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 Missing Number in Array using Sum Formula - Optimal Algorithm

Arrays

Contents

ProblemExamplesSolutionAlgorithmCodeTimeSpace

Problem Statement❯

You are given an array of N-1 distinct integers in the range 1 to N. The task is to find the one number that is missing from the array.

The array has no duplicates, and only one number is missing. The numbers can be in any order.

Your goal is to identify the missing number efficiently.

Examples❯

Input Array Expected N Missing Number Description
[1, 2, 4, 5] 5 3 All numbers from 1 to 5 are present except 3Visualization
[2, 3, 1, 5] 5 4 Missing number is in the middleVisualization
[1, 2, 3, 4] 5 5 Last number is missingVisualization
[2, 3, 4, 5] 5 1 First number is missingVisualization
[1] 2 2 Only one number present, second one is missingVisualization
[] 1 1 Empty array, so the only number 1 is missingVisualization
[2] 2 1 Only number 2 is present, so 1 is missingVisualization

Solution❯

To find the missing number in an array of size N-1 where elements are from 1 to N, we can take advantage of the fact that the sum of first N natural numbers can be calculated directly using the formula:

Expected Sum = N * (N + 1) / 2

Let’s say we compute this expected sum and compare it with the sum of the actual elements present in the array. The difference between the two will tell us which number is missing.

Let’s Discuss a Few Scenarios:

1. Normal Case: Suppose the array is [1, 2, 4, 5] and N = 5. The expected sum is 15, but the actual sum is 12. The missing number is 15 - 12 = 3.

2. First Number Missing: In [2, 3, 4, 5], the missing number is 1. The formula still works: Expected = 15, Actual = 14, Missing = 1.

3. Last Number Missing: For [1, 2, 3, 4] with N = 5, the missing number is 5. Again, Expected = 15, Actual = 10, Missing = 5.

4. Only One Element in Array: If array = [1] and N = 2, then Expected = 3, Actual = 1, Missing = 2.

5. Empty Array Case: This is a valid edge case. If N = 1 and the array is empty [], then the only number that could have been in the array is 1. So, 1 is the missing number.

6. Random Order: It doesn’t matter what the order of elements is. Even if the array is [3, 1, 5, 2], the total will still work out because the sum is independent of order.

Why This Works

Since we are not iterating over every possible number from 1 to N, but instead just comparing totals, the time complexity is O(N) (to sum the array) and the space complexity is O(1). This makes the solution optimal and suitable for large inputs.

This technique also avoids needing any extra memory like hash sets or frequency arrays.

Algorithm Steps❯

  1. Given an array of size N-1, containing unique numbers between 1 to N.
  2. Calculate the sum of the first N natural numbers using the formula N*(N+1)/2.
  3. Calculate the sum of all elements in the given array.
  4. The missing number is the difference between the expected sum and the actual sum.

Code

Python
JavaScript
Java
C++
C
def find_missing_number(arr, N):
    total = N * (N + 1) // 2
    actual_sum = sum(arr)
    return total - actual_sum

# Sample Input
arr = [1, 2, 4, 5, 6]
N = 6
print("Missing Number:", find_missing_number(arr, N))

Time Complexity❯

CaseTime ComplexityExplanation
Best CaseO(n)We need to iterate through the array once to calculate the actual sum, which takes linear time.
Average CaseO(n)Regardless of the input distribution, we always sum all elements, resulting in linear time complexity.
Worst CaseO(n)In the worst case, the missing number is at the end, but we still iterate the full array, so the time remains linear.

Space Complexity❯

O(1)

Explanation: Only a constant number of variables are used to store sums and the final result. No extra space proportional to input size is required.