Three way partitioning around an element
Last Updated :
10 Dec, 2024
Given an array arr[] of integers and a value pivot, the task is to partition the array around the pivot such that array is divided in three parts.
- All elements smaller than pivot come first.
- All elements equal to pivot come next.
- All elements greater than pivot appear in the end.
The individual elements of the three parts can appear in any order.
Note: We can assume that the pivot will always be present as an element in the array.
Examples:
Input: arr[] = [0, -1, 3, 2, -7, 0, -5, 6], pivot = 0
Output: [-1, -7, -5, 0, 0, 2, 3, 6]
Explanation: All elements smaller than pivot [-1, -7, -5] were arranged before it and all elements larger than pivot [2, 3, 6] were arranged after it. Note that [-1, -5, -7, 0, 0, 3, 2, 6] is also a valid 3 way partition.
Input: arr[] = [2, 8, 6, 9, 8, 5, 2, 1], pivot = 5
Output: [2, 1, 2, 5, 6, 8, 8, 9]
Explanation: All elements smaller than pivot element [2, 1, 2] were arranged before it and elements larger than pivot [6, 8, 8, 9] were arranged after it.
Using Naive Partition Approach (Stable Partitioning)
A simple approach to partition an array is to create a new temporary array which will store the rearranged elements. In this approach, we first iterate over the original array and add all elements that are smaller than the pivot to the temporary array. Then, we add elements equal to the pivot. Finally, we fill the remaining part of the temporary array with elements that are greater than the pivot.
This ensures that the smaller elements come before the pivot, and the larger elements come after it. Now, copy the elements from the temporary array back to the original array.
C++
// C++ program for 3 way partitioning of array
// using Naive Partition approach
#include <iostream>
#include <vector>
using namespace std;
// Function to three way partition an array
void partition(vector<int> &arr, int pivot) {
int n = arr.size();
// create a temp array to store the elements in order
vector<int> temp(n);
int idx = 0;
// First fill element smaller than pivot,
// into the temp array
for (int i = 0; i < n; i++) {
if (arr[i] < pivot)
temp[idx++] = arr[i];
}
// Now fill element equal to pivot,
// into the temp array
for (int i = 0; i < n; i++) {
if (arr[i] == pivot)
temp[idx++] = arr[i];
}
// Finally fill the elements greater than pivot
for (int i = 0; i < n; i++) {
if (arr[i] > pivot)
temp[idx++] = arr[i];
}
// copy the elements from temp to arr
arr = temp;
}
int main() {
vector<int> arr = {2, 8, 6, 9, 8, 5, 2, 1};
int pivot = 5;
partition(arr, pivot);
for (int i = 0; i < arr.size(); i++)
cout << arr[i] << " ";
return 0;
}
// C program for 3 way partitioning of array
// using Naive Partition approach
#include <stdio.h>
#include <stdlib.h>
// Function to three way partition an array
void partition(int* arr, int n, int pivot) {
// create a temp array to store the elements in order
int* temp = (int*)malloc(n * sizeof(int));
int idx = 0;
// First fill elements smaller than pivot into the temp array
for (int i = 0; i < n; i++) {
if (arr[i] < pivot)
temp[idx++] = arr[i];
}
// Now fill elements equal to pivot into the temp array
for (int i = 0; i < n; i++) {
if (arr[i] == pivot)
temp[idx++] = arr[i];
}
// Finally fill the elements greater than pivot
for (int i = 0; i < n; i++) {
if (arr[i] > pivot)
temp[idx++] = arr[i];
}
// copy the elements from temp to arr
for (int i = 0; i < n; i++) {
arr[i] = temp[i];
}
free(temp);
}
int main() {
int arr[] = {2, 8, 6, 9, 8, 5, 2, 1};
int n = sizeof(arr) / sizeof(arr[0]);
int pivot = 5;
partition(arr, n, pivot);
for (int i = 0; i < n; i++)
