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DSA Visualizations /

Overview of TechniquesOverview of Techniques14
❯

  1. 1Overview of DSA Problem Solving Techniques
  2. 2Brute Force Technique in DSA
  3. 3Greedy Algorithm Technique in DSA
  4. 4Divide and Conquer Technique in DSA
  5. 5Dynamic Programming Technique in DSA
  6. 6Backtracking Technique in DSA
  7. 7Recursion Technique in DSA
  8. 8Sliding Window Technique in DSA
  9. 9Two Pointers Technique
  10. 10Binary Search Technique
  11. 11Tree / Graph Traversal Technique in DSA
  12. 12Bit Manipulation Technique in DSA
  13. 13Hashing Technique
  14. 14Heaps Technique in DSA

SortingSorting10
❯

  1. 1Bubble Sort
  2. 2Selection Sort
  3. 3Insertion Sort
  4. 4Merge Sort
  5. 5Quick Sort
  6. 6Heap Sort
  7. 7Radix Sort
  8. 8Counting Sort
  9. 9Bucket Sort
  10. 10Shell Sort

ArraysArrays32
❯

  1. 1Find Maximum and Minimum in Array using Loop
  2. 2Find Second Largest in Array
  3. 3Find Second Smallest in Array
  4. 4Reverse Array using Two Pointers
  5. 5Check if Array is Sorted
  6. 6Remove Duplicates from Sorted Array
  7. 7Left Rotate an Array by One Place
  8. 8Left Rotate an Array by K Places
  9. 9Move Zeroes in Array to End
  10. 10Linear Search in Array
  11. 11Union of Two Arrays
  12. 12Find Missing Number in Array
  13. 13Max Consecutive Ones in Array
  14. 14Find Kth Smallest Element
  15. 15Longest Subarray with Given Sum (Positives)
  16. 16Longest Subarray with Given Sum (Positives and Negatives)
  17. 17Find Majority Element in Array (more than n/2 times)
  18. 18Find Majority Element in Array (more than n/3 times)
  19. 19Maximum Subarray Sum using Kadane's Algorithm
  20. 20Print Subarray with Maximum Sum
  21. 21Stock Buy and Sell
  22. 22Rearrange Array Alternating Positive and Negative Elements
  23. 23Next Permutation of Array
  24. 24Leaders in an Array
  25. 25Longest Consecutive Sequence in Array
  26. 26Count Subarrays with Given Sum
  27. 27Sort an Array of 0s, 1s, and 2s
  28. 28Two Sum Problem
  29. 29Three Sum Problem
  30. 304 Sum Problem
  31. 31Find Length of Largest Subarray with 0 Sum
  32. 32Find Maximum Product Subarray

Arrays - Binary SearchArrays - Binary Search28
❯

  1. 1Binary Search in Array<br><span class="highlight">Using Iteration</span>
  2. 2Find Lower Bound in Sorted Array
  3. 3Find Upper Bound in Sorted Array
  4. 4Search Insert Position in Sorted Array (Lower Bound Approach)
  5. 5Floor and Ceil in Sorted Array
  6. 6First Occurrence in a Sorted Array
  7. 7Last Occurrence in a Sorted Array
  8. 8Count Occurrences in Sorted Array
  9. 9Search Element in a Rotated Sorted Array
  10. 10Search in Rotated Sorted Array with Duplicates
  11. 11Minimum in Rotated Sorted Array
  12. 12Find Rotation Count in Sorted Array
  13. 13Search Single Element in Sorted Array
  14. 14Find Peak Element in Array
  15. 15Square Root using Binary Search
  16. 16Nth Root of a Number using Binary Search
  17. 17Koko Eating Bananas
  18. 18Minimum Days to Make M Bouquets
  19. 19Find the Smallest Divisor Given a Threshold
  20. 20Capacity to Ship Packages within D Days
  21. 21Kth Missing Positive Number
  22. 22Aggressive Cows Problem
  23. 23Allocate Minimum Number of Pages
  24. 24Split Array - Minimize Largest Sum
  25. 25Painter's Partition Problem
  26. 26Minimize Maximum Distance Between Gas Stations
  27. 27Median of Two Sorted Arrays of Different Sizes
  28. 28K-th Element of Two Sorted Arrays

