Dutch National Flag Three Way Partition in DSA Java - Time & Space Complexity

Time Complexity: Dutch National Flag Three Way Partition

O(n)

Understanding Time Complexity

We want to understand how the time needed to sort an array into three parts grows as the array gets bigger.

Specifically, how many steps does the Dutch National Flag algorithm take as input size increases?

Scenario Under Consideration

Analyze the time complexity of the following code snippet.

public void threeWayPartition(int[] arr, int lowVal, int highVal) {
    int start = 0, end = arr.length - 1;
    int i = 0;
    while (i <= end) {
        if (arr[i] < lowVal) {
            int temp = arr[start];
            arr[start] = arr[i];
            arr[i] = temp;
            start++;
            i++;
        } else if (arr[i] > highVal) {
            int temp = arr[end];
            arr[end] = arr[i];
            arr[i] = temp;
            end--;
        } else {
            i++;
        }
    }
}
    

This code sorts the array into three parts: less than lowVal, between lowVal and highVal, and greater than highVal.

Identify Repeating Operations

Identify the loops, recursion, array traversals that repeat.

• Primary operation: The while loop that moves through the array elements.
• How many times: Each element is visited at most once or twice, but the loop runs until the index crosses the end pointer.

How Execution Grows With Input

As the array size grows, the number of steps grows roughly in direct proportion to the number of elements.

Input Size (n)	Approx. Operations
10	About 10 to 20 steps
100	About 100 to 200 steps
1000	About 1000 to 2000 steps

Pattern observation: The steps grow roughly linearly with input size, sometimes a bit more due to swaps, but never more than a constant factor times n.

Final Time Complexity

Time Complexity: O(n)

This means the time to complete the partition grows in a straight line as the array gets bigger.

Common Mistake

[X] Wrong: "Because there are nested ifs and swaps, the time must be quadratic like O(n²)."

[OK] Correct: The loop moves forward or backward carefully so each element is handled only a few times, not repeatedly nested.

Interview Connect

Knowing this linear time partition helps you solve many sorting and grouping problems efficiently, a skill valued in coding challenges and interviews.

Self-Check

"What if we used a simple for-loop without adjusting the end pointer when swapping? How would the time complexity change?"

Practice

1. What is the main goal of the Dutch National Flag Three Way Partition algorithm?

easy

A. To reverse the elements of an array

B. To rearrange an array into three parts based on a pivot value

C. To find the maximum element in an array

D. To sort an array in ascending order

Solution

1. Step 1: Understand the algorithm's purpose

   The Dutch National Flag algorithm divides an array into three parts: elements less than, equal to, and greater than a pivot.

2. Step 2: Compare with other options

   Sorting or finding max are different tasks; reversing is unrelated.

3. Final Answer:

   To rearrange an array into three parts based on a pivot value -> Option B

4. Quick Check:

   Partitioning by pivot = D [OK]

Hint: Remember: It partitions into three groups around a pivot [OK]

Common Mistakes:

• Confusing partitioning with sorting
• Thinking it finds max or min values
• Assuming it reverses the array

2. Which of the following is the correct Java syntax to swap two elements arr[i] and arr[j] in an integer array?

easy

A. int temp = arr[i]; arr[i] = arr[j]; arr[j] = temp;

B. arr[i] = arr[j]; arr[j] = arr[i];

C. swap(arr[i], arr[j]);

D. int temp = arr[i]; arr[i] = arr[j]; arr[j] = arr[i];

Solution

1. Step 1: Understand swapping logic

   Swapping requires a temporary variable to hold one value while assigning the other.

2. Step 2: Check each option

   swap(arr[i], arr[j]); assumes a non-existent swap function; arr[i] = arr[j]; arr[j] = arr[i]; overwrites arr[i] without saving; int temp = arr[i]; arr[i] = arr[j]; arr[j] = arr[i]; loses original arr[i] value by assigning arr[j] = arr[i] after changing arr[i]; int temp = arr[i]; arr[i] = arr[j]; arr[j] = temp; correctly uses temp.

