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Treap
In computer science, the treap and the randomized binary search tree are two closely related forms of binary search tree data structures that maintain a dynamic set of ordered keys and allow binary searches among the keys. After any sequence of insertions and deletions of keys, the shape of the tree is a random variable with the same probability distribution as a random binary tree; in particular, with high probability its height is proportional to the logarithm of the number of keys, so that each search, insertion, or deletion operation takes logarithmic time to perform. Description The treap was first described by Raimund Seidel and Cecilia R. Aragon in 1989; its name is a portmanteau of tree and heap. It is a Cartesian tree in which each key is given a (randomly chosen) numeric priority. As with any binary search tree, the inorder traversal order of the nodes is the same as the sorted order of the keys. The structure of the tree is determined by the requirement that it ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Treap Split
In computer science, the treap and the randomized binary search tree are two closely related forms of binary search tree data structures that maintain a dynamic set of ordered keys and allow binary searches among the keys. After any sequence of insertions and deletions of keys, the shape of the tree is a random variable with the same probability distribution as a random binary tree; in particular, with high probability its height is proportional to the logarithm of the number of keys, so that each search, insertion, or deletion operation takes logarithmic time to perform. Description The treap was first described by Raimund Seidel and Cecilia R. Aragon in 1989; its name is a portmanteau of tree and heap. It is a Cartesian tree in which each key is given a (randomly chosen) numeric priority. As with any binary search tree, the inorder traversal order of the nodes is the same as the sorted order of the keys. The structure of the tree is determined by the requirement tha ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Treap Merge
In computer science, the treap and the randomized binary search tree are two closely related forms of binary search tree data structures that maintain a dynamic set of ordered keys and allow binary searches among the keys. After any sequence of insertions and deletions of keys, the shape of the tree is a random variable with the same probability distribution as a random binary tree; in particular, with high probability its height is proportional to the logarithm of the number of keys, so that each search, insertion, or deletion operation takes logarithmic time to perform. Description The treap was first described by Raimund Seidel and Cecilia R. Aragon in 1989; its name is a portmanteau of tree and heap. It is a Cartesian tree in which each key is given a (randomly chosen) numeric priority. As with any binary search tree, the inorder traversal order of the nodes is the same as the sorted order of the keys. The structure of the tree is determined by the requirement that it ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finger Search
In computer science, a finger search on a data structure is an extension of any search operation that structure supports, where a reference (finger) to an element in the data structure is given along with the query. While the search time for an element is most frequently expressed as a function of the number of elements in a data structure, finger search times are a function of the distance between the element and the finger. In a set of ''n'' elements, the distance ''d''(''x'',''y'') (or simply ''d'' when unambiguous) between two elements ''x'' and ''y'' is their difference in rank. If ''x'' and ''y'' are the ''i''th and ''j''th largest elements in the structure, then the difference in rank is , ''i'' - ''j'', . If a normal search in some structure would normally take O(f(''n'')) time, then a finger search for ''x'' with finger ''y'' should ideally take O(f(''d'')) time. Remark that since ''d'' ≤ ''n'', it follows that in the worst case, finger search is only as bad as normal se ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Heap (data Structure)
In computer science, a heap is a specialized tree-based data structure which is essentially an almost complete tree that satisfies the heap property: in a ''max heap'', for any given node C, if P is a parent node of C, then the ''key'' (the ''value'') of P is greater than or equal to the key of C. In a ''min heap'', the key of P is less than or equal to the key of C. The node at the "top" of the heap (with no parents) is called the ''root'' node. The heap is one maximally efficient implementation of an abstract data type called a priority queue, and in fact, priority queues are often referred to as "heaps", regardless of how they may be implemented. In a heap, the highest (or lowest) priority element is always stored at the root. However, a heap is not a sorted structure; it can be regarded as being partially ordered. A heap is a useful data structure when it is necessary to repeatedly remove the object with the highest (or lowest) priority, or when insertions need to be interspe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cartesian Tree
In computer science, a Cartesian tree is a binary tree derived from a sequence of numbers; it can be uniquely defined from the properties that it is heap-ordered and that a symmetric (in-order) traversal of the tree returns the original sequence. Introduced by in the context of geometric range searching data structures, Cartesian trees have also been used in the definition of the treap and randomized binary search tree data structures for binary search problems. The Cartesian tree for a sequence may be constructed in linear time using a stack-based algorithm for finding all nearest smaller values in a sequence. Definition The Cartesian tree for a sequence of distinct numbers can be uniquely defined by the following properties: #The Cartesian tree for a sequence has one node for each number in the sequence. Each node is associated with a single sequence value. #A symmetric (in-order) traversal of the tree results in the original sequence. That is, the left subtree consists of t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Random Binary Search Tree
