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Van Emde Boas Tree
A van Emde Boas tree (), also known as a vEB tree or van Emde Boas priority queue, is a tree data structure which implements an associative array with -bit integer keys. It was invented by a team led by Dutch computer scientist Peter van Emde Boas in 1975. It performs all operations in time (assuming that an m bit operation can be performed in constant time), or equivalently in O(\log \log M) time, where M = 2^m is the largest element that can be stored in the tree. The parameter M is not to be confused with the actual number of elements stored in the tree, by which the performance of other tree data-structures is often measured. The standard vEB tree has inadequate space efficiency. For example, for storing 32-bit integers (i.e., when m = 32), it requires M = 2^ bits of storage. However, similar data structures with equally good time efficiency and space efficiency of O(n) (where n is the number of stored elements) exist, and vEB trees can be modified to require only O(n \log ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tree Data Structure
In computer science, a tree is a widely used abstract data type that represents a hierarchical tree structure with a set of connected nodes. Each node in the tree can be connected to many children (depending on the type of tree), but must be connected to exactly one parent, except for the ''root'' node, which has no parent (i.e., the root node as the top-most node in the tree hierarchy). These constraints mean there are no cycles or "loops" (no node can be its own ancestor), and also that each child can be treated like the root node of its own subtree, making recursion a useful technique for tree traversal. In contrast to linear data structures, many trees cannot be represented by relationships between neighboring nodes (parent and children nodes of a node under consideration, if they exist) in a single straight line (called edge or link between two adjacent nodes). Binary trees are a commonly used type, which constrain the number of children for each parent to at most two. When ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Trie
In computer science, a trie (, ), also known as a digital tree or prefix tree, is a specialized search tree data structure used to store and retrieve strings from a dictionary or set. Unlike a binary search tree, nodes in a trie do not store their associated key. Instead, each node's ''position'' within the trie determines its associated key, with the connections between nodes defined by individual Character (computing), characters rather than the entire key. Tries are particularly effective for tasks such as autocomplete, spell checking, and IP routing, offering advantages over hash tables due to their prefix-based organization and lack of hash collisions. Every child node shares a common prefix (computer science), prefix with its parent node, and the root node represents the empty string. While basic trie implementations can be memory-intensive, various optimization techniques such as compression and bitwise representations have been developed to improve their efficiency. A n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Computer Science Articles Needing Expert Attention
A computer is a machine that can be programmed to automatically carry out sequences of arithmetic or logical operations (''computation''). Modern digital electronic computers can perform generic sets of operations known as ''programs'', which enable computers to perform a wide range of tasks. The term computer system may refer to a nominally complete computer that includes the hardware, operating system, software, and peripheral equipment needed and used for full operation; or to a group of computers that are linked and function together, such as a computer network or computer cluster. A broad range of industrial and consumer products use computers as control systems, including simple special-purpose devices like microwave ovens and remote controls, and factory devices like industrial robots. Computers are at the core of general-purpose devices such as personal computers and mobile devices such as smartphones. Computers power the Internet, which links billions of computers ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Systems Theory
''Theory of Computing Systems'' is a peer-reviewed scientific journal published by Springer Verlag. Published since 1967 as ''Mathematical Systems Theory'' and since volume 30 in 1997 under its current title, it is devoted to publishing original research from all areas of theoretical computer science, such as computational complexity, algorithms and data structures, or parallel and distributed algorithms and architectures. It is published 8 times per year since 2018, although the frequency varied in the past. References External links * Computer science journals Theoretical computer science Springer Science+Business Media academic journals 8 times per year journals {{Computer-science-journal-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tango Tree
A tango tree is a type of binary search tree proposed by Erik D. Demaine, Dion Harmon, John Iacono, and Mihai Pătrașcu in 2004. It is named after Buenos Aires, of which the tango is emblematic. It is an online binary search tree that achieves an O(\log \log n) competitive ratio relative to the offline optimal binary search tree, while only using O(\log \log n) additional bits of memory per node. This improved upon the previous best known competitive ratio, which was O(\log n). Structure Tango trees work by partitioning a binary search tree into a set of ''preferred paths'', which are themselves stored in auxiliary trees (so the tango tree is represented as a tree of trees). Reference tree To construct a tango tree, we simulate a complete binary search tree called the ''reference tree'', which is simply a traditional binary search tree containing all the elements. This tree never shows up in the actual implementation, but is the conceptual basis behind the following pieces of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integer Sorting
In computer science, integer sorting is the algorithmic problem of sorting a collection of data values by integer keys. Algorithms designed for integer sorting may also often be applied to sorting problems in which the keys are floating point numbers, rational numbers, or text strings.. The ability to perform integer arithmetic on the keys allows integer sorting algorithms to be faster than comparison sorting algorithms in many cases, depending on the details of which operations are allowed in the model of computing and how large the integers to be sorted are. Integer sorting algorithms including pigeonhole sort, counting sort, and radix sort are widely used and practical. Other integer sorting algorithms with smaller worst-case time bounds are not believed to be practical for computer architectures with 64 or fewer bits per word. Many such algorithms are known, with performance depending on a combination of the number of items to be sorted, number of bits per key, and number of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Standard ML
