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Enumeration Algorithm
In computer science, an enumeration algorithm is an algorithm that enumerates the answers to a computational problem. Formally, such an algorithm applies to problems that take an input and produce a list of solutions, similarly to function problems. For each input, the enumeration algorithm must produce the list of all solutions, without duplicates, and then halt. The performance of an enumeration algorithm is measured in terms of the time required to produce the solutions, either in terms of the total time required to produce all solutions, or in terms of the maximal delay between two consecutive solutions and in terms of a preprocessing time, counted as the time before outputting the first solution. This complexity can be expressed in terms of the size of the input, the size of each individual output, or the total size of the set of all outputs, similarly to what is done with output-sensitive algorithms. Formal definitions An enumeration problem P is defined as a relation R o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Computer Science
Computer science is the study of computation, information, and automation. Computer science spans Theoretical computer science, theoretical disciplines (such as algorithms, theory of computation, and information theory) to Applied science, applied disciplines (including the design and implementation of Computer architecture, hardware and Software engineering, software). Algorithms and data structures are central to computer science. The theory of computation concerns abstract models of computation and general classes of computational problem, problems that can be solved using them. The fields of cryptography and computer security involve studying the means for secure communication and preventing security vulnerabilities. Computer graphics (computer science), Computer graphics and computational geometry address the generation of images. Programming language theory considers different ways to describe computational processes, and database theory concerns the management of re ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Partition (mathematics)
Generally, a partition is a division of a whole into non-overlapping parts. Among the kinds of partitions considered in mathematics are * partition of a set or an ordered partition of a set, * partition of a graph, * partition of an integer, * partition of an interval, * partition of unity, * partition of a matrix; see block matrix, and * partition of the sum of squares in statistics problems, especially in the analysis of variance, * quotition and partition, two ways of viewing the operation of division of integers. Integer partitions * Composition (combinatorics) * Ewens's sampling formula * Ferrers graph * Glaisher's theorem * Landau's function * Partition function (number theory) * Pentagonal number theorem * Plane partition * Quotition and partition * Rank of a partition ** Crank of a partition * Solid partition * Young tableau * Young's lattice Set partitions {{main, Partition of a set * Bell number * Bell polynomials ** Dobinski's fo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Monotone Dualization
In theoretical computer science, monotone dualization is a computational problem of constructing the dual of a monotone Boolean function. Equivalent problems can also be formulated as constructing the transversal hypergraph of a given hypergraph, of listing all minimal hitting sets of a family of sets, or of listing all minimal set covers of a family of sets. These problems can be solved in quasi-polynomial time in the combined size of its input and output, but whether they can be solved in polynomial time is an open problem. Definitions A Boolean function takes as input an assignment of truth values to its arguments, and produces as output another truth value. It is monotone when changing an argument from false to true cannot change the output from true to false. Every monotone Boolean function can be expressed as a Boolean expression using only logical disjunction ("or") and logical conjunction ("and"), without using logical negation ("not"). Such an expression is called a monoton ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Hypergraph (mathematics)
In mathematics, a hypergraph is a generalization of a graph in which an edge can join any number of vertices. In contrast, in an ordinary graph, an edge connects exactly two vertices. Formally, a directed hypergraph is a pair (X,E), where X is a set of elements called ''nodes'', ''vertices'', ''points'', or ''elements'' and E is a set of pairs of subsets of X. Each of these pairs (D,C)\in E is called an ''edge'' or ''hyperedge''; the vertex subset D is known as its ''tail'' or ''domain'', and C as its ''head'' or ''codomain''. The order of a hypergraph (X,E) is the number of vertices in X. The size of the hypergraph is the number of edges in E. The order of an edge e=(D,C) in a directed hypergraph is , e, = (, D, ,, C, ): that is, the number of vertices in its tail followed by the number of vertices in its head. The definition above generalizes from a directed graph to a directed hypergraph by defining the head or tail of each edge as a set of vertices (C\subseteq X or D\ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Minimal Transversal
Minimal may refer to: * Minimal (music genre), art music that employs limited or minimal musical materials * "Minimal" (song), 2006 song by Pet Shop Boys * Minimal (supermarket) or miniMAL, a former supermarket chain in Germany and Poland * Minimal (''Dungeons & Dragons''), a creature of magically reduced size in the game ''Dungeons & Dragons'' * Minimal (chocolate), a bean to bar chocolate store in Japan, featured in '' Kantaro: The Sweet Tooth Salaryman'' * Minimal (clothing), an Indonesia clothing-retail company that worked with fashion model Ayu Gani * MINIMAL (restaurant), high end restaurant in Taichung, Taiwan See also * *Minimalism (other) * Maximal (other) *Minimisation (other) Minimisation or minimization may refer to: * Minimisation (psychology), downplaying the significance of an event or emotion * Minimisation (clinical trials) * Minimisation (code) or Minification, removing unnecessary characters from source code * ... * Minimal prime ( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Vertex (geometry)
