Four Sum Problem All Unique Quadruplets in DSA C# - Time & Space Complexity

Time Complexity: Four Sum Problem All Unique Quadruplets

O(n³)

Understanding Time Complexity

We want to understand how the time needed grows when finding all unique sets of four numbers that add up to a target.

How does the work increase as the list of numbers gets bigger?

Scenario Under Consideration

Analyze the time complexity of the following code snippet.

public IList> FourSum(int[] nums, int target) {
    Array.Sort(nums);
    var result = new List>();
    int n = nums.Length;
    for (int i = 0; i < n - 3; i++) {
        for (int j = i + 1; j < n - 2; j++) {
            int left = j + 1, right = n - 1;
            while (left < right) {
                int sum = nums[i] + nums[j] + nums[left] + nums[right];
                if (sum == target) {
                    result.Add(new List{nums[i], nums[j], nums[left], nums[right]});
                    left++; right--;
                } else if (sum < target) {
                    left++;
                } else {
                    right--;
                }
            }
        }
    }
    return result;
}
    

This code finds all unique quadruplets in the array that sum to the target by sorting and using two nested loops plus a two-pointer approach.

Identify Repeating Operations

Identify the loops, recursion, array traversals that repeat.

• Primary operation: The nested loops over i and j, and the two-pointer while loop inside.
• How many times: Outer loop runs about n times, second loop about n times, and the two-pointer loop runs up to n times in total for each pair.

How Execution Grows With Input

As the input size grows, the number of operations grows roughly with the cube of the input size.

Input Size (n)	Approx. Operations
10	About 1,000
100	About 1,000,000
1000	About 1,000,000,000

Pattern observation: When input size doubles, operations increase roughly eight times (cube growth).

Final Time Complexity

Time Complexity: O(n³)

This means the time needed grows roughly with the cube of the input size, so tripling the input makes the work about 27 times bigger.

Common Mistake

[X] Wrong: "The time complexity is O(n⁴) because there are four numbers to pick."

[OK] Correct: The two-pointer approach inside the loops reduces one loop, so the complexity is O(n³), not O(n⁴).

Interview Connect

Understanding this time complexity helps you explain how nested loops and pointers affect performance, a key skill in solving array problems efficiently.

Self-Check

"What if we replaced the two-pointer approach with another nested loop? How would the time complexity change?"

Practice

1. What is the main goal of the Four Sum Problem in data structures and algorithms?

easy

A. Sort the array in ascending order

B. Count the number of pairs with a given sum

C. Find the maximum element in the array

D. Find all unique quadruplets in an array that sum up to a target value

Solution

1. Step 1: Understand the problem statement

   The Four Sum Problem asks to find all unique sets of four numbers in an array that add up to a specific target sum.

2. Step 2: Differentiate from other problems

   Unlike sorting or finding maximum, this problem focuses on quadruplets and their sums, not individual elements or pairs.

3. Final Answer:

   Find all unique quadruplets in an array that sum up to a target value -> Option D

4. Quick Check:

   Four Sum Problem = Find quadruplets with target sum [OK]

Hint: Look for four numbers adding to target, not sorting or max [OK]

Common Mistakes:

• Confusing with sorting or max element problems
• Looking for pairs instead of quadruplets
• Ignoring uniqueness of quadruplets

2. Which of the following C# code snippets correctly initializes a list to store quadruplets for the Four Sum Problem?

easy

A. var quadruplets = new List<List<int>>();

B. var quadruplets = new List<int>();

C. var quadruplets = new Dictionary<int, int>();

D. var quadruplets = new int[4];

Solution

1. Step 1: Identify the data structure for storing quadruplets

   Each quadruplet is a list of 4 integers, so a list of lists is needed to store multiple quadruplets.

2. Step 2: Match with C# syntax

   Using List<List<int>> correctly represents multiple quadruplets, unlike a single list or dictionary.

