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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 bit operation can be performed in constant time), or equivalently in time, where is the largest element that can be stored in the tree. The parameter 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 vEB tree has poor space efficiency. For example, for storing 32-bit integers (i.e., when ), it requires bits of storage. However, similar data structures with equally good time efficiency and space exist, where is the number of stored elements. Supported operations A vEB supports the operations of an ''ordered associative array'', which includes the usual associative ... [...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. 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 in a single straight line. Binary trees are a commonly used type, which constrain the number of children for each parent to exactly two. When the order of the children is specified, this data structure corresponds to an ordered tree in graph theory. A value or pointer to other data may be associated with every node in the tre ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Word (data Type)
In computing, a word is the natural unit of data used by a particular processor design. A word is a fixed-sized datum handled as a unit by the instruction set or the hardware of the processor. The number of bits or digits in a word (the ''word size'', ''word width'', or ''word length'') is an important characteristic of any specific processor design or computer architecture. The size of a word is reflected in many aspects of a computer's structure and operation; the majority of the registers in a processor are usually word-sized and the largest datum that can be transferred to and from the working memory in a single operation is a word in many (not all) architectures. The largest possible address size, used to designate a location in memory, is typically a hardware word (here, "hardware word" means the full-sized natural word of the processor, as opposed to any other definition used). Documentation for older computers with fixed word size commonly states memory sizes in words ra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Computer Science Articles Needing Expert Attention
A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations (computation) automatically. Modern digital electronic computers can perform generic sets of operations known as programs. These programs enable computers to perform a wide range of tasks. A computer system is a nominally complete computer that includes the hardware, operating system (main software), and peripheral equipment needed and used for full operation. This term may also refer 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. Simple special-purpose devices like microwave ovens and remote controls are included, as are factory devices like industrial robots and computer-aided design, as well as general-purpose devices like personal computers and mobile devices like smartphones. Computers power the Internet, which links bil ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Systems Theory
Dynamical systems theory is an area of mathematics used to describe the behavior of complex dynamical systems, usually by employing differential equations or difference equations. When differential equations are employed, the theory is called ''continuous dynamical systems''. From a physical point of view, continuous dynamical systems is a generalization of classical mechanics, a generalization where the equations of motion are postulated directly and are not constrained to be Euler–Lagrange equations of a least action principle. When difference equations are employed, the theory is called ''discrete dynamical systems''. When the time variable runs over a set that is discrete over some intervals and continuous over other intervals or is any arbitrary time-set such as a Cantor set, one gets dynamic equations on time scales. Some situations may also be modeled by mixed operators, such as differential-difference equations. This theory deals with the long-term qualitative behavior ... [...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] |
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 integer has size less than 2w and is non-negative. When operating on a collection of 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, a model of circuit complexity that allows addition and bitwise Boolean operations but does not allow the multip ... [...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 less than or equal to * successor(x), which returns the smalles ... [...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 bi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Standard ML
Standard ML (SML) is a general-purpose, modular, functional programming language with compile-time type checking and type inference. It is popular among compiler writers and programming language researchers, as well as in the development of theorem provers. Standard ML is a modern dialect of 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 expressions as opposed to statements or commands, although some expressions of type unit are only evaluated for their side-effects. Functions Like all functional languages, a key feature of Standard ML is the function, which is used for abstraction. The factorial function can be expressed as follows: ... [...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 an 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 as well as implementations for code-generation, documentation, and specific support for a variety of formal methods. It can be seen as an 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 provides a meta-logic (a weak type theory), which is used to ... [...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 t ... [...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, s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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