Time Complexity of Binary Search Algorithm

The time complexity of an algorithm determines how the running time of the algorithm grows as the input size increases. In the case of the binary search algorithm, the time complexity is logarithmic, which means the running time grows at a slower rate compared to linear time complexity.

Overview of Binary Search Algorithm

Binary search is an efficient algorithm used to find a specific target value within a sorted array. It works by repeatedly dividing the search space in half until the target value is found or it is determined that the target value does not exist in the array.

Explanation of Time Complexity

The time complexity of the binary search algorithm is O(log n), where n represents the size of the input array. This means that the running time of the algorithm grows logarithmically with the input size.

Reasoning

The binary search algorithm operates by repeatedly dividing the search space in half. In each iteration, it compares the target value with the middle element of the current search space and determines whether to continue the search in the left or right half.

Due to the halving of the search space in each iteration, the algorithm quickly converges towards the target value. This logarithmic behavior is the reason for the O(log n) time complexity.

Example

Let's consider an example to illustrate the time complexity of the binary search algorithm. Suppose we have a sorted array of size n = 8. The binary search algorithm would perform the following steps:

1. Compare the target value with the middle element: arr[3] = 7.
2. Since the target value is greater than 7, we continue the search in the right half.
3. Compare the target value with the middle element: arr[5] = 9.
4. Since the target value is smaller than 9, we continue the search in the left half.
5. Compare the target value with the middle element: arr[4] = 8.
6. The target value is found at index 4.

In this example, the binary search algorithm successfully finds the target value in just three iterations. Even for larger input sizes, the algorithm's logarithmic time complexity ensures efficient search operations.

Conclusion

In conclusion, the time complexity of the binary search algorithm is O(log n). It is a highly efficient algorithm for searching within sorted arrays, as it quickly converges towards the target value by halving the search space in each iteration.