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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. If the search ends with the remaining half being empty, the target is not in the array. 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 constant O 1 space, meaning that the space taken by the algorithm is the same for any number of elements in the array. Although the idea is simple, implementing binary search correctly requires attention to some subtleties about its exit conditions and midpoint calculation.

There are 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.

Given an array A of n elements with values or records A 0 , A 1 , In the above procedure, the algorithm checks whether the middle element m is equal to the target t in every iteration.

Some implementations leave out this check during each iteration. This results in a faster comparison loop, as one comparison is eliminated per iteration.

However, it requires one more iteration on average. The above procedure only performs exact matches, finding the position of a target value. However, due to the ordered nature of sorted arrays, it is trivial to extend binary search to perform approximate matches. For example, binary search can be used to compute, for a given value, its rank the number of smaller elements , predecessor next-smallest element , successor next-largest element , and nearest neighbor.

Range queries seeking the number of elements between two values can be performed with two rank queries. The performance of binary search can be analyzed by reducing the procedure to a binary comparison tree, where the root node is the middle element of the array. The middle element of the lower half is the left child node of the root and the middle element of the upper half is the right child node of the root.

The rest of the tree is built in a similar fashion. This model represents binary search; starting from the root node, the left or right subtrees are traversed depending on whether the target value is less or more than the node under consideration, representing the successive elimination of elements.

The worst case is reached when the search reaches the deepest level of the tree, equivalent to a binary search that has reduced to one element and, in each iteration, always eliminates the smaller subarray out of the two if they are not of equal size. The worst case may also be reached when the target element is not in the array. In the best case, where the target value is the middle element of the array, its position is returned after one iteration. In terms of iterations, no search algorithm that works only by comparing elements can exhibit better average and worst-case performance than binary search.

This is because the comparison tree representing binary search has the fewest levels possible as each level is filled completely with nodes if there are enough. This is the case for other search algorithms based on comparisons, as while they may work faster on some target values, the average performance over all elements is affected.

This problem is solved by binary search, as dividing the array in half ensures that the size of both subarrays are as similar as possible. Fractional cascading can be used to speed up searches of the same value in multiple arrays. Each iteration of the binary search procedure defined above makes one or two comparisons, checking if the middle element is equal to the target in each iteration.

Again assuming that each element is equally likely to be searched, each iteration makes 1. A variation of the algorithm checks whether the middle element is equal to the target at the end of the search, eliminating on average half a comparison from each iteration.

This slightly cuts the time taken per iteration on most computers, while guaranteeing that the search takes the maximum number of iterations, on average adding one iteration to the search. For implementing associative arrays , hash tables , a data structure that maps keys to records using a hash function , are generally faster than binary search on a sorted array of records; [19] most implementations require only amortized constant time on average.

In addition, all operations possible on a sorted array can be performed—such as finding the smallest and largest key and performing range searches. A binary search tree is a binary tree data structure that works based on the principle of binary search. The records of the tree are arranged in sorted order, and each record in the tree can be searched using an algorithm similar to binary search, taking on average logarithmic time.

Insertion and deletion also require on average logarithmic time in binary search trees. This can be faster than the linear time insertion and deletion of sorted arrays, and binary trees retain the ability to perform all the operations possible on a sorted array, including range and approximate queries.

However, binary search is usually more efficient for searching as binary search trees will most likely be imperfectly balanced, resulting in slightly worse performance than binary search.

This applies even to balanced binary search trees , binary search trees that balance their own nodes—as they rarely produce optimally -balanced trees—but to a lesser extent. Binary search trees lend themselves to fast searching in external memory stored in hard disks, as binary search trees can effectively be structured in filesystems. The B-tree generalizes this method of tree organization; B-trees are frequently used to organize long-term storage such as databases and filesystems.

Linear search is a simple search algorithm that checks every record until it finds the target value. Linear search can be done on a linked list, which allows for faster insertion and deletion than an array. Binary search is faster than linear search for sorted arrays except if the array is short. Sorting the array also enables efficient approximate matches and other operations. A related problem to search is set membership.

Any algorithm that does lookup, like binary search, can also be used for set membership. There are other algorithms that are more specifically suited for set membership. For approximate results, Bloom filters , another probabilistic data structure based on hashing, store a set of keys by encoding the keys using a bit array and multiple hash functions.

