Four Sum Problem All Unique Quadruplets in DSA Python - Cheat Sheet & Quick Revision

Recall & Review

beginner

What is the Four Sum problem?

The Four Sum problem asks to find all unique groups of four numbers in an array that add up to a given target sum.

beginner

Why do we sort the array before solving the Four Sum problem?

Sorting helps to efficiently skip duplicates and use two-pointer technique to find pairs that sum to a target, reducing complexity.

intermediate

Explain the two-pointer technique used in the Four Sum problem.

After fixing two numbers, two pointers start from the ends of the remaining array segment and move inward to find pairs that sum to the remaining target.

intermediate

How do we avoid duplicate quadruplets in the Four Sum solution?

We skip over numbers that are the same as the previous one at each step (for first, second, and during two-pointer search) to ensure uniqueness.

intermediate

What is the time complexity of the Four Sum problem solution using sorting and two pointers?

The time complexity is O(n^3) because we use two nested loops and a two-pointer scan inside them.

What is the first step in solving the Four Sum problem efficiently?

A Use a hash map

B Sort the array

C Reverse the array

D Remove duplicates first

Sorting the array helps to use two-pointer technique and skip duplicates easily.

In the Four Sum problem, after fixing two numbers, what technique is used to find the other two?

A Two-pointer technique

B Recursion

C Binary search

D Brute force

Two-pointer technique efficiently finds pairs that sum to the remaining target.

How do we ensure quadruplets are unique in the Four Sum problem?

A By sorting only

B By using a set to store quadruplets

C By sorting and skipping duplicates

D By using recursion

Sorting and skipping duplicates at each step prevents repeated quadruplets.

What is the overall time complexity of the Four Sum solution using sorting and two pointers?

A O(n^3)

B O(n log n)

C O(n^4)

D O(n^2)

Two nested loops and a two-pointer scan inside give O(n^3) complexity.

Which of these is NOT a step in the Four Sum solution?

A Sort the array

B Fix two numbers with loops

C Use two pointers to find pairs

D Use dynamic programming

Dynamic programming is not used in the standard Four Sum solution.

Describe the step-by-step approach to solve the Four Sum problem with unique quadruplets.

Think about how sorting and two pointers help find quadruplets without repeats.

Explain how skipping duplicates works in the Four Sum problem to avoid repeated quadruplets.

Duplicates appear next to each other after sorting.

Practice

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

easy

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

B. Sort the array in ascending order

C. Find the maximum element in the array

D. Count the number of pairs that sum to a target

Solution

1. Step 1: Understand the problem statement

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

2. Step 2: Identify the goal

   The goal is not sorting or counting pairs but to find quadruplets with a specific sum without duplicates.

3. Final Answer:

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

4. Quick Check:

   Four Sum = unique quadruplets summing to target [OK]

Hint: Look for four numbers adding to target, no duplicates [OK]

Common Mistakes:

• Confusing Four Sum with Two Sum or Three Sum problems
• Thinking the problem is about sorting or counting pairs
• Ignoring the uniqueness of quadruplets

2. Which of the following Python code snippets correctly initializes an empty list to store quadruplets for the Four Sum problem?

easy

A. quadruplets = ()

B. quadruplets = []

C. quadruplets = {}

D. quadruplets = set()

Solution

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

   Quadruplets are multiple groups of four numbers, so a list is suitable to store them.

2. Step 2: Check the initialization syntax

   Using [] initializes an empty list, which is correct. Curly braces {} create a dictionary, parentheses () create a tuple, and set() creates a set which is unordered and not ideal here.