StringsStrings2
❯

  1. 1Remove Outermost Parentheses in String
  2. 2Edit Distance Problem

MatrixMatrix3
❯

  1. 1Set Matrix Zeroes
  2. 2Rotate Matrix by 90 Degrees Clockwise
  3. 3Print Matrix in Spiral Manner

Matrix - Binary SearchMatrix - Binary Search5
❯

  1. 1Row with Maximum Number of 1's
  2. 2Search in a Sorted 2D Matrix
  3. 3Search in Row and Column-wise Sorted Matrix
  4. 4Find Target in 2D Matrix
  5. 5Matrix Median

Binary TreesBinary Trees36
❯

  1. 1Preorder Traversal of a Binary Tree using Recursion
  2. 2Preorder Traversal of a Binary Tree using Iteration
  3. 3Postorder Traversal of a Binary Tree Using Recursion
  4. 4Postorder Traversal of a Binary Tree using Iteration
  5. 5Level Order Traversal of a Binary Tree using Recursion
  6. 6Level Order Traversal of a Binary Tree using Iteration
  7. 7Reverse Level Order Traversal of a Binary Tree using Iteration
  8. 8Reverse Level Order Traversal of a Binary Tree using Recursion
  9. 9Find Height of a Binary Tree
  10. 10Find Diameter of a Binary Tree
  11. 11Find Mirror of a Binary Tree - Todo
  12. 12Inorder Traversal of a Binary Tree using Recursion
  13. 13Inorder Traversal of a Binary Tree using Iteration
  14. 14Left View of a Binary Tree
  15. 15Right View of a Binary Tree
  16. 16Top View of a Binary Tree
  17. 17Bottom View of a Binary Tree
  18. 18Zigzag Traversal of a Binary Tree
  19. 19Check if a Binary Tree is Balanced
  20. 20Diagonal Traversal of a Binary Tree
  21. 21Boundary Traversal of a Binary Tree
  22. 22Construct a Binary Tree from a String with Bracket Representation
  23. 23Convert a Binary Tree into a Doubly Linked List
  24. 24Convert a Binary Tree into a Sum Tree
  25. 25Find Minimum Swaps Required to Convert a Binary Tree into a BST
  26. 26Check if a Binary Tree is a Sum Tree
  27. 27Check if All Leaf Nodes are at the Same Level in a Binary Tree
  28. 28Lowest Common Ancestor (LCA) in a Binary Tree
  29. 29Solve the Tree Isomorphism Problem
  30. 30Check if a Binary Tree Contains Duplicate Subtrees of Size 2 or More
  31. 31Check if Two Binary Trees are Mirror Images
  32. 32Calculate the Sum of Nodes on the Longest Path from Root to Leaf in a Binary Tree
  33. 33Print All Paths in a Binary Tree with a Given Sum
  34. 34Find the Distance Between Two Nodes in a Binary Tree
  35. 35Find the kth Ancestor of a Node in a Binary Tree
  36. 36Find All Duplicate Subtrees in a Binary Tree

Binary Search TreesBinary Search Trees14
❯

  1. 1Find a Value in a Binary Search Tree
  2. 2Delete a Node in a Binary Search Tree
  3. 3Find the Minimum Value in a Binary Search Tree
  4. 4Find the Maximum Value in a Binary Search Tree
  5. 5Find the Inorder Successor in a Binary Search Tree
  6. 6Find the Inorder Predecessor in a Binary Search Tree
  7. 7Check if a Binary Tree is a Binary Search Tree
  8. 8Find the Lowest Common Ancestor of Two Nodes in a Binary Search Tree
  9. 9Convert a Binary Tree into a Binary Search Tree
  10. 10Balance a Binary Search Tree
  11. 11Merge Two Binary Search Trees
  12. 12Find the kth Largest Element in a Binary Search Tree
  13. 13Find the kth Smallest Element in a Binary Search Tree
  14. 14Flatten a Binary Search Tree into a Sorted List

4 Sum Problem
Optimal Solution using Two Pointers

Topic Contents

ProblemExamplesSolutionCodeTime ComplexitySpace Complexity

Problem Statement

Given an array of integers arr and an integer target, find all unique quadruplets (four numbers) in the array that sum up to the given target value.

If no such quadruplets exist, return an empty list.

