3-Way QuickSort (Dutch National Flag)

Difficulty Level : Hard
Last Updated : 02 Oct, 2020

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 a pivot in Simple Quick Sort, we fix only one 4 and recursively process remaining occurrences.
The idea of 3 way Quick Sort is to process all occurrences of the pivot and is based on Dutch National Flag algorithm.

In 3 Way QuickSort, an array arr[l..r] is divided in 3 parts:
a) arr[l..i] elements less than pivot.
b) arr[i+1..j-1] elements equal to pivot.
c) arr[j..r] elements greater than pivot.

Below is the implementation of the above algorithm.

C++

// C++ program for 3-way quick sort
#include <bits/stdc++.h>
using namespace std;
 
/* This function partitions a[] in three parts
   a) a[l..i] contains all elements smaller than pivot
   b) a[i+1..j-1] contains all occurrences of pivot
   c) a[j..r] contains all elements greater than pivot */
void partition(int a[], int l, int r, int& i, int& j)
{
    i = l - 1, j = r;
    int p = l - 1, q = r;
    int v = a[r];
 
    while (true) {
        // From left, find the first element greater than
        // or equal to v. This loop will definitely
        // terminate as v is last element
        while (a[++i] < v)
            ;
 
        // From right, find the first element smaller than
        // or equal to v
        while (v < a[--j])
            if (j == l)
                break;
 
        // If i and j cross, then we are done
        if (i >= j)
            break;
 
        // Swap, so that smaller goes on left greater goes
        // on right
        swap(a[i], a[j]);
 
        // Move all same left occurrence of pivot to
        // beginning of array and keep count using p
        if (a[i] == v) {
            p++;
            swap(a[p], a[i]);
        }
 
        // Move all same right occurrence of pivot to end of
        // array and keep count using q
        if (a[j] == v) {
            q--;
            swap(a[j], a[q]);
        }
    }
 
    // Move pivot element to its correct index
    swap(a[i], a[r]);
 
    // Move all left same occurrences from beginning
    // to adjacent to arr[i]
    j = i - 1;
    for (int k = l; k < p; k++, j--)
        swap(a[k], a[j]);
 
    // Move all right same occurrences from end
    // to adjacent to arr[i]
    i = i + 1;
    for (int k = r - 1; k > q; k--, i++)
        swap(a[i], a[k]);
}
 
// 3-way partition based quick sort
void quicksort(int a[], int l, int r)
{
    if (r <= l)
        return;
 
    int i, j;
 
    // Note that i and j are passed as reference
    partition(a, l, r, i, j);
 
    // Recur
    quicksort(a, l, j);
    quicksort(a, i, r);
}
 
// A utility function to print an array
void printarr(int a[], int n)
{
    for (int i = 0; i < n; ++i)
        printf("%d  ", a[i]);
    printf("\n");
}
 
// Driver code
int main()
{
    int a[] = { 4, 9, 4, 4, 1, 9, 4, 4, 9, 4, 4, 1, 4 };
    int size = sizeof(a) / sizeof(int);
   
      // Function Call
    printarr(a, size);
    quicksort(a, 0, size - 1);
    printarr(a, size);
    return 0;
}

C#

// C# program for 3-way quick sort
using System;
 
class GFG {
    // A function which is used to swap values
    static void swap<T>(ref T lhs, ref T rhs)
    {
        T temp;
        temp = lhs;
        lhs = rhs;
        rhs = temp;
    }
    /* This function partitions a[] in three parts
       a) a[l..i] contains all elements smaller than pivot
       b) a[i+1..j-1] contains all occurrences of pivot
       c) a[j..r] contains all elements greater than pivot
     */
    public static void partition(int[] a, int l, int r,
                                 ref int i, ref int j)
    {
        i = l - 1;
        j = r;
        int p = l - 1, q = r;
        int v = a[r];
 
        while (true) {
            // From left, find the first element greater
            // than or equal to v. This loop will definitely
            // terminate as v is last element
            while (a[++i] < v)
                ;
 
            // From right, find the first element smaller
            // than or equal to v
            while (v < a[--j])
                if (j == l)
                    break;
 
            // If i and j cross, then we are done
            if (i >= j)
                break;
 
            // Swap, so that smaller goes on left greater
            // goes on right
            swap(ref a[i], ref a[j]);
 
            // Move all same left occurrence of pivot to
            // beginning of array and keep count using p
            if (a[i] == v) {
                p++;
                swap(ref a[p], ref a[i]);
            }
 
            // Move all same right occurrence of pivot to
            // end of array and keep count using q
            if (a[j] == v) {
                q--;
                swap(ref a[j], ref a[q]);
            }
        }
 
        // Move pivot element to its correct index
        swap(ref a[i], ref a[r]);
 
        // Move all left same occurrences from beginning
        // to adjacent to arr[i]
        j = i - 1;
        for (int k = l; k < p; k++, j--)
            swap(ref a[k], ref a[j]);
 
        // Move all right same occurrences from end
        // to adjacent to arr[i]
        i = i + 1;
        for (int k = r - 1; k > q; k--, i++)
            swap(ref a[i], ref a[k]);
    }
 
    // 3-way partition based quick sort
    public static void quicksort(int[] a, int l, int r)
    {
        if (r <= l)
            return;
 
        int i = 0, j = 0;
 
        // Note that i and j are passed as reference
        partition(a, l, r, ref i, ref j);
 
        // Recur
        quicksort(a, l, j);
        quicksort(a, i, r);
    }
 
    // A utility function to print an array
    public static void printarr(int[] a, int n)
    {
        for (int i = 0; i < n; ++i)
            Console.Write(a[i] + " ");
        Console.Write("\n");
    }
 
    // Driver code
    static void Main()
    {
        int[] a = { 4, 9, 4, 4, 1, 9, 4, 4, 9, 4, 4, 1, 4 };
        int size = a.Length;
         
          // Function Call
          printarr(a, size);
        quicksort(a, 0, size - 1);
        printarr(a, size);
    }
    // This code is contributed by DrRoot_
}

Output

4  9  4  4  1  9  4  4  9  4  4  1  4  
1  1  4  4  4  4  4  4  4  4  9  9  9

Thanks to Utkarsh for suggesting above implementation.