3. Final Answer:

   var quadruplets = new List<List<int>>(); -> Option A

4. Quick Check:

   Use List of List for quadruplets [OK]

Hint: Use List<List<int>> to store multiple quadruplets [OK]

Common Mistakes:

• Using List<int> for multiple quadruplets
• Using Dictionary instead of List
• Using fixed-size array for dynamic quadruplets

3. For the input array nums = [1, 0, -1, 0, -2, 2] and target 0, what does FourSum(nums, 0) return?
Assuming FourSum returns all unique quadruplets that sum to target.

medium

A. []

B. [[1, 0, -1, 0]]

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

D. [[0, 0, 0, 0]]

Solution

1. Step 1: Sort and find quadruplets summing to 0

   Sorting nums gives [-2, -1, 0, 0, 1, 2]. The unique quadruplets summing to 0 are [-2, -1, 1, 2], [-2, 0, 0, 2], and [-1, 0, 0, 1].

2. Step 2: Verify uniqueness and completeness

   All quadruplets are unique and sum exactly to 0, matching the expected output.

3. Final Answer:

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

4. Quick Check:

   Unique quadruplets sum to 0 = [[-2, -1, 1, 2], [-2, 0, 0, 2], [-1, 0, 0, 1]] [OK]

Hint: Sort array first, then find quadruplets with two pointers [OK]

Common Mistakes:

• Not sorting array before processing
• Including duplicate quadruplets
• Missing some valid quadruplets

4. Identify the bug in this C# snippet for the Four Sum Problem:

for (int i = 0; i < nums.Length - 3; i++) {
    for (int j = i + 1; j < nums.Length - 2; j++) {
        int left = j + 1, right = nums.Length - 1;
        while (left < right) {
            int sum = nums[i] + nums[j] + nums[left] + nums[right];
            if (sum == target) {
                result.Add(new List<int> { nums[i], nums[j], nums[left], nums[right] });
                left++;
                right--;
            } else if (sum < target) {
                left++;
            } else {
                right--;
            }
        }
    }
}

medium

A. No check to skip duplicate quadruplets causing duplicates in result

B. Incorrect initialization of left and right pointers

C. Wrong loop ranges for i and j causing index out of range

D. Missing sorting of the input array before processing

Solution

1. Step 1: Analyze duplicate handling

   The code adds quadruplets without skipping duplicates, so repeated quadruplets can appear in the result.

2. Step 2: Check other parts

   Left and right pointers are correctly initialized; loop ranges are valid; sorting is assumed done before this snippet.

3. Final Answer:

   No check to skip duplicate quadruplets causing duplicates in result -> Option A

4. Quick Check:

   Missing duplicate skip = No check to skip duplicate quadruplets causing duplicates in result [OK]

Hint: Always skip duplicates after finding quadruplets [OK]

Common Mistakes:

• Not skipping duplicates for i, j, left, right
• Assuming sorting is done inside the loop
• Incorrect pointer updates causing infinite loops

5. Given the array nums = [2, 2, 2, 2, 2] and target 8, what is the correct output of the Four Sum Problem algorithm that finds all unique quadruplets?

medium

A. [[2, 2, 2, 2], [2, 2, 2, 2]]

B. [[2, 2, 2, 2]]

C. []

D. [[2, 2, 2]]

Solution

1. Step 1: Understand the input and target

   All elements are 2, and target is 8, so quadruplets of four 2's sum to 8.

2. Step 2: Identify unique quadruplets

   Only one unique quadruplet [2, 2, 2, 2] exists despite duplicates in input.

3. Final Answer:

   [[2, 2, 2, 2]] -> Option B

4. Quick Check:

   Only one unique quadruplet sums to 8 [OK]

Hint: Duplicates form one unique quadruplet if sum matches target [OK]

Common Mistakes:

• Counting duplicate quadruplets multiple times
• Returning empty list when quadruplet exists
• Returning triplets instead of quadruplets