Dutch National Flag Algorithm

Dutch National Flag Algorithm

In the field of computer science, algorithms often take inspiration from real-world events, and the Dutch national flag algorithm is a prime example. This algorithm, although seemingly abstract, has its roots deeply embedded in the vibrant tricolor of the Dutch national flag. Let’s embark on a journey to understand its origins, design, and the historical context that gave it its unique name.

The Dutch national flag, characterized by its three horizontal stripes of red, white and blue, serves as a visual metaphor for this algorithm. Just as the flag has three different colors, the algorithm deals with three unique elements, often represented as 0, 1, and 2. Objective? Sorting an array consisting of these three elements in a manner reminiscent of the bands of a flag.

Historical Context: Edsger W. Dijkstra’s Brainchild

The Dutch national flag algorithm was pioneered by computer science giant Edsger W. Dijkstra. Hailing from the Netherlands, Dijkstra’s Dutch heritage played a role in the naming of this algorithm. In the 1970s, while considering computational problems, Dijkstra conceived of an algorithmic challenge that mirrored the tricolor of his country’s flag. This thought experiment led to the formulation of the “three-way partition problem”, which we now recognize as the Dutch national flag problem.

Dijkstra’s primary goal was not just to sort the elements but to do it efficiently. He aimed for an algorithm that could rearrange elements in linear time by making a single pass through the array, using minimal additional space. His solution was both elegant and efficient, and it soon gained popularity in the computer science community.

Why the Name Matters

In computer science, especially for algorithms, names are often chosen for their descriptive nature, helping practitioners understand the core concept at a glance. The name “Dutch National Flag Algorithm” does exactly this. It encapsulates the essence of the problem – sorting out three distinct elements – while paying homage to its historical and cultural origins. The name acts as a bridge, connecting the abstract world of algorithms to tangible real-world imagery, making the concept more relevant and easier to imagine.

Problem Statement

Among the vast landscape of algorithmic challenges, the Dutch national flag problem stands out for its simplicity in premise but depth in implementation. At its core, it is a sorting problem, but its constraints and objectives lead it to a unique place in the world of algorithms.

Defining the Problem

Imagine an array, a linear data structure, filled with three distinct elements: 0, 1, and 2. These numbers, while arbitrary, can represent anything from colors to categories. . The Dutch national flag problem poses a simple question: How can we rearrange this table so that all 0’s appear first, then all 1’s, and finally, all 2’s?

To draw a parallel line, imagine the three horizontal stripes of the Dutch flag: red (0), white (1), and blue (2). The task is to transform our array into a representation of this flag, with each “color” or number grouped together.

Objective and Constraints

While the goal is clear – to sort the array – the constraints complicate the problem:

Linear time complexity: The algorithm must traverse the array only once. This means that as the size of the array (n) increases, the number of operations must increase linearly, which is represented as O(n). Multiple passes or nested loops are off the table.

Constant space complexity: External storage or data structures that grow with input size are not allowed. The sorting must happen in place, within the array itself, ensuring that the space used remains constant, which is denoted as O(1).

The Significance of Efficiency

Why insist on linear time and constant space? In the real world, data can be huge. An algorithm that makes multiple passes over large datasets may be excessively slow, making it impractical for real-time applications. Similarly, if an algorithm requires additional storage disproportionate to the data size, it can quickly exhaust system memory, especially when working with large-scale datasets.

Achieving O(n) time complexity ensures that our algorithm remains scalable and can handle large datasets efficiently. Constant space complexity, O(1), guarantees minimal memory usage, making the algorithm resource-friendly.

Delving into the Dutch National Flag Algorithm: A Comprehensive Guide

In the world of algorithms, the Dutch national flag stands as a symbol of algorithmic beauty and efficiency. While its primary goal is to sort an array with three distinct elements, the method it uses is both interesting and practical. Let us embark on a journey to understand its inner workings.

The Tri-Element Sorting Challenge

Before diving into the algorithm, it is important to understand why it is important to sort an array with only three unique elements. In most sorting algorithms, the focus is on sorting elements based on their inherent value, from smallest to largest or vice versa. However, when we know that there are only three distinct elements, we can take advantage of this knowledge to optimize our sorting process, making it faster and more efficient than normal sorting algorithms.

Embracing the Two-Pointer Approach

At the heart of the Dutch national flag algorithm lies the two-pointer technique, although, in this case, it is more appropriate to call it a “three-pointer” approach. The algorithm employs three indicators:

  1. Low indicator: This indicator marks the range where 0 ends. Everything to the left of this pointer (except itself) is 0.
  2. Middle pointer: This is our active pointer, which scans the array. It represents the current element under inspection.
  3. High indicator: This indicator shows where the 2 starts. Everything to the right of it (except itself) is 2.

