In C programming, what is the key idea behind the iterative binary search algorithm on a sorted array?

Introduction / Context:
Binary search is a fundamental search algorithm used when the input array is already sorted. It is much more efficient than linear search for large datasets. In C, binary search is often implemented manually in algorithm problems and also provided through functions like bsearch in the standard library.

Given Data / Assumptions:

• The array elements are sorted in ascending order.
• The array is stored in random access memory so that indexing by position is efficient.
• We want to search for a target key value in this sorted array.

Concept / Approach:
Binary search works by repeatedly halving the search interval. Instead of checking each element in sequence, it compares the key to the element in the middle of the current interval. If the key is equal to the middle element, the search is successful. If the key is smaller, all elements to the right of the middle can be discarded. If the key is larger, all elements to the left can be discarded. This halving process continues until the key is found or the interval becomes empty.

Step-by-Step Solution:
Step 1: Initialize low to 0 and high to n - 1, where n is the number of elements.Step 2: While low is less than or equal to high, compute mid = (low + high) / 2.Step 3: Compare key with a[mid]. If key equals a[mid], return mid as the index.Step 4: If key is less than a[mid], set high = mid - 1 to search in the left half.Step 5: If key is greater than a[mid], set low = mid + 1 to search in the right half.Step 6: If the loop ends without a match, report that the key is not present.

Verification / Alternative check:
For example, searching for 15 in [3, 8, 12, 15, 20, 25] starts with mid at index 2 (value 12). Since 15 is greater than 12, the search continues in the right half [15, 20, 25]. The next mid is 15, which matches the key. This behaviour matches option B.

Why Other Options Are Wrong:
Option A is linear search, which checks each element in order instead of halving the search space.Option C is a sorting method, not a search algorithm.Option D describes hashing, which is a different technique that requires building a hash table and is not the same as binary search on a sorted array.

Common Pitfalls:
Students sometimes apply binary search to unsorted arrays, which breaks the algorithm. Another frequent mistake is incorrect handling of mid, low, and high indices, leading to infinite loops or off by one errors.

Final Answer:
The key idea is It repeatedly compares the key with the middle element, discards the half in which the key cannot lie, and continues searching in the remaining half.