Yandex

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

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

TriesTries1

Search Insert Position in Sorted Array
Lower Bound Binary Search



Problem Statement

Given a sorted array of distinct integers and a target number x, your task is to find the insert position of x.

This means you must return the index where x is located in the array. If it is not present, return the index where it should be inserted in order to keep the array sorted.

This problem is also known as finding the lower bound — the first index where an element is greater than or equal to x.

Examples

Input Array Target (x) Insert Position Description
[10, 20, 30, 40, 50] 35 3 35 should be inserted before 40
[10, 20, 30, 40, 50] 20 1 Exact match at index 1
[10, 20, 30, 40, 50] 5 0 Smaller than all elements, inserted at beginning
[10, 20, 30, 40, 50] 55 5 Larger than all elements, inserted at end
[5, 15, 25, 35] 25 2 Exact match at index 2
[5] 2 0 Inserted before the only element
[5] 5 0 Exact match at index 0
[5] 8 1 Inserted after the only element

Solution

To find the position where a target value x should be inserted in a sorted array, we use a binary search approach known as the lower bound.

Understanding the Goal

The goal is to find the first index where the value in the array is greater than or equal to x. If such a number exists, we return its index. If not, we return the position after the last element.

Step-by-Step Explanation

We start by setting up two pointers, low and high, at the start and end of the array respectively. We also initialize an answer variable ans as the length of the array. This variable keeps track of the best candidate index where x could be inserted.

In each step of the binary search:

  • We calculate the mid index between low and high.
  • If arr[mid] is greater than or equal to x, it means this position might be a valid place to insert x, so we update ans = mid and search the left half.
  • If arr[mid] is less than x, then we need to move right to find a higher or equal value, so we set low = mid + 1.

The loop continues until low > high. At this point, the ans holds the index where the target should be inserted to keep the array sorted.

Why Binary Search?

Since the array is already sorted, binary search helps us find the insert position in O(log n) time, which is much more efficient than scanning every element linearly.

Visualization

Algorithm Steps

  1. Given a sorted array arr of distinct integers and a target value x.
  2. Initialize: low = 0, high = arr.length - 1, ans = arr.length.
  3. While low ≤ high:
  4. → Compute mid = Math.floor((low + high) / 2).
  5. → If arr[mid] ≥ x, update ans = mid and move left: high = mid - 1.
  6. → Else, move right: low = mid + 1.
  7. Return ans as the correct insert position.

Code

Python
JavaScript
Java
C++
C
def search_insert_position(arr, x):
    n = len(arr)
    low, high = 0, n - 1
    ans = n
    while low <= high:
        mid = (low + high) // 2
        if arr[mid] >= x:
            ans = mid
            high = mid - 1
        else:
            low = mid + 1
    return ans

# Sample Input
arr = [1, 3, 5, 6]
target = 2
print("Insert Position:", search_insert_position(arr, target))

Time Complexity

CaseTime ComplexityExplanation
Best CaseO(1)The target is found at the middle index on the first comparison, requiring only one iteration.
Average CaseO(log n)Binary search halves the search space in each iteration, so on average, it takes log n steps to find the insert position.
Worst CaseO(log n)In the worst case, the entire array must be searched logarithmically to determine where the target should be inserted.

Space Complexity

O(1)

Explanation: The algorithm operates in-place using only a constant amount of memory — just a few integer variables for low, high, mid, and ans.



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