printf("%d ", arr[i]);
return 0;
}
// Java program for 3 way partitioning of array
// using Naive Partition approach
import java.util.Arrays;
class GFG {
// Function to three way partition an array
static void partition(int[] arr, int pivot) {
int n = arr.length;
// create a temp array to store the elements in order
int[] temp = new int[n];
int idx = 0;
// First fill elements smaller than pivot into the temp array
for (int i = 0; i < n; i++) {
if (arr[i] < pivot)
temp[idx++] = arr[i];
}
// Now fill elements equal to pivot into the temp array
for (int i = 0; i < n; i++) {
if (arr[i] == pivot)
temp[idx++] = arr[i];
}
// Finally fill the elements greater than pivot
for (int i = 0; i < n; i++) {
if (arr[i] > pivot)
temp[idx++] = arr[i];
}
// copy the elements from temp to arr
System.arraycopy(temp, 0, arr, 0, n);
}
public static void main(String[] args) {
int[] arr = {2, 8, 6, 9, 8, 5, 2, 1};
int pivot = 5;
partition(arr, pivot);
for (int value : arr)
System.out.print(value + " ");
}
}
# Python program for 3 way partitioning of array
# using Naive Partition approach
# Function to three way partition an array
def partition(arr, pivot):
n = len(arr)
# create a temp array to store the elements in order
temp = [0] * n
idx = 0
# First fill elements smaller than pivot into the temp array
for i in range(n):
if arr[i] < pivot:
temp[idx] = arr[i]
idx += 1
# Now fill elements equal to pivot into the temp array
for i in range(n):
if arr[i] == pivot:
temp[idx] = arr[i]
idx += 1
# Finally fill the elements greater than pivot
for i in range(n):
if arr[i] > pivot:
temp[idx] = arr[i]
idx += 1
# copy the elements from temp to arr
for i in range(n):
arr[i] = temp[i]
if __name__ == "__main__":
arr = [2, 8, 6, 9, 8, 5, 2, 1]
pivot = 5
partition(arr, pivot)
print(" ".join(map(str, arr)))
// C# program for 3 way partitioning of array
// using Naive Partition approach
using System;
class GFG {
// Function to three way partition an array
static void partition(int[] arr, int pivot) {
int n = arr.Length;
// create a temp array to store the elements in order
int[] temp = new int[n];
int idx = 0;
// First fill elements smaller than pivot
// into the temp array
for (int i = 0; i < n; i++) {
if (arr[i] < pivot)
temp[idx++] = arr[i];
}
// Now fill elements equal to pivot into the temp array
for (int i = 0; i < n; i++) {
if (arr[i] == pivot)
temp[idx++] = arr[i];
}
// Finally fill the elements greater than pivot
for (int i = 0; i < n; i++) {
if (arr[i] > pivot)
temp[idx++] = arr[i];
}
// copy the elements from temp to arr
Array.Copy(temp, arr, n);
}
static void Main() {
int[] arr = { 2, 8, 6, 9, 8, 5, 2, 1 };
int pivot = 5;
partition(arr, pivot);
foreach (int val in arr)
Console.Write(val + " ");
}
}
// JavaScript program for 3 way partitioning of array
// using Naive Partition approach
// Function to three way partition an array
function partition(arr, pivot) {
let n = arr.length;
// create a temp array to store the elements in order
let temp = new Array(n);
let idx = 0;
// First fill elements smaller than pivot into the temp array
for (let i = 0; i < n; i++) {
if (arr[i] < pivot)
temp[idx++] = arr[i];
}
// Now fill elements equal to pivot into the temp array
for (let i = 0; i < n; i++) {
if (arr[i] === pivot)
temp[idx++] = arr[i];
}
// Finally fill the elements greater than pivot
for (let i = 0; i < n; i++) {
if (arr[i] > pivot)
temp[idx++] = arr[i];
}
// copy the elements from temp to arr
for (let i = 0; i < n; i++) {
arr[i] = temp[i];
}
}
// Driver Code
let arr = [2, 8, 6, 9, 8, 5, 2, 1];
let pivot = 5;
partition(arr, pivot);
console.log(arr.join(" "));
Time Complexity: O(n), for array traversal.