Bloom filters are much more space-efficient than bit arrays in most cases and not much slower: However, Bloom filters suffer from false positives. There exist data structures that may improve on binary search in some cases for both searching and other operations available for sorted arrays.

For example, searches, approximate matches, and the operations available to sorted arrays can be performed more efficiently than binary search on specialized data structures such as van Emde Boas trees , fusion trees , tries , and bit arrays. However, while these operations can always be done at least efficiently on a sorted array regardless of the keys, such data structures are usually only faster because they exploit the properties of keys with a certain attribute usually keys that are small integers , and thus will be time or space consuming for keys that lack that attribute.

Uniform binary search stores, instead of the lower and upper bounds, the index of the middle element and the change in the middle element from the current iteration to the next iteration. Each step reduces the change by about half.

For example, if the array to be searched was [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11] , the middle element would be 6. Uniform binary search works on the basis that the difference between the index of middle element of the array and the left and right subarrays is the same. In this case, the middle element of the left subarray [1, 2, 3, 4, 5] is 3 and the middle element of the right subarray [7, 8, 9, 10, 11] is 9.

Uniform binary search would store the value of 3 as both indices differ from 6 by this same amount. The main advantage of uniform binary search is that the procedure can store a table of the differences between indices for each iteration of the procedure, which may improve the algorithm's performance on some systems. It starts by finding the first element with an index that is both a power of two and greater than the target value.

Afterwards, it sets that index as the upper bound, and switches to binary search. Exponential search works on bounded lists, but becomes an improvement over binary search only if the target value lies near the beginning of the array. Instead of calculating the midpoint, interpolation search estimates the position of the target value, taking into account the lowest and highest elements in the array as well as length of the array. This is only possible if the array elements are numbers.

It works on the basis that the midpoint is not the best guess in many cases. For example, if the target value is close to the highest element in the array, it is likely to be located near the end of the array. In practice, interpolation search is slower than binary search for small arrays, as interpolation search requires extra computation. Although its time complexity grows more slowly than binary search, this only compensates for the extra computation for large arrays.

Fractional cascading is a technique that speeds up binary searches for the same element for both exact and approximate matching in "catalogs" arrays of sorted elements associated with vertices in graphs.

Fractional cascading was originally developed to efficiently solve various computational geometry problems, but it also has been applied elsewhere, in domains such as data mining and Internet Protocol routing. Fibonacci search is a method similar to binary search that successively shortens the interval in which the maximum of a unimodal function lies. Given a finite interval, a unimodal function, and the maximum length of the resulting interval, Fibonacci search finds a Fibonacci number such that if the interval is divided equally into that many subintervals, the subintervals would be shorter than the maximum length.

After dividing the interval, it eliminates the subintervals in which the maximum cannot lie until one or more contiguous subintervals remain. Noisy binary search algorithms solve the case where the algorithm cannot reliably compare elements of the array. For each pair of elements, there is a certain probability that the algorithm makes the wrong comparison. Noisy binary search can find the correct position of the target with a given probability that controls the reliability of the yielded position.

In , John Mauchly made the first mention of binary search as part of the Moore School Lectures , the first ever set of lectures regarding any computer-related topic. Guibas introduced fractional cascading as a method to solve numerous search problems in computational geometry.

Although the basic idea of binary search is comparatively straightforward, the details can be surprisingly tricky When Jon Bentley assigned binary search as a problem in a course for professional programmers, he found that ninety percent failed to provide a correct solution after several hours of working on it, [56] and another study published in shows that accurate code for it is only found in five out of twenty textbooks.

The Java programming language library implementation of binary search had the same overflow bug for more than nine years. In a practical implementation, the variables used to represent the indices will often be of fixed size, and this can result in an arithmetic overflow for very large arrays. If the target value is greater than the greatest value in the array, and the last index of the array is the maximum representable value of L , the value of L will eventually become too large and overflow.

A similar problem will occur if the target value is smaller than the least value in the array and the first index of the array is the smallest representable value of R. In particular, this means that R must not be an unsigned type if the array starts with index 0. An infinite loop may occur if the exit conditions for the loop are not defined correctly.

Once L exceeds R , the search has failed and must convey the failure of the search. In addition, the loop must be exited when the target element is found, or in the case of an implementation where this check is moved to the end, checks for whether the search was successful or failed at the end must be in place.

Bentley found that, in his assignment of binary search, most of the programmers who implemented binary search incorrectly made an error defining the exit conditions. Many languages' standard libraries include binary search routines:.