Sorting 0s, 1s, and 2s with the Dutch National Flag Algorithm

Overview:

The Dutch National Flag Algorithm is a highly efficient technique to sort an array containing only the elements 0, 1, and 2 in linear time. It utilizes a three-pointer strategy and performs sorting directly on the array, optimizing both time and space.

Key Features:

• Three Pointers Strategy:
  The algorithm operates using three pointers (low, mid, and high), each having a specific role:

  • low for positioning 0s
  • mid for traversal
  • high for positioning 2s

• In-Place Sorting:
  This approach does not require additional memory for sorting, making it space-efficient as the array is sorted in-place.

• Optimized for Specific Data:
  Designed specifically for arrays with three distinct elements (0, 1, and 2), the algorithm runs in linear time, making it optimal for such cases.

Time Complexity:

• Best Case: $O(N)$
  The array is traversed once using the mid pointer, providing a linear runtime.

• Average Case: $O(N)$
  Even in random input order, the algorithm processes each element a maximum of one time, maintaining linear time.

• Worst Case: $O(N)$
  Although multiple swaps may be needed, the time complexity remains linear at $O(N)$, where $N$ is the array length.

Space Complexity:

• Space Efficiency: $O(1)$
  The algorithm uses constant extra space as it sorts the array in-place without requiring any auxiliary data structures.

Algorithm Steps:

The algorithm follows these steps to partition the array using three pointers (low, mid, high):

• The range nums[0] to nums[low-1] holds 0s.
• The range nums[low] to nums[mid-1] holds 1s.
• The range nums[high+1] to nums[n-1] holds 2s.

Process:

1. Initialize low and mid at index 0 and high at the last index of the array.
2. While mid is less than or equal to high, follow these rules:
   • If nums[mid] == 0, swap it with nums[low], increment both low and mid.
   • If nums[mid] == 1, increment mid only.
   • If nums[mid] == 2, swap it with nums[high] and decrement high without changing mid.

C++ Code Implementation:

#include <bits/stdc++.h>
using namespace std;

class Solution {
public:
    // Function to sort an array with only 0s, 1s, and 2s
    void sortZeroOneTwo(vector<int>& nums) {
        int low = 0, mid = 0, high = nums.size() - 1;

        while (mid <= high) {
            if (nums[mid] == 0) {
                swap(nums[low], nums[mid]);
                low++;
                mid++;
            } else if (nums[mid] == 1) {
                mid++;
            } else {
                swap(nums[mid], nums[high]);
                high--;
            }
        }
    }
};


int main() {
    vector<int> nums = {0, 2, 1, 2, 0, 1};
    Solution sol;
    sol.sortZeroOneTwo(nums);

    cout << "After sorting: ";
    for (int i = 0; i < nums.size(); i++) {
        cout << nums[i] << " ";
    }
    cout << endl;

    return 0;
}