Auxiliary Space: O(n), used for temporary array.
Using Dutch National Flag Algorithm
The idea is to partition the array using three pointers: lo = 0, mid = 0 and hi = n – 1 such that the array is divided into three parts –
- arr[0 ... lo-1]: This part will have all the elements smaller than pivot.
- arr[lo ... hi – 1]: This part will have all the elements equal to pivot.
- arr[hi ... n – 1]: This part will have all the elements greater than pivot.
Here, lo indicates the position where next element smaller than pivot should be placed, mid is used to traverse through the array and hi indicates the position where next element greater than pivot should be placed.
Traverse over the array till mid <= hi, according to the value of arr[mid] we can have three cases:
- If arr[mid] < pivot, then swap arr[lo] and arr[mid] and increment lo by 1 because all the values till index lo, are lesser than pivot. Also move to the next element by incrementing mid by 1.
- If arr[mid] = pivot, then move to the next element by incrementing mid by 1.
- If arr[mid] > pivot, then swap arr[mid] and arr[hi] and decrement hi by 1 because all the values in arr[hi ... n-1] are greater than pivot. Now, we don’t move to the next element because the element which is now at index mid might be smaller than pivot and therefore needs to be checked again.
C++
// C++ program for Three way partitioning around
// a pivot using Dutch National Flag Algorithm
#include <iostream>
#include <vector>
using namespace std;
// Function to three-way partition an array around a pivot
void partition(vector<int> &arr, int pivot) {
int n = arr.size();
// lo: boundary for elements less than pivot
// hi: boundary for elements greater than pivot
// mid: current element being considered
int lo = 0;
int hi = n - 1;
int mid = 0;
// Iterate over the array until mid crosses hi
while (mid <= hi) {
if (arr[mid] < pivot) {
// Element is less than pivot;
// move it to the beginning
swap(arr[lo++], arr[mid++]);
} else if (arr[mid] == pivot) {
// Element is equal to pivot;
// move mid pointer ahead
mid++;
} else {
// Element is greater than pivot; move it to the end
swap(arr[mid], arr[hi--]);
}
}
}
int main() {
vector<int> arr = {2, 8, 6, 9, 8, 5, 2, 1};
int pivot = 5;
partition(arr, pivot);
for (int i = 0; i < arr.size(); i++)
cout << arr[i] << " ";
return 0;
}
// C program for Three way partitioning around
// a pivot using Dutch National Flag Algorithm
#include <stdio.h>
void swap(int *a, int *b) {
int temp = *a;
*a = *b;
*b = temp;
}
// Function to three-way partition an array around a pivot
void partition(int array[], int n, int pivot) {
// lo: boundary for elements less than pivot
// hi: boundary for elements greater than pivot
// mid: current element being considered
int lo = 0;
int hi = n - 1;
int mid = 0;
// Iterate over the array until mid crosses hi
while (mid <= hi) {
if (array[mid] < pivot) {
// Element is less than pivot;
// move it to the beginning
swap(&array[lo++], &array[mid++]);
} else if (array[mid] == pivot) {
// Element is equal to pivot;
// move mid pointer ahead
mid++;
} else {
// Element is greater than pivot; move it to the end
swap(&array[mid], &array[hi--]);
}
}
}
int main() {
int array[] = {2, 8, 6, 9, 8, 5, 2, 1};
int n = sizeof(array) / sizeof(array[0]);
int pivot = 5;
partition(array, n, pivot);
for (int i = 0; i < n; i++)
printf("%d ", array[i]);
return 0;
}
// Java program for Three way partitioning around
// a pivot using Dutch National Flag Algorithm
import java.util.*;
class GfG {
// Function to three-way partition an array around a pivot
