia.org/wiki/Quicksort#Hoare_partition_scheme
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Quick Sort using Multi-threadingQuickSort is a popular sorting technique based on divide and conquer algorithm. In this technique, an element is chosen as a pivot and the array is partitioned around it. The target of partition is, given an array and an element x of the array as a pivot, put x at its correct position in a sorted ar
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Stable QuickSortA sorting algorithm is said to be stable if it maintains the relative order of records in the case of equality of keys.Input : (1, 5), (3, 2) (1, 2) (5, 4) (6, 4) We need to sort key-value pairs in the increasing order of keys of first digit There are two possible solution for the two pairs where th
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Dual pivot QuicksortAs we know, the single pivot quick sort takes a pivot from one of the ends of the array and partitioning the array, so that all elements are left to the pivot are less than or equal to the pivot, and all elements that are right to the pivot are greater than the pivot.The idea of dual pivot quick sor
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3-Way QuickSort (Dutch National Flag)In simple QuickSort algorithm, we select an element as pivot, partition the array around a pivot and recur for subarrays on the left and right of the pivot. Consider an array which has many redundant elements. For example, {1, 4, 2, 4, 2, 4, 1, 2, 4, 1, 2, 2, 2, 2, 4, 1, 4, 4, 4}. If 4 is picked as
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Visualization of QuickSort
Partitions in QuickSort
Some problems on QuickSort
Is Quick Sort Algorithm Adaptive or not Adaptive sorting algorithms are designed to take advantage of existing order in the input data. This means, if the array is already sorted or partially sorted, an adaptive algorithm will recognize that and sort the array faster than it would for a completely random array.Quick Sort is not an adaptiv
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