Dutch National Flag Algorithm

Sort an array of 0s, 1s and 2s

Given an array A[] consisting 0s, 1s and 2s, write a function that sorts A[]. The functions should put all 0s first, then all 1s and all 2s in last.

Example
Input = {0, 1, 1, 0, 1, 2, 1, 2, 0, 0, 0, 1};
Output = {0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 2, 2}

We strongly recommend that you click here and practice it, before moving on to the solution.

The problem is similar to our old post Segregate 0s and 1s in an array, and both of these problems are variation of famous Dutch national flag problem.

The problem was posed with three colours, here `0′, `1′ and `2′. The array is divided into four sections:

1. a[1..Lo-1] zeroes (red)
2. a[Lo..Mid-] ones (white)
3. a[Mid..Hi] unknown
4. a[Hi+1..N] twos (blue)

The unknown region is shrunk while maintaining these conditions

1. Lo := 1; Mid := 1; Hi := N;
2. while Mid <= Hi do
   1. Invariant: a[1..Lo-1]=0 and a[Lo..Mid-1]=1 and a[Hi+1..N]=2; a[Mid..Hi] are unknown.
   2. case a[Mid] in
      • 0: swap a[Lo] and a[Mid]; Lo++; Mid++
      • 1: Mid++
      • 2: swap a[Mid] and a[Hi]; Hi–

— Dutch National Flag Algorithm, or 3-way Partitioning —

Part way through the process, some red, white and blue elements are known and are in the “right” place. The section of unknown elements, a[Mid..Hi], is shrunk by examining a[Mid]:

Examine a[Mid]. There are three possibilities: 
a[Mid] is (0) red, (1) white or (2) blue.
Case (0) a[Mid] is red, swap a[Lo] and a[Mid]; Lo++; Mid++




Case (1) a[Mid] is white, Mid++




Case (2) a[Mid] is blue, swap a[Mid] and a[Hi]; Hi--



Continue until Mid>Hi. 

Below is C implementation of above algorithm.

• C
• Java

// C program to sort an array with 0,1 and 2 // in a single pass #include   /* Function to swap *a and *b */ void swap(int *a, int *b);   // Sort the input array, the array is assumed to // have values in {0, 1, 2} void sort012(int a[], int arr_size) {     int lo = 0;     int hi = arr_size - 1;     int mid = 0;       while (mid <= hi)     {         switch (a[mid])         {         case 0:             swap(&a[lo++], &a[mid++]);             break;         case 1:             mid++;             break;         case 2:             swap(&a[mid], &a[hi--]);             break;         }     } }   /* UTILITY FUNCTIONS */ void swap(int *a, int *b) {     int temp = *a;     *a = *b;     *b = temp; }   /* Utility function to print array arr[] */ void printArray(int arr[], int arr_size) {     int i;     for (i = 0; i < arr_size; i++)         printf("%d ", arr[i]);     printf("\n"); }   /* driver program to test */ int main() {     int arr[] = {0, 1, 1, 0, 1, 2, 1, 2, 0, 0, 0, 1};     int arr_size = sizeof(arr)/sizeof(arr[0]);     int i;       sort012(arr, arr_size);       printf("array after segregation ");     printArray(arr, arr_size);       getchar();     return 0; }