static void partition(int[] array, int pivot) {
int n = array.length;
// lo: boundary for elements less than pivot
// hi: boundary for elements greater than pivot
// mid: current element being considered
int lo = 0;
int hi = n - 1;
int mid = 0;
// Iterate over the array until mid crosses hi
while (mid <= hi) {
if (array[mid] < pivot) {
// Element is less than pivot;
// move it to the beginning
swap(array, lo++, mid++);
} else if (array[mid] == pivot) {
// Element is equal to pivot;
// move mid pointer ahead
mid++;
} else {
// Element is greater than pivot; move it to the end
swap(array, mid, hi--);
}
}
}
static void swap(int[] array, int i, int j) {
int temp = array[i];
array[i] = array[j];
array[j] = temp;
}
public static void main(String[] args) {
int[] array = {2, 8, 6, 9, 8, 5, 2, 1};
int pivot = 5;
partition(array, pivot);
for (int i : array)
System.out.print(i + " ");
}
}
# Python program for Three way partitioning around
# a pivot using Dutch National Flag Algorithm
def partition(array, pivot):
# lo: boundary for elements less than pivot
# hi: boundary for elements greater than pivot
# mid: current element being considered
lo = 0
hi = len(array) - 1
mid = 0
# Iterate over the array until mid crosses hi
while mid <= hi:
if array[mid] < pivot:
# Element is less than pivot;
# move it to the beginning
array[lo], array[mid] = array[mid], array[lo]
lo += 1
mid += 1
elif array[mid] == pivot:
# Element is equal to pivot;
# move mid pointer ahead
mid += 1
else:
# Element is greater than pivot; move it to the end
array[mid], array[hi] = array[hi], array[mid]
hi -= 1
if __name__ == "__main__":
array = [2, 8, 6, 9, 8, 5, 2, 1]
pivot = 5
partition(array, pivot)
print(" ".join(map(str, array)))
using System;
class GfG {
// Function to three-way partition an array around a pivot
static void Partition(int[] arr, int pivot) {
int n = arr.Length;
// lo: boundary for elements less than pivot
// hi: boundary for elements greater than pivot
// mid: current element being considered
int lo = 0;
int hi = n - 1;
int mid = 0;
// Iterate over the array until mid crosses hi
while (mid <= hi) {
if (arr[mid] < pivot) {
// Element is less than pivot;
// move it to the beginning
Swap(arr, lo++, mid++);
} else if (arr[mid] == pivot) {
// Element is equal to pivot;
// move mid pointer ahead
mid++;
} else {
// Element is greater than pivot; move it to the end
Swap(arr, mid, hi--);
}
}
}
static void Swap(int[] arr, int i, int j) {
int temp = arr[i];
arr[i] = arr[j];
arr[j] = temp;
}
static void Main() {
int[] arr = {2, 8, 6, 9, 8, 5, 2, 1};
int pivot = 5;
Partition(arr, pivot);
foreach (int val in arr) {
Console.Write(val + " ");
}
}
}
// JavaScript program for Three way partitioning around
// a pivot using Dutch National Flag Algorithm
function partition(array, pivot) {
let lo = 0;
let hi = array.length - 1;
let mid = 0;
// Iterate over the array until mid crosses hi
while (mid <= hi) {
if (array[mid] < pivot) {
// Element is less than pivot;
// move it to the beginning
[array[lo], array[mid]] = [array[mid], array[lo]];
lo++;
mid++;
} else if (array[mid] == pivot) {
// Element is equal to pivot;
// move mid pointer ahead
mid++;
} else {
// Element is greater than pivot; move it to the end
[array[mid], array[hi]] = [array[hi], array[mid]];
hi--;
}
}
}
// Driver code
const array = [2, 8, 6, 9, 8, 5, 2, 1];
const pivot = 5;
partition(array, pivot);
console.log(array.join(" "));
Time Complexity: O(n), for array traversal.
Auxiliary Space: O(1)
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