3. Final Answer:

   int temp = arr[i]; arr[i] = arr[j]; arr[j] = temp; -> Option A

4. Quick Check:

   Use temp variable for swap = C [OK]

Hint: Use a temp variable to swap two array elements safely [OK]

Common Mistakes:

• Overwriting one element before saving it
• Using a non-existent swap function
• Swapping without a temporary variable

3. What will be the output array after applying the Dutch National Flag algorithm on arr = [2, 0, 2, 1, 1, 0] with pivot = 1?

medium

A. [0, 1, 2, 0, 1, 2]

B. [2, 2, 1, 1, 0, 0]

C. [1, 1, 0, 0, 2, 2]

D. [0, 0, 1, 1, 2, 2]

Solution

1. Step 1: Partition array by pivot 1

   Elements less than 1: 0,0; equal to 1: 1,1; greater than 1: 2,2.

2. Step 2: Arrange in order less, equal, greater

   So the array becomes [0, 0, 1, 1, 2, 2].

3. Final Answer:

   [0, 0, 1, 1, 2, 2] -> Option D

4. Quick Check:

   Partitioned array = A [OK]

Hint: Group elements less, equal, then greater than pivot [OK]

Common Mistakes:

• Not grouping equal elements in the middle
• Mixing order of groups
• Ignoring pivot value placement

4. Identify the error in this Java code snippet for Dutch National Flag partition:

int low = 0, mid = 0, high = arr.length - 1;
while (mid <= high) {
  if (arr[mid] < pivot) {
    swap(arr[low], arr[mid]);
    low++;
    mid++;
  } else if (arr[mid] == pivot) {
    mid++;
  } else {
    swap(arr[mid], arr[high]);
    high--;
  }
}

medium

A. The pivot comparison should use > instead of <

B. The loop condition should be mid < high

C. swap function is called incorrectly with array elements instead of indices

D. The variables low, mid, high are not initialized properly

Solution

1. Step 1: Check swap function usage

   swap(arr[low], arr[mid]) passes values, not indices; swap needs indices to swap elements in array.

2. Step 2: Verify other parts

   Loop condition and variable initialization are correct; pivot comparisons are standard.

3. Final Answer:

   swap function is called incorrectly with array elements instead of indices -> Option C

4. Quick Check:

   Swap needs indices, not values = A [OK]

Hint: Swap array elements by indices, not by passing values [OK]

Common Mistakes:

• Passing values instead of indices to swap
• Wrong loop boundary condition
• Incorrect pivot comparison operators

5. Given an array arr = [3, 5, 2, 1, 4, 3, 2] and pivot = 3, how many swaps will the Dutch National Flag algorithm perform to partition the array?

hard

A. 5

B. 4

C. 3

D. 6

Solution

1. Step 1: Trace swaps during partition

   Initial: low=0, mid=0, high=6 - mid=0 (3 == pivot): mid++ (0 swaps) - mid=1 (5 > pivot): swap arr[mid] and arr[high] -> swap arr[1] and arr[6] (swap 1), high--=5 - mid=1 (now 2 < pivot): swap arr[low] and arr[mid] -> swap arr[0] and arr[1] (swap 2), low++=1, mid++=2 - mid=2 (2 < pivot): swap arr[low] and arr[mid] -> swap arr[1] and arr[2] (swap 3), low++=2, mid++=3 - mid=3 (1 < pivot): swap arr[low] and arr[mid] -> swap arr[2] and arr[3] (swap 4), low++=3, mid++=4 - mid=4 (4 > pivot): swap arr[mid] and arr[high] -> swap arr[4] and arr[5] (swap 5), high--=4 - mid=4 (now 3 == pivot): mid++=5 Loop ends as mid > high.

2. Step 2: Count total swaps

   Total swaps counted = 5.

3. Final Answer:

   5 -> Option A

4. Quick Check:

   Count swaps carefully during partition = B [OK]

Hint: Count swaps when elements move across low, mid, high pointers [OK]

Common Mistakes:

• Counting swaps incorrectly when pivot equals element
• Ignoring swaps after high pointer moves
• Assuming one swap per iteration