Another Implementation using Dutch National Flag Algorithm

C++

// C++ program for 3-way quick sort
#include <bits/stdc++.h>
using namespace std;
 
void swap(int* a, int* b)
{
    int temp = *a;
    *a = *b;
    *b = temp;
}
 
// A utility function to print an array
void printarr(int a[], int n)
{
    for (int i = 0; i < n; ++i)
        printf("%d ", a[i]);
    printf("\n");
}
 
/* This function partitions a[] in three parts
a) a[l..i] contains all elements smaller than pivot
b) a[i+1..j-1] contains all occurrences of pivot
c) a[j..r] contains all elements greater than pivot */
 
// It uses Dutch National Flag Algorithm
void partition(int a[], int low, int high, int& i, int& j)
{
    // To handle 2 elements
    if (high - low <= 1) {
        if (a[high] < a[low])
            swap(&a[high], &a[low]);
        i = low;
        j = high;
        return;
    }
 
    int mid = low;
    int pivot = a[high];
    while (mid <= high) {
        if (a[mid] < pivot)
            swap(&a[low++], &a[mid++]);
        else if (a[mid] == pivot)
            mid++;
        else if (a[mid] > pivot)
            swap(&a[mid], &a[high--]);
    }
 
    // update i and j
    i = low - 1;
    j = mid; // or high+1
}
 
// 3-way partition based quick sort
void quicksort(int a[], int low, int high)
{
    if (low >= high) // 1 or 0 elements
        return;
 
    int i, j;
 
    // Note that i and j are passed as reference
    partition(a, low, high, i, j);
 
    // Recur two halves
    quicksort(a, low, i);
    quicksort(a, j, high);
}
 
// Driver Code
int main()
{
    int a[] = { 4, 9, 4, 4, 1, 9, 4, 4, 9, 4, 4, 1, 4 };
    // int a[] = {4, 39, 54, 14, 31, 89, 44, 34, 59, 64, 64,
    // 11, 41}; int a[] = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10};
    // int a[] = {91, 82, 73, 64, 55, 46, 37, 28, 19, 10};
    // int a[] = {4, 9, 4, 4, 9, 1, 1, 1};
    int size = sizeof(a) / sizeof(int);
 
    // Function Call
    printarr(a, size);
    quicksort(a, 0, size - 1);
    printarr(a, size);
    return 0;
}

C#

// C# program for 3-way quick sort
using System;
 
class GFG {
    // A function which is used to swap values
    static void swap<T>(ref T lhs, ref T rhs)
    {
        T temp;
        temp = lhs;
        lhs = rhs;
        rhs = temp;
    }
 
    // A utility function to print an array
    public static void printarr(int[] a, int n)
    {
        for (int i = 0; i < n; ++i)
            Console.Write(a[i] + " ");
        Console.Write("\n");
    }
 
    /* This function partitions a[] in three parts
    a) a[l..i] contains all elements smaller than pivot
    b) a[i+1..j-1] contains all occurrences of pivot
    c) a[j..r] contains all elements greater than pivot */
 
    // It uses Dutch National Flag Algorithm
    public static void partition(int[] a, int low, int high,
                                 ref int i, ref int j)
    {
        // To handle 2 elements
        if (high - low <= 1) {
            if (a[high] < a[low])
                swap(ref a[high], ref a[low]);
            i = low;
            j = high;
            return;
        }
 
        int mid = low;
        int pivot = a[high];
        while (mid <= high) {
            if (a[mid] < pivot)
                swap(ref a[low++], ref a[mid++]);
            else if (a[mid] == pivot)
                mid++;
            else if (a[mid] > pivot)
                swap(ref a[mid], ref a[high--]);
        }
 
        // update i and j
        i = low - 1;
        j = mid; // or high+1
    }
 
    // 3-way partition based quick sort
    public static void quicksort(int[] a, int low, int high)
    {
        if (low >= high) // 1 or 0 elements
            return;
 
        int i = 0, j = 0;
 
        // Note that i and j are passed as reference
        partition(a, low, high, ref i, ref j);
 
        // Recur two halves
        quicksort(a, low, i);
        quicksort(a, j, high);
    }
 
    // Driver code
    static void Main()
    {
        int[] a = { 4, 9, 4, 4, 1, 9, 4, 4, 9, 4, 4, 1, 4 };
        // int[] a = {4, 39, 54, 14, 31, 89, 44, 34, 59, 64,
        // 64, 11, 41}; int[] a = {1, 2, 3, 4, 5, 6, 7, 8, 9,
        // 10}; int[] a = {91, 82, 73, 64, 55, 46, 37, 28,
        // 19, 10}; int[] a = {4, 9, 4, 4, 9, 1, 1, 1};
        int size = a.Length;
         
          // Function Call
        printarr(a, size);
        quicksort(a, 0, size - 1);
        printarr(a, size);
    }
    // This code is contributed by DrRoot_
}

Output

4 9 4 4 1 9 4 4 9 4 4 1 4 
1 1 4 4 4 4 4 4 4 4 9 9 9

Thanks Aditya Goel for this implementation.
Reference:
http://algs4.cs.princeton.edu/lectures/23DemoPartitioning.pdf
http://www.sorting-algorithms.com/quick-sort-3-way
Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above.

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