Step-by-Step Breakdown of the Algorithm

Initialization of pointers:
  1. Low is set to the beginning of the array.
  2. The middle also starts at the beginning, ready to traverse the array.
  3. High is initialized to the end of the array.
Recurrence/Itteration conditions:

The core of the algorithm lies in a loop where the middle is less than or equal to the high. This ensures that the entire array is scanned, and all elements are positioned correctly.

Handling different element values:

  1. If the middle points to 0: it means that this 0 is out of its place and should be moved to the beginning. Swap the low and middle elements, then move both low and middle one step.
  2. If the middle points to 1: it is in its correct area. Just take a step forward in the middle.
  3. If the middle points to 2: This 2 should be at the end. Swap elements on the middle and high, then take a step back to the high position. Note that we do not move to the middle here because the new element in the middle (after the swap) has not been inspected yet.

Swap arguments and pointer movements:

  1. Swapping is a fundamental operation in this algorithm. Thus we rearrange the elements. When we get a 0 or 2 on the middle pointer, we swap it with the element on the low or high pointer respectively.
  2. The movement of indicators is important. After each operation, we adjust our pointers to continue our inspection and ensure that each element lands in its correct position.

C++ Implementation of the Dutch National Flag Algorithm

Diving into the Code

To truly grasp the essence of the Dutch National Flag Algorithm, let’s delve into its C++ implementation. This hands-on approach will provide a tangible understanding of the algorithm’s mechanics.   

using namespace std;

void dutchFlagSort(int arr[], int n) {
    int low = 0, mid = 0, high = n - 1;

    while (mid <= high) {
        switch (arr[mid]) {
            case 0:
                swap(arr[low++], arr[mid++]);
            case 1:
            case 2:
                swap(arr[mid], arr[high--]);

int main() {
    int arr[] = {1, 2, 0, 2, 1, 0, 1, 2, 0};
    int n = sizeof(arr) / sizeof(arr[0]);

    cout << "Original Array: ";
    for (int i = 0; i < n; i++) {
        cout << arr[i] << " ";
    cout << endl;

    dutchFlagSort(arr, n);

    cout << "Sorted Array: ";
    for (int i = 0; i < n; i++) {
        cout << arr[i] << " ";
    cout << endl;

    return 0;


Code Explanation

  • Initialization: Three pointers low, mid, and high are initialized to represent the three regions of the array.

  • Loop Iteration: The while loop continues until the mid pointer surpasses the high pointer, ensuring every element is inspected.

  • Switch-Case Logic: Depending on the value at the mid pointer, we either swap the element to its correct position or move the pointer forward.

Sample Output:

					Original Array: 1 2 0 2 1 0 1 2 0 
Sorted Array: 0 0 0 1 1 1 2 2 2 


Time and Space Complexity Analysis

Time Complexity: O(n)

The beauty of the Dutch national flag algorithm lies in its efficiency. This inspects each element of the array exactly once, yielding a linear time complexity of O(n), where n is the number of elements in the array.

Space Complexity: O(1)

The algorithm also shines in its space efficiency. It operates in-house, meaning it does not require any additional storage that grows with the input size. This results in a constant space complexity of O(1).

Why does complexity matter?

Time and space complexities are important metrics in algorithm analysis. They provide insight into the scalability and resource efficiency of algorithms. An algorithm like the Dutch national flag, with its O(n) time and O(1) space complexities, ensures fast execution even for large datasets while being memory-efficient.

Practical Applications of the Dutch National Flag Algorithm


Of course, the primary application is to sort arrays with three distinct elements. This is especially useful when these elements represent ranges rather than numerical values.


Algorithms can be employed for data partitioning tasks. For example, in a dataset where data points may belong to one of three categories, the algorithm can separate these efficiently.

Data Cleansing:

During preprocessing in data analytics or machine learning tasks, algorithms can help clean datasets. It can group data points based on specific characteristics, making further processing more streamlined.

String Manipulation:

Beyond numbers, the algorithm can be adapted to strings. If there are three different patterns or characters in a string that need to be ordered, the Dutch national flag algorithm can be a valuable tool.

Lets check how professor can ask you for Problem based on Dutch National Flag Algorithm?

Imagine you’re presented with a sequence, A[], exclusively populated by the trio: 0s, 1s, and 2s. Your mission, should you choose to accept it, is to conjure a function that meticulously rearranges this sequence. The catch? All 0s must lead the parade, followed closely by the 1s, with the 2s bringing up the rear.

