Binary search

Definition

Let $(X, \le)$ be a totally ordered set, let $a_1 \le a_2 \le \cdots \le a_n$ be a sorted array over $X$ , and let $t \in X$ be a target. Binary search is the algorithm that keeps two indices $\ell$ and $r$ , starting with $\ell = 1$ and $r = n$ , and repeats the following:

if $\ell > r$ , stop and report that $t$ does not occur;
otherwise let $m = \floor{(\ell + r)/2}$ ;
if $a_m = t$ , stop and return $m$ ; if $a_m < t$ , set $\ell = m + 1$ ; if $a_m > t$ , set $r = m - 1$ .

Remarks

The invariant is: if $t$ occurs in the array, then it occurs among $a_\ell, \dots, a_r$ . It holds at the start, and step 3 keeps it because the array is sorted. The length $r - \ell + 1$ of the range drops from $s$ to at most $\floor{s/2}$ at each round, so there are at most $\floor{\log_2 n} + 1$ rounds.

Example: for $(1, 3, 5, 7, 9, 11)$ and $t = 9$ : $m = 3$ and $a_3 = 5 < 9$ , so $\ell = 4$ ; then $m = 5$ and $a_5 = 9$ , so the search returns $5$ .

Non-example: take the unsorted array $(5, 1, 3)$ and $t = 5$ . The first round finds $a_2 = 1 < 5$ and sets $\ell = 3$ . The second finds $a_3 = 3 < 5$ , and the search wrongly reports that $5$ does not occur. So the hypothesis that the array is sorted is needed.

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Algorithm concept
Total order concept
Binary search definition

Tag	`8YTFTSRRFS`
URI	`m2://8YTFTSRRFS`
Kind	definition
Writing	W0 stub
MSC	68P10, 68W40
Defines	Binary search
Uses	Algorithm, Total order
Changed	3 November 2026

Binary search

Definition

Remarks

Reading path

Further reading

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