Standard ML (SML) is a General-purpose programming language, general-purpose, High-level programming language, high-level, Modular programming, modular, Functional programming, functional programming language with compile-time type checking and type inference. It is popular for writing compilers, for programming language research, and for developing automated theorem proving, theorem provers. Standard ML is a modern dialect of ML (programming language), ML, the language used in the Logic for Computable Functions (LCF) theorem-proving project. It is distinctive among widely used languages in that it has a formal specification, given as typing rules and operational semantics in ''The Definition of Standard ML''. Language Standard ML is a functional programming language with some impure features. Programs written in Standard ML consist of Expression (computer science), expressions in contrast to statements or commands, although some expressions of type Unit type, unit are only eva ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Isabelle (proof Assistant)
The Isabelle automated theorem prover is a higher-order logic (HOL) theorem prover, written in Standard ML and Scala. As a Logic for Computable Functions (LCF) style theorem prover, it is based on a small logical core (kernel) to increase the trustworthiness of proofs without requiring, yet supporting, explicit proof objects. Isabelle is available inside a flexible system framework allowing for logically safe extensions, which comprise both theories and implementations for code-generating, documenting, and specific support for a variety of formal methods. It can be seen as an integrated development environment (IDE) for formal methods. In recent years, a substantial number of theories and system extensions have been collected in the Isabelle ''Archive of Formal Proofs'' (Isabelle AFP). Isabelle was named by Lawrence Paulson after Gérard Huet's daughter. The Isabelle theorem prover is free software, released under the revised BSD license. Features Isabelle is generic: it ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Predecessor Problem
In computer science, the predecessor problem involves maintaining a set of items to, given an element, efficiently query which element precedes or succeeds that element in an order. Data structures used to solve the problem include balanced binary search trees, van Emde Boas trees, and fusion trees. In the static predecessor problem, the set of elements does not change, but in the dynamic predecessor problem, insertions into and deletions from the set are allowed. The predecessor problem is a simple case of the nearest neighbor problem, and data structures that solve it have applications in problems like integer sorting. Definition The problem consists of maintaining a set , which contains a subset of integers. Each of these integers can be stored with a word size of , implying that U \le 2^w. Data structures that solve the problem support these operations: * predecessor(x), which returns the largest element in strictly smaller than * successor(x), which returns the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fusion Tree
In computer science, a fusion tree is a type of tree data structure that implements an associative array on -bit integers on a finite universe, where each of the input integers has size less than 2w and is non-negative. When operating on a collection of Attribute–value pair, key–value pairs, it uses space and performs searches in time, which is asymptotically faster than a traditional self-balancing binary search tree, and also better than the van Emde Boas tree for large values of . It achieves this speed by using certain constant-time operations that can be done on a machine word. Fusion trees were invented in 1990 by Michael Fredman and Dan Willard. Several advances have been made since Fredman and Willard's original 1990 paper. In 1999. it was shown how to implement fusion trees under a model of computation in which all of the underlying operations of the algorithm belong to AC0, AC0, a model of circuit complexity that allows addition and bitwise Boolean operations ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Y-fast Trie
In computer science, a y-fast trie is a data structure for storing integers from a bounded domain. It supports exact and predecessor or successor queries in time ''O''(log log ''M''), using ''O''(''n'') space, where ''n'' is the number of stored values and ''M'' is the maximum value in the domain. The structure was proposed by Dan Willard in 1982 to decrease the ''O''(''n'' log ''M'') space used by an x-fast trie. Structure A y-fast trie consists of two data structures: the top half is an x-fast trie and the lower half consists of a number of balanced binary trees. The keys are divided into groups of ''O''(log ''M'') consecutive elements and for each group a balanced binary search tree is created. To facilitate efficient insertion and deletion, each group contains at least (log ''M'')/4 and at most 2 log ''M'' elements. For each balanced binary search tree a representative ''r'' is chosen. These representatives are stored in the x-fast ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
X-fast Trie
In computer science, an x-fast trie is a data structure for storing integers from a bounded domain. It supports exact and predecessor or successor queries in time ''O''(log log ''M''), using ''O''(''n'' log ''M'') space, where ''n'' is the number of stored values and ''M'' is the maximum value in the domain. The structure was proposed by Dan Willard in 1982, along with the more complicated y-fast trie, as a way to improve the space usage of van Emde Boas trees, while retaining the ''O''(log log ''M'') query time. Structure An x-fast trie is a bitwise trie: a binary tree where each subtree stores values whose binary representations start with a common prefix. Each internal node is labeled with the common prefix of the values in its subtree and typically, the left child adds a 0 to the end of the prefix, while the right child adds a 1. The binary representation of an integer between 0 and ''M'' − 1 uses ⌈log2 ''M''⌉ bits, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
