Binary TreesBinary Trees36
  1. 1Preorder Traversal of a Binary Tree using Recursion
  2. 2Preorder Traversal of a Binary Tree using Iteration
  3. 3Inorder Traversal of a Binary Tree using Recursion
  4. 4Inorder Traversal of a Binary Tree using Iteration
  5. 5Postorder Traversal of a Binary Tree Using Recursion
  6. 6Postorder Traversal of a Binary Tree using Iteration
  7. 7Level Order Traversal of a Binary Tree using Recursion
  8. 8Level Order Traversal of a Binary Tree using Iteration
  9. 9Reverse Level Order Traversal of a Binary Tree using Iteration
  10. 10Reverse Level Order Traversal of a Binary Tree using Recursion
  11. 11Find Height of a Binary Tree
  12. 12Find Diameter of a Binary Tree
  13. 13Find Mirror of a Binary Tree
  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
GraphsGraphs46
  1. 1Breadth-First Search in Graphs
  2. 2Depth-First Search in Graphs
  3. 3Number of Provinces in an Undirected Graph
  4. 4Connected Components in a Matrix
  5. 5Rotten Oranges Problem - BFS in Matrix
  6. 6Flood Fill Algorithm - Graph Based
  7. 7Detect Cycle in an Undirected Graph using DFS
  8. 8Detect Cycle in an Undirected Graph using BFS
  9. 9Distance of Nearest Cell Having 1 - Grid BFS
  10. 10Surrounded Regions in Matrix using Graph Traversal
  11. 11Number of Enclaves in Grid
  12. 12Word Ladder - Shortest Transformation using Graph
  13. 13Word Ladder II - All Shortest Transformation Sequences
  14. 14Number of Distinct Islands using DFS
  15. 15Check if a Graph is Bipartite using DFS
  16. 16Topological Sort Using DFS
  17. 17Topological Sort using Kahn's Algorithm
  18. 18Cycle Detection in Directed Graph using BFS
  19. 19Course Schedule - Task Ordering with Prerequisites
  20. 20Course Schedule 2 - Task Ordering Using Topological Sort
  21. 21Find Eventual Safe States in a Directed Graph
  22. 22Alien Dictionary Character Order
  23. 23Shortest Path in Undirected Graph with Unit Distance
  24. 24Shortest Path in DAG using Topological Sort
  25. 25Dijkstra's Algorithm Using Set - Shortest Path in Graph
  26. 26Dijkstra’s Algorithm Using Priority Queue
  27. 27Shortest Distance in a Binary Maze using BFS
  28. 28Path With Minimum Effort in Grid using Graphs
  29. 29Cheapest Flights Within K Stops - Graph Problem
  30. 30Number of Ways to Reach Destination in Shortest Time - Graph Problem
  31. 31Minimum Multiplications to Reach End - Graph BFS
  32. 32Bellman-Ford Algorithm for Shortest Paths
  33. 33Floyd Warshall Algorithm for All-Pairs Shortest Path
  34. 34Find the City With the Fewest Reachable Neighbours
  35. 35Minimum Spanning Tree in Graphs
  36. 36Prim's Algorithm for Minimum Spanning Tree
  37. 37Disjoint Set (Union-Find) with Union by Rank and Path Compression
  38. 38Kruskal's Algorithm - Minimum Spanning Tree
  39. 39Minimum Operations to Make Network Connected
  40. 40Most Stones Removed with Same Row or Column
  41. 41Accounts Merge Problem using Disjoint Set Union
  42. 42Number of Islands II - Online Queries using DSU
  43. 43Making a Large Island Using DSU
  44. 44Bridges in Graph using Tarjan's Algorithm
  45. 45Articulation Points in Graphs
  46. 46Strongly Connected Components using Kosaraju's Algorithm

Find All Unique Triplets That Sum to Zero Optimal Solution using Two Pointers

Problem Statement

Given an array of integers, your task is to find all the unique triplets (a, b, c) in the array such that:

  • a + b + c = 0 (the sum of the three elements is zero)
  • Each triplet should be returned in sorted order and must be unique (i.e., no duplicate triplets in the result)

If no such triplets exist, return an empty list.

Examples

Input Array Output Triplets Description
[-1, 0, 1, 2, -1, -4] [[-1, -1, 2], [-1, 0, 1]] Two unique triplets sum up to 0
[0, 1, 1] [] No triplet sums to zero
[0, 0, 0] [[0, 0, 0]] Three zeroes sum to zero, one valid triplet
[3, -2, 1, 0] [[-2, -1, 3]] Triplet [-2, -1, 3] gives zero
[1, -1] [] Only two elements, not enough for a triplet
[] [] Empty input, return empty list
[-2, 0, 1, 1, 2] [[-2, 0, 2], [-2, 1, 1]] Two triplets, handles duplicates correctly

Visualization Player

Solution

Understanding the Problem

The Three Sum problem asks us to find all unique triplets in an array such that the sum of the three numbers is zero. In other words, we need to find all combinations of three numbers (a, b, c) from the array where a + b + c = 0.

It’s important to avoid returning duplicate triplets. For example, [-1, 0, 1] and [0, -1, 1] are considered the same triplet and should appear only once in the output.