Examples❯

Input ArrayTargetQuadrupletsDescription
[1, 0, -1, 0, -2, 2]0[[-2, -1, 1, 2], [-2, 0, 0, 2], [-1, 0, 0, 1]]Three unique quadruplets sum to 0
[2, 2, 2, 2, 2]8[[2, 2, 2, 2]]Duplicates allowed in input, but result is unique
[1, 2, 3, 4]50[]No combination of four elements adds to 50
[0, 0, 0, 0]0[[0, 0, 0, 0]]All elements are zero and they form one valid result
[1, -2, -5, -4, -3, 3, 3, 5]-11[[-5, -4, -3, 1]]Only one valid quadruplet with negative target
[1, 2, 3]6[]Less than 4 elements; no quadruplet is possible
[]0[]Empty input array returns an empty list

Solution❯

The 4 Sum problem asks us to find sets of four numbers in an array that add up to a given target. A brute force method would try every possible combination of four numbers, but this becomes inefficient very quickly as the array grows. Instead, we use a more optimal and structured approach.

Understanding the Problem

We are given an array that may contain positive, negative, or duplicate numbers. We need to find all unique sets of four numbers (quadruplets) such that their sum equals the given target. The key challenge is not just finding these combinations, but doing so efficiently while avoiding duplicates in the result.

How the Efficient Approach Works

First, we sort the array. Sorting helps in two ways:

  • It lets us skip duplicate numbers easily
  • It enables the use of the two-pointer technique

We fix the first two numbers using two nested loops. For each pair, we use two pointers — left and right — to find the remaining two numbers that make the total sum equal to the target.

Handling Duplicates

Since the array can contain duplicates, we must make sure that we don’t return the same set more than once. To handle this:

  • We skip duplicate elements while selecting the first and second numbers in the outer loops.
  • We also skip duplicates while moving the left and right pointers after finding a valid quadruplet.

What Happens in Special Cases?

  • Empty array: No elements means no quadruplets can be formed → return [].
  • Fewer than 4 elements: Again, not enough numbers → return [].
  • All elements are the same: If four or more elements match the target when summed, return one quadruplet only.
  • No valid combination exists: If no group of 4 numbers can meet the target, return an empty list.

Why This Works

The sorted array and two-pointer technique ensure that we don’t waste time checking every possible combination. This reduces the time complexity to O(n3), which is significantly better than the naive O(n4) approach.

By combining sorting, nested loops, and careful duplicate skipping, we ensure the solution is both efficient and accurate.

Algorithm Steps

  1. Given an array arr of size n and a target value target.
  2. Sort the array in ascending order.
  3. Fix the first element with index i (0 to n-4).
  4. Fix the second element with index j (i+1 to n-3).
  5. Use two pointers left = j+1 and right = n-1 to find two other numbers.
  6. If the sum of the four elements equals target, store the quadruplet.
  7. Move pointers intelligently while skipping duplicates to find all unique quadruplets.

Code

Python
JavaScript
Java
C++
C
def four_sum(arr, target):
    arr.sort()
    n = len(arr)
    res = []

    for i in range(n-3):
        if i > 0 and arr[i] == arr[i-1]:
            continue
        for j in range(i+1, n-2):
            if j > i+1 and arr[j] == arr[j-1]:
                continue
            left, right = j+1, n-1
            while left < right:
                total = arr[i] + arr[j] + arr[left] + arr[right]
                if total == target:
                    res.append([arr[i], arr[j], arr[left], arr[right]])
                    left += 1
                    right -= 1
                    while left < right and arr[left] == arr[left-1]:
                        left += 1
                    while left < right and arr[right] == arr[right+1]:
                        right -= 1
                elif total < target:
                    left += 1
                else:
                    right -= 1
    return res

# Sample Input
arr = [1, 0, -1, 0, -2, 2]
target = 0
print("Quadruplets:", four_sum(arr, target))

Time Complexity

CaseTime ComplexityExplanation
Best CaseO(n^3)Even in the best case, we use two nested loops and a two-pointer approach, leading to cubic time complexity when scanning all possible quadruplets.
Average CaseO(n^3)For each pair of the first two elements (i and j), we use a two-pointer scan through the remaining array. This results in O(n³) combinations on average.
Average CaseO(n^3)The algorithm always uses two nested loops plus a two-pointer traversal, so the worst-case complexity remains cubic.

Space Complexity

O(1) (excluding output)

Explanation: Only a few pointers and loop variables are used for the computation. However, the space needed to store the resulting quadruplets depends on the number of valid combinations found.

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