Mastering Efficient Searches: Finding the Correct Insertion Point

Description:

In this edition, we focus on a critical problem in algorithm design: finding the correct index for inserting a target value into a sorted array. This problem is a great exercise for understanding binary search and achieving efficient O(log n) performance.

Problem Statement:

Given a sorted array of distinct integers and a target value, return the index if the target is found. If not, return the index where it would be if it were inserted in order.

Step-by-Step Explanation:

How It Works:

1. Initialization: We start by setting two pointers, left and right, to the beginning (0) and end (len(nums) - 1) of the array.
2. Binary Search Loop: We enter a loop that continues as long as left is less than or equal to right.
3. Calculate Midpoint: Inside the loop, calculate the midpoint mid using integer division. This helps us split the array into halves.
4. Compare Middle Element: Check if the element at mid equals the target:If True: We found the target and return the index mid. If nums[mid] is less than the target: Move the left pointer to mid + 1 to search the right half. If nums[mid] is greater than the target: Move the right pointer to mid - 1 to search the left half.
5. Return Insertion Point: When the loop exits, it means the target is not in the array. The left pointer will be at the index where the target should be inserted.

Example Walkthroughs:

Example 1:

• Input: nums = [1, 3, 5, 6], target = 5
• Output: 2
• Explanation: The target is found at index 2.

Example 2:

• Input: nums = [1, 3, 5, 6], target = 2
• Output: 1
• Explanation: The target is not found. The correct insertion index is 1.

Example 3:

• Input: nums = [1, 3, 5, 6], target = 7
• Output: 4
• Explanation: The target is not found. The correct insertion index is 4.

Key Takeaways:

• Binary Search: A powerful algorithm for efficient searching in sorted arrays.
• Time Complexity: O(log n), ensuring fast performance even with large arrays.
• Insertion Point: Useful for maintaining order when adding new elements.

Implementing this approach ensures you handle search and insertion efficiently!