We are going to solve this step-by-step, with a beginner-friendly explanation, and handle all edge cases carefully.

Step-by-Step Solution Using an Example

Let’s take the example array: [-1, 0, 1, 2, -1, -4]

Step 1: Sort the Array

We start by sorting the array. This helps in skipping duplicates and using the two-pointer method effectively.

Sorted array: [-4, -1, -1, 0, 1, 2]

Step 2: Fix One Element and Use Two Pointers

We loop through each element (let’s call the index i) and treat it as the first element of a potential triplet.

Then, we use two pointers: left starting from i + 1 and right starting from the end of the array.

We calculate the sum of the elements at positions i, left, and right and act based on the result:

  • If the sum is zero: we’ve found a valid triplet. Add it to the result.
  • If the sum is less than zero: we need a bigger number, so we move the left pointer forward.
  • If the sum is more than zero: we need a smaller number, so we move the right pointer backward.

Step 3: Skip Duplicates

To avoid duplicate triplets:

  • Skip repeated values at the i position.
  • After finding a valid triplet, skip repeated values at left and right positions.

Example Walkthrough

Let’s go through the sorted array step by step:

  • i = 0: value = -4, left = 1 (-1), right = 5 (2) → sum = -3 → move left
  • left = 2 (-1), right = 5 (2) → sum = -3 → move left
  • left = 3 (0), right = 5 (2) → sum = -2 → move left
  • left = 4 (1), right = 5 (2) → sum = -1 → move left
  • i = 1: value = -1, left = 2 (-1), right = 5 (2) → sum = 0 → valid triplet [-1, -1, 2]
  • Skip duplicate -1 at left, move left → left = 3 (0), right = 4 (1) → sum = 0 → valid triplet [-1, 0, 1]
  • i = 2: value = -1 (duplicate), skip
  • i = 3: value = 0, left = 4 (1), right = 5 (2) → sum = 3 → move right

Final result: [[-1, -1, 2], [-1, 0, 1]]

Handling Edge Cases

  • Less than 3 elements: No triplets possible, return an empty list.
  • All zeros: If the array is like [0, 0, 0, 0], return [[0, 0, 0]] only once.
  • Empty array: Return an empty list.
  • Duplicates: Skip over duplicate values during iteration to prevent redundant triplets.

Why This Approach Works

This method uses sorting + two pointers and avoids brute-force triplet checking. The time complexity is O(n²) compared to O(n³) in brute-force, making it efficient for larger inputs.

Algorithm Steps

  1. Given an array arr of integers.
  2. Sort the array in ascending order.
  3. Traverse the array with index i from 0 to n-2:
  4. → If i > 0 and arr[i] == arr[i-1], skip to avoid duplicates.
  5. → Initialize two pointers: left = i + 1 and right = n - 1.
  6. → While left < right:
  7. → → Calculate the sum: arr[i] + arr[left] + arr[right].
  8. → → If sum is zero, add triplet to result and move left and right to skip duplicates.
  9. → → If sum < 0, move left forward.
  10. → → If sum > 0, move right backward.
  11. Return the list of unique triplets.

Code

C
C++
Python
Java
JS
Go
Rust
Kotlin
Swift
TS
#include <stdio.h>
#include <stdlib.h>

int cmp(const void *a, const void *b) {
    return (*(int*)a - *(int*)b);
}

void threeSum(int* nums, int n) {
    qsort(nums, n, sizeof(int), cmp);
    for (int i = 0; i < n - 2; i++) {
        if (i > 0 && nums[i] == nums[i-1]) continue;
        int left = i + 1, right = n - 1;
        while (left < right) {
            int total = nums[i] + nums[left] + nums[right];
            if (total == 0) {
                printf("[%d, %d, %d]\n", nums[i], nums[left], nums[right]);
                while (left < right && nums[left] == nums[left+1]) left++;
                while (left < right && nums[right] == nums[right-1]) right--;
                left++;
                right--;
            } else if (total < 0) {
                left++;
            } else {
                right--;
            }
        }
    }
}

int main() {
    int arr[] = {-1, 0, 1, 2, -1, -4};
    int n = sizeof(arr) / sizeof(arr[0]);
    printf("Triplets:\n");
    threeSum(arr, n);
    return 0;
}

Time Complexity

CaseTime ComplexityExplanation
Best CaseO(n^2)In the best case, the input is small or contains many duplicate values that allow early skips, but the dominant term is still the nested loop structure with two pointers.
Average CaseO(n^2)We sort the array in O(n log n) and then use a loop with a two-pointer technique inside, which takes O(n^2) in total.
Worst CaseO(n^2)Even in the worst case with no duplicate skipping and all combinations needing to be checked, the time complexity remains O(n^2) due to the nested traversal using two pointers.

Space Complexity

O(1)

Explanation: The algorithm uses constant extra space apart from the output list. Sorting is done in-place and only a few variables (i, left, right) are used.


Comments

💬 Please keep your comment relevant and respectful. Avoid spamming, offensive language, or posting promotional/backlink content.
All comments are subject to moderation before being published.


Loading comments...