In geometry, a vertex (: vertices or vertexes), also called a corner, is a point (geometry), point where two or more curves, line (geometry), lines, or line segments Tangency, meet or Intersection (geometry), intersect. For example, the point where two lines meet to form an angle and the point where edge (geometry), edges of polygons and polyhedron, polyhedra meet are vertices. Definition Of an angle The ''vertex'' of an angle is the point where two Line (mathematics)#Ray, rays begin or meet, where two line segments join or meet, where two lines intersect (cross), or any appropriate combination of rays, segments, and lines that result in two straight "sides" meeting at one place. :(3 vols.): (vol. 1), (vol. 2), (vol. 3). Of a polytope A vertex is a corner point of a polygon, polyhedron, or other higher-dimensional polytope, formed by the intersection (Euclidean geometry), intersection of Edge (geometry), edges, face (geometry), faces or facets of the object. In a polygon, a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Linear Inequality
In mathematics a linear inequality is an inequality (mathematics), inequality which involves a linear function. A linear inequality contains one of the symbols of inequality: * greater than * ≤ less than or equal to * ≥ greater than or equal to * ≠ not equal to A linear inequality looks exactly like a linear equation, with the inequality sign replacing the equality sign. Linear inequalities of real numbers Two-dimensional linear inequalities Two-dimensional linear inequalities, are expressions in two variables of the form: :ax + by < c \text ax + by \geq c, where the inequalities may either be strict or not. The solution set of such an inequality can be graphically represented by a half-plane (all the points on one "side" of a fixed line) in the Euclidean plane. The line that determines the half-planes (''ax'' + ''by'' = ''c'') is not included in the solution set when the inequality is strict. A simple procedure to determine which half-plane is in the solution se ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
System Of Equations
In mathematics, a set of simultaneous equations, also known as a system of equations or an equation system, is a finite set of equations for which common solutions are sought. An equation system is usually classified in the same manner as single equations, namely as a: * System of linear equations, * System of nonlinear equations, * System of bilinear equations, * System of polynomial equations, * System of differential equations, or a * System of difference equations See also * Simultaneous equations model, a statistical model in the form of simultaneous linear equations * Elementary algebra Elementary algebra, also known as high school algebra or college algebra, encompasses the basic concepts of algebra. It is often contrasted with arithmetic: arithmetic deals with specified numbers, whilst algebra introduces variable (mathematics ..., for elementary methods {{set index article Equations Broad-concept articles de:Gleichung#Gleichungssysteme ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Polytope
In elementary geometry, a polytope is a geometric object with flat sides ('' faces''). Polytopes are the generalization of three-dimensional polyhedra to any number of dimensions. Polytopes may exist in any general number of dimensions as an -dimensional polytope or -polytope. For example, a two-dimensional polygon is a 2-polytope and a three-dimensional polyhedron is a 3-polytope. In this context, "flat sides" means that the sides of a -polytope consist of -polytopes that may have -polytopes in common. Some theories further generalize the idea to include such objects as unbounded apeirotopes and tessellations, decompositions or tilings of curved manifolds including spherical polyhedra, and set-theoretic abstract polytopes. Polytopes of more than three dimensions were first discovered by Ludwig Schläfli before 1853, who called such a figure a polyschem. The German term ''Polytop'' was coined by the mathematician Reinhold Hoppe, and was introduced to English mathematic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Vertex Enumeration Problem
In mathematics, the vertex enumeration problem for a polytope, a polyhedral cell complex, a hyperplane arrangement, or some other object of discrete geometry, is the problem of determination of the object's vertices given some formal representation of the object. A classical example is the problem of enumeration of the vertices of a convex polytope specified by a set of linear inequalities: :Ax \leq b where ''A'' is an ''m''×''n'' matrix, ''x'' is an ''n''×1 column vector of variables, and ''b'' is an ''m''×1 column vector of constants. The inverse (dual) problem of finding the bounding inequalities given the vertices is called '' facet enumeration'' (see convex hull algorithms). Computational complexity The computational complexity of the problem is a subject of research in computer science. For unbounded polyhedra, the problem is known to be NP-hard, more precisely, there is no algorithm that runs in polynomial time in the combined input-output size, unless P=NP. A 1992 art ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Cartesian Product
In mathematics, specifically set theory, the Cartesian product of two sets and , denoted , is the set of all ordered pairs where is an element of and is an element of . In terms of set-builder notation, that is A\times B = \. A table can be created by taking the Cartesian product of a set of rows and a set of columns. If the Cartesian product is taken, the cells of the table contain ordered pairs of the form . One can similarly define the Cartesian product of sets, also known as an -fold Cartesian product, which can be represented by an -dimensional array, where each element is an -tuple. An ordered pair is a 2-tuple or couple. More generally still, one can define the Cartesian product of an indexed family of sets. The Cartesian product is named after René Descartes, whose formulation of analytic geometry gave rise to the concept, which is further generalized in terms of direct product. Set-theoretic definition A rigorous definition of the Cartesian product re ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Hash Table
In computer science, a hash table is a data structure that implements an associative array, also called a dictionary or simply map; an associative array is an abstract data type that maps Unique key, keys to Value (computer science), values. A hash table uses a hash function to compute an ''index'', also called a ''hash code'', into an array of ''buckets'' or ''slots'', from which the desired value can be found. During lookup, the key is hashed and the resulting hash indicates where the corresponding value is stored. A map implemented by a hash table is called a hash map. Most hash table designs employ an Perfect hash function, imperfect hash function. Hash collision, Hash collisions, where the hash function generates the same index for more than one key, therefore typically must be accommodated in some way. In a well-dimensioned hash table, the average time complexity for each lookup is independent of the number of elements stored in the table. Many hash table designs also ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
