binary search

The Problem

Let be a total orderingPlanetmathPlanetmath on the set S. Given a sequence of n elements, L={x1x2xn}, and a value yS, locate the position of any elements in L that are equal to y, or determine that none exist.

The Algorithm

The binary searchMathworldPlanetmath technique is a fundamental method for locating an element of a particular value within a sequence of sorted elements (see Sorting Problem). The idea is to eliminate half of the search space with each comparison.

First, the middle element of the sequence is compared to the value we are searching for. If this element matches the value we are searching for, we are done. If, however, the middle element is “less than” the value we are chosen for (as specified by the relationMathworldPlanetmathPlanetmath ( used to specify a total order over the set of elements), then we know that, if the value exists in the sequence, it must exist somewhere after the middle element. Therefore we can eliminate the first half of the sequence from our search and simply repeat the search in the exact same manner on the remaining half of the sequence. If, however, the value we are searching for comes before the middle element, then we repeat the search on the first half of the sequence.


AlgorithmMathworldPlanetmath Binary_Search(L, n, key)
Input: A list L of n elements, and key (the search key)
Output: Position (such that X[Position]=key)

          if bottom=top then

if L[bottom]=key then







if key<L[middle] then






We can specify the runtime complexity of this binary search algorithm by counting the number of comparisons required to locate some element y in L. Since half of the list is eliminated with each comparison, there can be no more than log2n comparisons before either the positions of all the y elements are found or the entire list is eliminated and y is determined to not exist in L. Thus the worst-case runtime complexity of the binary search is 𝒪(logn). It can also be shown that the average-case runtime complexity of the binary search is approximately log2n-1 comparisons. This means that any single entry in a phone book containing one million entries can be located with at most 20 comparisons (and on average 19).

Title binary search
Canonical name BinarySearch
Date of creation 2013-03-22 11:44:34
Last modified on 2013-03-22 11:44:34
Owner mathcam (2727)
Last modified by mathcam (2727)
Numerical id 18
Author mathcam (2727)
Entry type Algorithm
Classification msc 68P10
Classification msc 16S40
Classification msc 20G42
Classification msc 16W35
Classification msc 81R50
Classification msc 16W30
Classification msc 57T05
Classification msc 54-01
Related topic TotalOrder
Related topic PartialOrder
Related topic SortingProblem
Related topic InsertionSort