What is the time complexity of a binary search algorithm?

The time complexity of a binary search algorithm is O(log n).

Binary search is an efficient algorithm for finding an item from a sorted list of items. It works by repeatedly dividing in half the portion of the list that could contain the item, until you've narrowed down the possible locations to just one. The time complexity of this algorithm is O(log n), where n is the number of elements in the list. This is because with each comparison, the algorithm eliminates half of the remaining possibilities.

To understand why the time complexity is O(log n), let's consider how the binary search algorithm works. The algorithm starts by comparing the middle element of the list with the target value. If the target value matches the middle element, its position in the list is returned. If the target value is less than the middle element, the search continues on the lower half of the list. Alternatively, if the target value is greater than the middle element, the search continues on the upper half.

With each comparison, the size of the list is halved. This is because we know that the target value, if it exists in the list, must be in the half that we have chosen to continue the search. This halving of the list continues until the target value is found or until the size of the list is reduced to zero, which means the target is not in the list.

The number of steps it takes to reduce the list of n elements to just 1 element is log base 2 of n, which is written as log n. This is because each step of the algorithm halves the number of elements, which is the inverse of doubling. Doubling is a base 2 logarithmic operation, hence the log base 2 in the time complexity.

In conclusion, the time complexity of a binary search algorithm is O(log n) because with each step, the algorithm halves the number of elements it needs to search through. This makes binary search a very efficient algorithm when dealing with large data sets.