## Tree rotation

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In computer sciencebinary searchalso known as half-interval search[1] logarithmic search[2] or binary chop[3] is a search binary search tree algorithm wikipedia 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 timemaking O log n comparisons, where n is the number of elements in the array, the O is Big O notationand 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 binary search tree algorithm wikipedia 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 0A 1In 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 tree algorithm wikipedia 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 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 binary search tree algorithm wikipedia 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 binary search tree algorithm wikipedia 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 binary search tree algorithm wikipedia 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 arrayshash tablesa data structure that maps keys to records using a hash functionare 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 binary search tree algorithm wikipedia 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 treesbinary 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 binary search tree algorithm wikipedia storage such as databases and filesystems. Linear search binary search tree algorithm wikipedia a simple search algorithm that checks every record binary search tree algorithm wikipedia 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.

Binary search tree algorithm wikipedia the array also enables efficient approximate matches binary search tree algorithm wikipedia 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 filtersanother 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 treesfusion treestriesand bit arrays. However, while these binary search tree algorithm wikipedia 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 integersand 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 binary search tree algorithm wikipedia 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 binary search tree algorithm wikipedia 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 binary search tree algorithm wikipedia 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 binary search tree algorithm wikipedia 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 binary search tree algorithm wikipedia elsewhere, in domains such as binary search tree algorithm wikipedia 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 binary search tree algorithm wikipedia 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 binary search tree algorithm wikipedia the correct position of the target with a given probability that controls the reliability of the yielded position.

InJohn Mauchly made the first mention of binary search as part of the Moore School Lecturesthe 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 binary search tree algorithm wikipedia, [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 Lthe 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 Rthe 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:.

## Opzioni binarie anyoption

In computer science , a search tree is a tree data structure used for locating specific keys from within a set. In order for a tree to function as a search tree, the key for each node must be greater than any keys in subtrees on the left and less than any keys in subtrees on the right. The advantage of search trees is their efficient search time given the tree is reasonably balanced, which is to say the leaves at either end are of comparable depths.

Various search-tree data structures exist, several of which also allow efficient insertion and deletion of elements, which operations then have to maintain tree balance. Search trees are often used to implement an associative array. The search tree algorithm uses the key from the key-value pair to find a location, and then the application stores the entire key—value pair at that location.

A Binary Search Tree is a node-based data structure where each node contains a key and two subtrees, the left and right. For all nodes, the left subtree's key must be less than the node's key, and the right subtree's key must be greater than the node's key. These subtrees must all qualify as binary search trees. The worst-case time complexity for searching a binary search tree is the height of the tree , which can be as small as O log n for a tree with n elements.

B-trees are generalizations of binary search trees in that they can have a variable number of subtrees at each node. While child-nodes have a pre-defined range, they will not necessarily be filled with data, meaning B-trees can potentially waste some space.

The advantage is that B-trees do not need to be re-balanced as frequently as other self-balancing trees. Due to the variable range of their node length, B-trees are optimized for systems that read large blocks of data. They are also commonly used in databases. An a,b -tree is a search tree where all of its leaves are the same depth. Each node has at least a children and at most b children, while the root has at least 2 children and at most b children.

A ternary search tree is a type of trie that can have 3 nodes: Each node stores a single character and the tree itself is ordered the same way a binary search tree is, with the exception of a possible third node. Searching a ternary search tree involves passing in a string to test whether any path contains it. Assuming the tree is ordered, we can take a key and attempt to locate it within the tree.

The following algorithms are generalized for binary search trees, but the same idea can be applied to trees of other formats. In a sorted tree, the minimum is located at the node farthest left, while the maximum is located at the node farthest right.