# 3-Way QuickSort (Dutch National Flag)

In simple QuickSort algorithm, we select an element as pivot, partition the array around pivot and recur for subarrays on left and right of 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 pivot in Simple QuickSort, we fix only one 4 and recursively process remaining occurrences.

The idea of 3 way QuickSort is to process all occurrences of 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 above algorithm.

## C++

 `// C++ program for 3-way quick sort ` `#include ` `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 program ` `int` `main() ` `{ ` `    ``int` `a[] = {4, 9, 4, 4, 1, 9, 4, 4, 9, 4, 4, 1, 4}; ` `    ``int` `size = ``sizeof``(a) / ``sizeof``(``int``); ` `    ``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(``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 program  ` `    ``static` `void` `Main()  ` `    ``{  ` `        ``int``[] a = {4, 9, 4, 4, 1, 9, 4, 4, 9, 4, 4, 1, 4};  ` `        ``int` `size = a.Length;  ` `        ``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 ` `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[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 program ` `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``); ` ` `  `    ``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(``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[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 program ` `    ``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;  ` `       `  `        ``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.

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Improved By : DrRoot_, arnavvarshney

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