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

Find Kth Missing Positive Number in Sorted Array
Binary Search Approach



Problem Statement

You're given a sorted array of unique positive integers and an integer k. The array is strictly increasing, but it may be missing some positive numbers. Your task is to find the kth missing positive number in the sequence.

The array only contains some of the positive integers starting from 1, and some numbers are missing. You must figure out what the kth missing number would be if we continued counting all the positive numbers in order.

If the array is empty, then the answer is simply k because all numbers are missing.

Examples

Input ArraykOutputDescription
[2, 3, 4, 7, 11]59Missing numbers: [1, 5, 6, 8, 9, 10], 5th one is 9
[1, 2, 3, 4]26No missing until 4, missing numbers are [5, 6], 2nd is 6
[1, 3, 5, 10]46Missing: [2, 4, 6, 7, 8, 9], 4th one is 6
[2, 3, 4]111 is missing before array starts
[10, 11, 12]33Missing: [1, 2, 3, 4...], 3rd one is 3
[1, 2, 3, 4]00No missing required
[]44Empty array: all numbers are missing, return k = 4
[2]111 is missing before the first element
[5]22Missing: [1, 2, 3, 4], 2nd is 2

Solution

To solve the problem of finding the kth missing positive number, we need to understand where and how numbers go missing in the given array.

Understanding the Missing Numbers

Suppose we have an array like [2, 3, 4, 7]. Normally, if no numbers were missing, we would expect the first 4 numbers to be [1, 2, 3, 4]. But here, 1 is missing at the beginning. Then after 4, we expect 5 and 6, but we jump directly to 7, so 5 and 6 are missing too. This way, we can count how many numbers are missing up to any point.

Different Scenarios to Consider

  • Missing numbers at the beginning: If the array starts from a number greater than 1, then everything before that number is missing. For example, in [4, 5], the missing numbers are [1, 2, 3].
  • Missing in the middle: If numbers jump from one to a much bigger next number, then there are several missing numbers in between.
  • Empty array: If no numbers are given at all, then we assume all numbers are missing. So the kth missing is just k itself.
  • All initial values are continuous: If the array starts at 1 and continues in sequence (e.g., [1, 2, 3, 4]), then missing numbers only start after the last element.

How We Find the Kth Missing

As we go through the array, we can check how many numbers are missing up to each index. For example, at index i, the expected value is i + 1, but the actual value is arr[i]. So the number of missing values up to this point is arr[i] - (i + 1).

We continue this process until the number of missing values becomes at least k. If that happens, we know the kth missing value must be before or at that point.

If we reach the end of the array and still haven't found enough missing numbers, it means the missing number lies beyond the last element. In that case, the answer is arr.length + k if we’ve found no missing elements, or more accurately, last array value + (k - total missing).

Efficiency

This can be solved efficiently using binary search because the missing count increases with the index. We search for the first position where the number of missing elements becomes at least k and calculate the answer accordingly.

This approach ensures we avoid scanning all numbers one-by-one and still get the correct answer even for large arrays.

Visualization

Algorithm Steps

  1. Initialize low = 0 and high = arr.length - 1.
  2. While low ≤ high:
  3. → Calculate mid = low + (high - low) / 2.
  4. → Compute missing = arr[mid] - (mid + 1).
  5. → If missing < k, search right side by setting low = mid + 1.
  6. → Else, search left side by setting high = mid - 1.
  7. Finally, return low + k as the answer.

Code

Java
Python
JavaScript
C
C++
C#
Kotlin
Swift
Go
Php
public class KthMissingNumber {
  public static int findKthPositive(int[] arr, int k) {
    int low = 0;
    int high = arr.length - 1;

    while (low <= high) {
      int mid = low + (high - low) / 2;
      int missing = arr[mid] - (mid + 1); // How many numbers are missing before index mid

      if (missing < k) {
        low = mid + 1; // We need to move right to find more missing numbers
      } else {
        high = mid - 1; // Too many missing, move left
      }
    }

    return low + k; // kth missing number is after low numbers in total
  }

  public static void main(String[] args) {
    int[] arr = {2, 3, 4, 7, 11};
    int k = 5;
    System.out.println("Kth missing number is: " + findKthPositive(arr, k));
  }
}

Time Complexity

CaseTime ComplexityExplanation
Best CaseO(1)The target is found in the first binary search step.
Average CaseO(log n)Binary search halves the search space at each step, making it efficient even for large arrays.
Average CaseO(log n)At most, the search checks log(n) elements until the missing count reaches or exceeds k.

Space Complexity

O(1)

Explanation: The algorithm only uses a constant number of variables regardless of input size.



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