Binary search runs in at worst logarithmic timemaking O log n comparisons, where n is the number of elements in the array and log is the logarithm. Binary search takes only constant O 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 exist 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. A 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 wiki than or greater than the middle element, the search continues in the lower or upper half of the tree, respectively, eliminating the other half from consideration. Some implementations may place the comparison for equality search the end of the algorithm, resulting in a faster comparison loop but costing one more iteration on average. 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 elementspredecessor next-smallest elementsuccessor next-largest elementand nearest neighbor. Range queries seeking the number of elements between two values can be performed with two rank queries. 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. This 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. However, the tree may be unbalanced, with the deepest level partially filled, and equivalently, the array may not be divided perfectly by the search in search iterations, half of the time resulting in the smaller subarray being eliminated. Again assuming that each element is equally likely to be searched, each iteration makes comparisons on average. A variation of the algorithm instead checks for equality at the very end of the search, eliminating on average half a comparison from each iteration. This decreases the time taken per iteration very slightly on most computers, while guaranteeing that the search takes the maximum number of iterations, on average adding one iteration to the search. In addition, all operations possible on a sorted array can be performed—such as finding the smallest and largest key and performing range searches. Insertion and deletion also require logarithmic time in binary search trees. This is faster tree the linear time insertion and deletion of sorted arrays, and binary trees retain the ability to perform all the operations possible on a binary array, including range and approximate queries. Tree applies even to balanced binary search treesbinary search trees that balance their own nodes—as binary rarely produce optimally -balanced trees—but to a lesser extent. By splitting the tree into pages of some number of elements, each storing in turn a section of the tree, searching in a binary search tree costs wiki disk seeks, improving its overall performance. Notice that this effectively creates a multiway tree, as each page is connected to each other. 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 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. The Judy1 type of Judy array handles 64-bit keys efficiently. For approximate results, Bloom filtersanother probabilistic data structure based on hashing, store a set of keys by encoding the keys using a bit array and multiple hash functions. However, Bloom filters suffer from false positives. 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 treesfusion treestriesand bit arrays. However, while these operations can always be done at least efficiently on binary 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 integersand thus will be time or space consuming for keys that lack that attribute. Each step reduces the width by about half. This variation is uniform because the difference between the indices of middle elements and the preceding middle elements chosen remains constant between searches of arrays of the same length. 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. 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 beginning 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. Guibas introduced fractional cascadinga technique used to speed up binary searches in multiple arrays. The Java programming language library implementation of binary search had the same overflow bug for more than nine years. If the midpoint of the span is calculated as L Rthen the value of L R may exceed the range of integers of the data type used to store the midpoint, even if L and R are within the range. 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 search the exit conditions for the loop are not defined correctly. Once L exceeds Rthe search has failed and must convey the failure of the search. In addition, the loop must be exited when search target element is found, or in tree 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. The first iteration will select the midpoint of 8. On the left subarray are eight elements, but on the right are nine. If the search takes the right path, wiki is a higher chance that the search will make the maximum number of comparisons. The time complexity for this variation grows slightly more slowly, but at the cost wiki higher initial complexity. Linear search has lower initial complexity because it requires minimal computation, but it quickly outgrows binary search in complexity. Proceedings of the 14th ACM Southeast Conference. August "Dynamic Perfect Hashing: Upper and Lower Bounds". Proceedings of the 10th ACM International on Conference on emerging Networking Experiments and Technologies. Java Platform Standard Edition 8 Documentation. BinarySearch Method T ". The Python Standard Library. The Go Programming Language. Retrieved 1 May Alexandrescu, Andrei The D Programming Language. Upper Saddle River, Binary Addison-Wesley Professional. Programming Pearls 2nd ed Addison-Wesley. ISBN Chang, Shi-Kuo Data Structures and Algorithms. Software Engineering and Knowledge Engineering Singapore: World Scientific. ISBN Cormen, Thomas H. Introduction to Algorithms rd ed. MIT Press and McGraw-Hill. ISBN Fitzgerald, Michael Ruby Pocket Reference. ISBN Goldman, Sally A. Boca Raton: CRC Press. ISBN Knuth, Donald Sorting and Searching. The Art of Computer Programming nd ed. Boca Raton, FL: CRC Press. ISBN Moffat, Alistair; Turpin, Andrew Compression and Coding Algorithms. Hamburg, Germany: Kluwer Academic Publishers. 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