This enigma might ring a bell, echoing the renowned “Dutch National Flag conundrum” postulated by the legendary Edsger Dijkstra. To paint a vivid picture:

Visualize N orbs, each drenched in one of three hues: crimson (red), alabaster (white), or azure (blue), haphazardly aligned. Your challenge is to orchestrate these orbs so that those sharing the same shade huddle together. The sequence of shades is non-negotiable: crimson must pave the way, succeeded by alabaster, with azure concluding the procession. Can you master this chromatic ballet?

					#include <iostream>
using namespace std;

// Function to sort the array containing only 0s, 1s, and 2s
void sortColors(int A[], int N) {
    // 'low' pointer initialized at the start of the array.
    // It represents the position where the next '0' should be placed.
    int low = 0;

    // 'mid' pointer initialized at the start of the array.
    // It is used to traverse the array and inspect each element.
    int mid = 0;

    // 'high' pointer initialized at the end of the array.
    // It represents the position where the next '2' should be placed.
    int high = N - 1;

    // Loop until 'mid' pointer crosses the 'high' pointer
    while (mid <= high) {
        // Use switch-case to handle different values at 'mid' pointer
        switch (A[mid]) {
            // If the element is '0'
            case 0:
                // Swap the element at 'mid' with the element at 'low'
                swap(A[low], A[mid]);
                // Increment both 'low' and 'mid' pointers

            // If the element is '1'
            case 1:
                // Simply move the 'mid' pointer forward

            // If the element is '2'
            case 2:
                // Swap the element at 'mid' with the element at 'high'
                swap(A[mid], A[high]);
                // Decrement the 'high' pointer

int main() {
    int N;
    cout << "Enter the size of the array: ";
    cin >> N;

    int A[N];
    cout << "Enter the elements of the array (0, 1, or 2): ";
    for (int i = 0; i < N; i++) {
        cin >> A[i];

    // Call the function to sort the array
    sortColors(A, N);

    cout << "Sorted Array: ";
    for (int i = 0; i < N; i++) {
        cout << A[i] << " ";
    cout << endl;

    return 0;

Sample Input
					Enter the size of the array: 9
Enter the elements of the array (0, 1, or 2): 1 2 0 2 1 0 1 2 0

Sample Output
					Sorted Array: 0 0 0 1 1 1 2 2 2


The Dutch national flag algorithm, which has its roots in a simple sorting challenge, has far-reaching applications in the real world. Its efficiency, both in terms of time and space, makes it the preferred choice for various computational tasks. Whether you’re sorting, partitioning, cleaning data, or manipulating strings, this algorithm has you covered.


FAQ and Viva Voce Questions on the Dutch National Flag Algorithm Tutorial

The Dutch national flag algorithm is a sorting algorithm used to rearrange an array containing three distinct elements (0, 1, 2) in a specific order. It is named after the three horizontal stripes of the colors of the Dutch national flag.

The Dutch National Flag Algorithm was introduced by Edsger W. Dijkstra.

The algorithm uses three pointers: low, mid, and high. The low pointer represents the end boundary of 0s, the mid pointer actively scans the array, and the high pointer delineates the start boundary of 2s.

Achieving O(n) time complexity ensures that the algorithm can handle larger datasets efficiently, making a single pass through the array. O(1) space complexity ensures minimal memory usage, making the algorithm resource-friendly.

Yes, the algorithm can be adapted for strings. If a string has three distinct patterns or characters that need ordering, the Dutch National Flag Algorithm can be employed.

The primary objective is to rearrange an array such that all 0s appear first, followed by all 1s, and finally, all 2s.

When the middle pointer encounters ‘1’, it simply moves forward, as ‘1’ is considered to be in its correct area.

The algorithm is named after the Dutch national flag because it sorts an array in a manner reminiscent of the flag’s three horizontal bands of colors: red (0), white (1), and blue (2).

The algorithm has practical applications in sorting arrays with three distinct elements, data partitioning, data cleaning during preprocessing tasks, and string manipulation.

The algorithm uses a while loop that continues until the mid pointer surpasses the high pointer, ensuring that every element in the array is inspected.

When the mid pointer encounters a ‘2’, it swaps this value with the value at the high pointer and then decrements the high pointer. The mid pointer doesn’t move forward immediately because the new element at mid (after the swap) hasn’t been inspected yet.

The Dutch National Flag Algorithm is considered efficient because it can sort the array in a single pass (linear time complexity O(n)) and operates in-place without requiring additional storage (constant space complexity O(1)).

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