Question

What is the binary search algorithm and prove its running time and correctness: Say we have a sorted sequence S of n numbers a₀, a₁,...aₙ-₁ stored in an array A[0, 1, ... n - 1]

Answer 1

The binary search algorithm is a divide-and-conquer algorithm used to find a specific element in a sorted sequence. It has a running time of O(log n) and guarantees that the target element will be found if it exists in the sequence.

Step-by-step explanation:

The binary search algorithm is a divide-and-conquer algorithm used to find a specific element in a sorted sequence. It repeatedly divides the search space in half until the target element is found. Here's how it works:

1. Set the starting index low to 0 and the ending index high to length of the array - 1.
2. While low is less than or equal to high:

• Calculate the middle index (mid) by taking the average of low and high.
• If the element at index mid equals the target element, return mid.
• If the element at index mid is greater than the target element, set high to mid - 1.
• If the element at index mid is less than the target element, set low to mid + 1.

If the loop exits without finding the target element, return -1 to indicate that the element is not in the sequence.

The running time of the binary search algorithm is logarithmic, specifically O(log n), where n is the size of the sequence. This means that as the size of the sequence doubles, the number of steps needed to find the target element only increases by one. The algorithm is correct because it guarantees that the target element will be found if it exists in the sequence.