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

Search in Rotated Sorted Array
Using Binary Search



Problem Statement

Given a rotated sorted array and a target value key, your task is to find the index of the target using an optimal binary search approach.

A rotated sorted array is a sorted array that has been rotated at some unknown pivot. For example, [4, 5, 6, 7, 0, 1, 2] is a rotation of [0, 1, 2, 4, 5, 6, 7].

If the element is found, return its index. Otherwise, return -1.

Examples

Input ArrayKeyOutput IndexDescription
[4, 5, 6, 7, 0, 1, 2]040 is present after the pivot point
[4, 5, 6, 7, 0, 1, 2]626 is found before the rotation point
[1]10Single element array, match found
[1]0-1Single element array, target not present
[1, 3]31Simple rotated array of two elements
[6, 7, 8, 1, 2, 3, 4, 5]35Target lies in the rotated right half
[]10-1Empty array, element cannot be found
[30, 40, 50, 10, 20]103Rotation in the middle, target found
[30, 40, 50, 10, 20]60-1Target does not exist

Solution

To efficiently search in a rotated sorted array, we use a modified version of binary search. Although the array isn't fully sorted due to the rotation, we can still take advantage of the fact that at least one half of the array (either left or right) is always sorted.

Understanding the Problem

In a regular binary search, we look for the target by checking if it lies in the left or right half, since the array is sorted. In a rotated sorted array, that logic needs adjustment. At every step, we still check the middle element, but also identify which half is sorted, and whether the target lies within that sorted half.

What Cases Can Occur?

  • Case 1: The element at mid is the key — In this case, we simply return the index.
  • Case 2: The left half is sorted — This means arr[start] ≤ arr[mid]. Now:
    • If the key lies between arr[start] and arr[mid], then we discard the right half by moving end = mid - 1.
    • Otherwise, we move start = mid + 1 to search in the unsorted right half.
  • Case 3: The right half is sorted — This means arr[mid] ≤ arr[end]. Then:
    • If the key lies between arr[mid] and arr[end], we search the right half by setting start = mid + 1.
    • Otherwise, move end = mid - 1 to check the left half.
  • Case 4: Empty array — If the array is empty, we immediately return -1 because there's nothing to search.
  • Case 5: Single element — We directly compare it with the key. If it matches, return 0; otherwise, return -1.

Why This Works

This approach works in O(log n) time because we effectively discard half the array in every step, just like standard binary search. By checking which half is sorted and narrowing the range intelligently, we find the target if it exists.

It’s a very efficient way to deal with rotated arrays, and works well even for large datasets.

Visualization

Algorithm Steps

  1. Initialize start = 0, end = n - 1.
  2. While start ≤ end, calculate mid = start + (end - start) / 2.
  3. If arr[mid] == key, return mid.
  4. If the left half arr[start] ≤ arr[mid] is sorted:
    • Check if key lies between arr[start] and arr[mid].
    • If yes, move end = mid - 1.
    • Else, move start = mid + 1.
  5. If the right half is sorted:
    • Check if key lies between arr[mid] and arr[end].
    • If yes, move start = mid + 1.
    • Else, move end = mid - 1.
  6. If not found, return -1.

Code

Java
Python
JavaScript
C
C++
C#
Kotlin
Swift
Go
Php
public class RotatedArraySearch {
  public static int search(int[] arr, int key) {
    int start = 0;
    int end = arr.length - 1;

    while (start <= end) {
      int mid = start + (end - start) / 2;

      if (arr[mid] == key) return mid;

      // Left half is sorted
      if (arr[start] <= arr[mid]) {
        if (key >= arr[start] && key < arr[mid]) {
          end = mid - 1;
        } else {
          start = mid + 1;
        }
      }
      // Right half is sorted
      else {
        if (key > arr[mid] && key <= arr[end]) {
          start = mid + 1;
        } else {
          end = mid - 1;
        }
      }
    }
    return -1;
  }

  public static void main(String[] args) {
    int[] arr = {4, 5, 6, 7, 0, 1, 2};
    int key = 0;
    int index = search(arr, key);
    System.out.println("Index of key: " + index);
  }
}

Time Complexity

CaseTime ComplexityExplanation
Best CaseO(1)The key is found at the first mid-point without needing further comparisons.
Average CaseO(log n)Each step eliminates half the search space using binary search logic.
Average CaseO(log n)In the worst case, the search continues until one element remains.

Space Complexity

O(1)

Explanation: The algorithm uses a constant number of variables, no extra memory.



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