Binary Search Time Complexity Analysis

Understanding Binary Search

Binary search is a highly efficient algorithm for finding an element in a sorted array. It works by repeatedly dividing the search interval in half, significantly reducing the search space with each iteration.

Time Complexity Analysis

The time complexity of binary search is O(log n), where n is the number of elements in the array. Here's why:

• In each step, the search space is halved
• The maximum number of steps is log₂(n) + 1
• This logarithmic growth makes binary search extremely efficient for large datasets

Comparison with Linear Search

Compared to linear search (O(n)), binary search is much faster for large datasets:

• For n = 1,000,000, linear search takes up to 1,000,000 steps
• Binary search takes at most 20 steps (log₂(1,000,000) ≈ 19.93)

Implementing Binary Search

Here's a simple implementation of binary search in Python:

def binary_search(arr, target):
    left, right = 0, len(arr) - 1
    while left <= right:
        mid = (left + right) // 2
        if arr[mid] == target:
            return mid
        elif arr[mid] < target:
            left = mid + 1
        else:
            right = mid - 1
    return -1

Conclusion

Binary search's O(log n) time complexity makes it an invaluable algorithm for searching large sorted datasets. Understanding its efficiency can help you make better decisions in algorithm design and optimization.