3. Final Answer:

   quadruplets = [] -> Option B

4. Quick Check:

   Empty list for quadruplets = [] [OK]

Hint: Use [] to store multiple quadruplets in a list [OK]

Common Mistakes:

• Using {} which creates a dictionary, not a list
• Using () which creates an immutable tuple
• Using set() which is unordered and may lose duplicates

3. What will be the output of the following Python code snippet for the Four Sum problem?

nums = [1, 0, -1, 0, -2, 2]
target = 0
# After sorting
nums.sort()
quadruplets = []
for i in range(len(nums)-3):
    for j in range(i+1, len(nums)-2):
        left, right = j+1, len(nums)-1
        while left < right:
            total = nums[i] + nums[j] + nums[left] + nums[right]
            if total == target:
                quadruplets.append([nums[i], nums[j], nums[left], nums[right]])
                left += 1
                right -= 1
            elif total < target:
                left += 1
            else:
                right -= 1
print(quadruplets)

medium

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

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

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

D. []

Solution

1. Step 1: Sort the array and understand the loops

   Sorted nums = [-2, -1, 0, 0, 1, 2]. The nested loops and two-pointer approach check all quadruplets summing to target 0.

2. Step 2: Identify quadruplets that sum to zero

   Valid quadruplets are [-2, -1, 1, 2], [-2, 0, 0, 2], and [-1, 0, 0, 1]. These are appended to quadruplets list.

3. Final Answer:

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

4. Quick Check:

   Sum quadruplets = 0 matches output [OK]

Hint: Sort array, use two pointers inside nested loops [OK]

Common Mistakes:

• Not sorting the array before processing
• Missing some quadruplets due to pointer errors
• Appending duplicates without checks

4. Identify the bug in the following code snippet for the Four Sum problem and select the correct fix.

nums = [2, 2, 2, 2, 2]
target = 8
nums.sort()
quadruplets = []
for i in range(len(nums)-3):
    for j in range(i+1, len(nums)-2):
        left, right = j+1, len(nums)-1
        while left < right:
            total = nums[i] + nums[j] + nums[left] + nums[right]
            if total == target:
                quadruplets.append([nums[i], nums[j], nums[left], nums[right]])
                left += 1
                right -= 1
            elif total < target:
                left += 1
            else:
                right -= 1
print(quadruplets)

medium

A. Add checks to skip duplicates for i, j, left, and right indices

B. Change quadruplets to a set to avoid duplicates

C. Remove sorting to improve performance

D. Replace while loop with a for loop

Solution

1. Step 1: Identify the problem with duplicates

   The code appends quadruplets but does not skip duplicates, so it will add the same quadruplet multiple times.

2. Step 2: Fix by skipping duplicates

   Adding checks to skip duplicate values for i, j, left, and right indices after finding a quadruplet prevents repeated quadruplets.

3. Final Answer:

   Add checks to skip duplicates for i, j, left, and right indices -> Option A

4. Quick Check:

   Skipping duplicates avoids repeated quadruplets [OK]

Hint: Skip duplicates after finding quadruplets to avoid repeats [OK]

Common Mistakes:

• Not skipping duplicates causing repeated quadruplets
• Using set without sorting can cause unordered results
• Removing sorting breaks two-pointer logic

5. Given the array nums = [1, 0, -1, 0, -2, 2, 2] and target 2, which of the following lists contains all unique quadruplets that sum to the target?

hard

A. [[-2, -1, 2, 3], [0, 0, 1, 1]]

B. [[-2, 0, 1, 3], [-1, 0, 0, 3]]

C. [[1, 0, 0, 1], [-2, -1, 2, 3]]

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

Solution

1. Step 1: Sort the array and find quadruplets summing to 2

   Sorted nums = [-2, -1, 0, 0, 1, 2, 2]. We look for quadruplets where sum equals 2.

2. Step 2: Identify valid quadruplets

   Valid quadruplets are [-2, 0, 2, 2] (sum = 2) and [-1, 0, 1, 2] (sum = 2). Other options contain invalid or impossible sums.

3. Final Answer:

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

4. Quick Check:

   Sum quadruplets = 2 matches output [OK]

Hint: Sort and check quadruplets summing exactly to target [OK]

Common Mistakes:

• Including quadruplets with wrong sums
• Ignoring duplicates or invalid numbers
• Not sorting before searching quadruplets