Binary search

Binary search algorithm

Visualization of the binary search algorithm where 7 is the target value.
Class	Search algorithm
Data structure	Array
Worst-case performance	O(log n)
Best-case performance	O(1)
Average performance	O(log n)
Worst-case space complexity	O(1)

In computer science, binary search, also known as half-interval search or logarithmic search, is a search algorithm that finds the position of a target value within a sorted array. Binary search compares the target value to the middle element of the array; if they are unequal, the half in which the target cannot lie is eliminated and the search continues on the remaining half until it is successful or the remaining half is empty.

Binary search runs in at worst logarithmic time, making O(log n) comparisons, where n is the number of elements in the array, the O is Big O notation, and log is the logarithm. Binary search takes only constant (O(1)) space, meaning that the space taken by the algorithm is the same for any number of elements in the array. Although specialized data structures designed for fast searching—such as hash tables—can be searched more efficiently, binary search applies to a wider range of search problems.

Although the idea is simple, implementing binary search correctly requires attention to some subtleties about its exit conditions and midpoint calculation.

There exist numerous variations of binary search. In particular, fractional cascading speeds up binary searches for the same value in multiple arrays, efficiently solving a series of search problems in computational geometry and numerous other fields. Exponential search extends binary search to unbounded lists. The binary search tree and B-tree data structures are based on binary search.

Binary search works on sorted arrays. Binary search begins by comparing the middle element of the array with the target value. If the target value matches the middle element, its position in the array is returned. If the target value is less than or greater than the middle element, the search continues in the lower or upper half of the array, respectively, eliminating the other half from consideration.