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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)
Optimal	Yes

In computer science, binary search, also known as half-interval search,^[1] logarithmic search,^[2] or binary chop,^[3] is a search algorithm that finds the position of a target value within a sorted array.^[4][5] Binary search compares the target value to the middle element of the array. If they are not equal, the half in which the target cannot lie is eliminated and the search continues on the remaining half, again taking the middle element to compare to the target value, and repeating this until the target value is found. If the search ends with the remaining half being empty, the target is not in the array.

Binary search runs in logarithmic time in the worst case, making ${\displaystyle O(\log n)}$ comparisons, where ${\displaystyle n}$ is the number of elements in the array.^[a][6] Binary search is faster than linear search except for small arrays. However, the array must be sorted first to be able to apply binary search. There are specialized data structures designed for fast searching, such as hash tables, that can be searched more efficiently than binary search. However, binary search can be used to solve a wider range of problems, such as finding the next-smallest or next-largest element in the array relative to the target even if it is absent from the array.

There are numerous variations of binary search. In particular, fractional cascading speeds up binary searches for the same value in multiple arrays. Fractional cascading efficiently solves a number of search problems in computational geometry and in 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.

Algorithm

Binary search works on sorted arrays. Binary search begins by comparing an element in the middle of the array with the target value. If the target value matches the element, its position in the array is returned. If the target value is less than the element, the search continues in the lower half of the array. If the target value is greater than the element, the search continues in the upper half of the array. By doing this, the algorithm eliminates the half in which the target value cannot lie in each iteration.^[7]

Procedure

Given an array ${\displaystyle A}$ of ${\displaystyle n}$ elements with values or records ${\displaystyle A_{0},A_{1},A_{2},\ldots ,A_{n-1}}$sorted such that ${\displaystyle A_{0}\leq A_{1}\leq A_{2}\leq \cdots \leq A_{n-1}}$, and target value ${\displaystyle T}$, the following subroutine uses binary search to find the index of ${\displaystyle T}$ in ${\displaystyle A}$.^[7]

1. Set ${\displaystyle L}$ to ${\displaystyle 0}$ and ${\displaystyle R}$ to ${\displaystyle n-1}$.
2. If ${\displaystyle L>R}$, the search terminates as unsuccessful.
3. Set ${\displaystyle m}$ (the position of the middle element) to the floor of ${\displaystyle {\frac {L+R}{2}}}$, which is the greatest integer less than or equal to ${\displaystyle {\frac {L+R}{2}}}$.
4. If ${\displaystyle A_{m}<T}$, set ${\displaystyle L}$ to ${\displaystyle m+1}$ and go to step 2.
5. If ${\displaystyle A_{m}>T}$, set ${\displaystyle R}$ to ${\displaystyle m-1}$ and go to step 2.
6. Now ${\displaystyle A_{m}=T}$, the search is done; return ${\displaystyle m}$.

This iterative procedure keeps track of the search boundaries with the two variables ${\displaystyle L}$ and ${\displaystyle R}$. The procedure may be expressed in pseudocode as follows, where the variable names and types remain the same as above, floor is the floor function, and unsuccessful refers to a specific value that conveys the failure of the search.^[7]

function binary_search(A, n, T) is
    L := 0
    R := n − 1
    while L ≤ R do
        m := floor((L + R) / 2)
        if A < T then
            L := m + 1
        else if A > T then
            R := m − 1
        else:
            return m
    return unsuccessful

Alternatively, the algorithm may take the ceiling of ${\displaystyle {\frac {L+R}{2}}}$Zdroj:https://en.wikipedia.org?pojem=Binary_search
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