What is the Big O notation of a binary search algorithm on a sorted array of n elements?O(n^2)O(log n)O(n)O(n log n)

Break Down the Problem

1. Identify the algorithm being analyzed: binary search.
2. Determine the nature of the input: a sorted array of $n$ elements.
3. Understand the aim: to find the time complexity using Big O notation.

Relevant Concepts

1. Binary Search Algorithm: This algorithm works by repeatedly dividing the search interval in half.
2. Time Complexity: We need to establish how many times we can halve the array until we reach a single element.

Analysis and Detail

1. In each step of the binary search, we:
   • Compare the target value to the middle element of the array.
   • If the target value is equal to the middle element, we have found the value.
   • If the target is less than the middle element, we search the left half of the array.
   • If the target is more than the middle element, we search the right half of the array.
2. This halving process continues until the size of the array is reduced to 1.

For an array of size $n$:

• 1st comparison: $n$
• 2nd comparison: $\frac{n}{2}$
• 3rd comparison: $\frac{n}{4}$
• ...
• Last comparison: $1$

The number of times you can halve $n$ until you reach 1 is given by $\log_2(n)$.

Verify and Summarize

The time complexity of the binary search is $O(\log n)$ since each comparison reduces the search space by half.

Final Answer

The Big O notation of a binary search algorithm on a sorted array of $n$ elements is $O(\log n)$.