In computer science and probability theory, a random binary tree is a binary tree selected at random from some probability distribution on binary trees. Two different distributions are commonly used: binary trees formed by inserting nodes one at a time according to a random permutation, and binary trees chosen from a uniform discrete distribution in which all distinct trees are equally likely. It is also possible to form other distributions, for instance by repeated splitting. Adding and removing nodes directly in a random binary tree will in general disrupt its random structure, but the treap and related randomized binary search tree data structures use the principle of binary trees formed from a random permutation in order to maintain a balanced binary search tree dynamically as nodes are inserted and deleted. For random trees that are not necessarily binary, see random tree. Binary trees from random permutations For any set of numbers (or, more generally, values from some t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Binary Search Tree
In computer science, a binary search tree (BST), also called an ordered or sorted binary tree, is a rooted binary tree data structure with the key of each internal node being greater than all the keys in the respective node's left subtree and less than the ones in its right subtree. The time complexity of operations on the binary search tree is directly proportional to the height of the tree. Binary search trees allow binary search for fast lookup, addition, and removal of data items. Since the nodes in a BST are laid out so that each comparison skips about half of the remaining tree, the lookup performance is proportional to that of binary logarithm. BSTs were devised in the 1960s for the problem of efficient storage of labeled data and are attributed to Conway Berners-Lee and David Wheeler. The performance of a binary search tree is dependent on the order of insertion of the nodes into the tree since arbitrary insertions may lead to degeneracy; several variations of the b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
With High Probability
In mathematics, an event that occurs with high probability (often shortened to w.h.p. or WHP) is one whose probability depends on a certain number ''n'' and goes to 1 as ''n'' goes to infinity, i.e. the probability of the event occurring can be made as close to 1 as desired by making ''n'' big enough. Applications The term WHP is especially used in computer science, in the analysis of probabilistic algorithms. For example, consider a certain probabilistic algorithm on a graph with ''n'' nodes. If the probability that the algorithm returns the correct answer is 1-1/n, then when the number of nodes is very large, the algorithm is correct with a probability that is very near 1. This fact is expressed shortly by saying that the algorithm is correct WHP. Some examples where this term is used are: * Miller–Rabin primality test: a probabilistic algorithm for testing whether a given number ''n'' is prime or composite. If ''n'' is composite, the test will detect ''n'' as composite WHP. T ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Raimund Seidel
Raimund G. Seidel is a German and Austrian theoretical computer scientist and an expert in computational geometry. Seidel was born in Graz, Austria, and studied with Hermann Maurer at the Graz University of Technology. He received his M. Sc. in 1981 from University of British Columbia under David G. Kirkpatrick. He received his Ph.D. in 1987 from Cornell University under the supervision of John Gilbert. After teaching at the University of California, Berkeley, he moved in 1994 to Saarland University. In 1997 he and Christoph M. Hoffmann were program chairs for the Symposium on Computational Geometry. In 2014, he took over as Scientific Director of the Leibniz Center for Informatics (LZI) from Reinhard Wilhelm. Seidel invented backwards analysis of randomized algorithms and used it to analyze a simple linear programming algorithm that runs in linear time for problems of bounded dimension. With his student Cecilia R. Aragon in 1989 he devised the treap data structure, . an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tree Rotation
In discrete mathematics, tree rotation is an operation on a binary tree that changes the structure without interfering with the order of the elements. A tree rotation moves one node up in the tree and one node down. It is used to change the shape of the tree, and in particular to decrease its height by moving smaller subtrees down and larger subtrees up, resulting in improved performance of many tree operations. There exists an inconsistency in different descriptions as to the definition of the direction of rotations. Some say that the direction of rotation reflects the direction that a node is moving upon rotation (a left child rotating into its parent's location is a right rotation) while others say that the direction of rotation reflects which subtree is rotating (a left subtree rotating into its parent's location is a left rotation, the opposite of the former). This article takes the approach of the directional movement of the rotating node. Illustration The right rotation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pseudorandom Number Generator
A pseudorandom number generator (PRNG), also known as a deterministic random bit generator (DRBG), is an algorithm for generating a sequence of numbers whose properties approximate the properties of sequences of random numbers. The PRNG-generated sequence is not truly random, because it is completely determined by an initial value, called the PRNG's ''seed'' (which may include truly random values). Although sequences that are closer to truly random can be generated using hardware random number generators, ''pseudorandom number generators'' are important in practice for their speed in number generation and their reproducibility. PRNGs are central in applications such as simulations (e.g. for the Monte Carlo method), electronic games (e.g. for procedural generation), and cryptography. Cryptographic applications require the output not to be predictable from earlier outputs, and more elaborate algorithms, which do not inherit the linearity of simpler PRNGs, are needed. Good statis ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Join-based Tree Algorithms
In computer science, join-based tree algorithms are a class of algorithms for self-balancing binary search trees. This framework aims at designing highly-parallelized algorithms for various balanced binary search trees. The algorithmic framework is based on a single operation ''join''. Under this framework, the ''join'' operation captures all balancing criteria of different balancing schemes, and all other functions ''join'' have generic implementation across different balancing schemes. The ''join-based algorithms'' can be applied to at least four balancing schemes: AVL trees, red–black trees, weight-balanced trees and treaps. The ''join''(L,k,R) operation takes as input two binary balanced trees L and R of the same balancing scheme, and a key k, and outputs a new balanced binary tree t whose in-order traversal is the in-order traversal of L, then k then the in-order traversal of R. In particular, if the trees are search trees, which means that the in-order of the trees maintain ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
