# binary search

(algorithm)

**Definition:**
Search a *sorted array* by repeatedly dividing the search interval in half. Begin with an interval covering the whole array. If the value of the search *key* is less than the item in the middle of the interval, narrow the interval to the lower half. Otherwise narrow it to the upper half. Repeatedly check until the value is found or the interval is empty.

**Generalization** (I am a kind of ...)

*dichotomic search*, *search*.

**Aggregate parent** (I am a part of or used in ...)

*binary insertion sort*, *ideal merge*, *suffix array*, *inversion list*.

**Aggregate child** (... is a part of or used in me.)

*divide and conquer*.

**See also**
*linear search*, *interpolation search*, *Fibonaccian search*, *jump search*.

*Note:
Run time is **O(ln n)*.

*
* Finding the middle is often coded as

mid = (high + low)/2;

This overflows if high and low are close to the largest expressible integer. The following gives the same result and never overflows, if high and low are non-negative.
mid = low + (high - low)/2;

Thanks to Colin D. Wright, 1 June 2005.
* Binary search may be effective with an **ordered linked list*. It makes O(n) traversals, as does *linear search*, but it only performs O(log n) comparisons. For more explanation, see Tim Rolfe's Searching in a Sorted Linked List.

Author: PEB

## Implementation

Mukundan's animation (Java). search (C), which uses *0-based indexing*. Flower Brackets explanation (Java). Algorithms and Data Structures' explanation (Java and C++). Worst-case behavior annotated for real time (WOOP/ADA), including bibliography.
## More information

explanation

Go to the
Dictionary of Algorithms and Data
Structures home page.

If you have suggestions, corrections, or comments, please get in touch
with Paul Black.

Entry modified 12 February 2019.

HTML page formatted Tue Feb 12 10:57:42 2019.

Cite this as:

Paul E. Black, "binary search", in
*Dictionary of Algorithms and Data Structures* [online], Vreda Pieterse and Paul E. Black, eds. 12 February 2019. (accessed TODAY)
Available from: https://www.nist.gov/dads/HTML/binarySearch.html