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Orthogonal Range Searching
In computer science, the range searching problem consists of processing a set ''S'' of objects, in order to determine which objects from ''S'' intersect with a query object, called the ''range''. For example, if ''S'' is a set of points corresponding to the coordinates of several cities, find the subset of cities within a given range of latitudes and longitudes. The range searching problem and the data structures that solve it are a fundamental topic of computational geometry. Applications of the problem arise in areas such as geographical information systems (GIS), computer-aided design (CAD) and databases. Variations There are several variations of the problem, and different data structures may be necessary for different variations. In order to obtain an efficient solution, several aspects of the problem need to be specified: * Object types: Algorithms depend on whether ''S'' consists of points, lines, line segments, boxes, polygons.... The simplest and most studied ob ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sphere
A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the centre of the sphere, and is the sphere's radius. The earliest known mentions of spheres appear in the work of the ancient Greek mathematicians. The sphere is a fundamental object in many fields of mathematics. Spheres and nearly-spherical shapes also appear in nature and industry. Bubbles such as soap bubbles take a spherical shape in equilibrium. The Earth is often approximated as a sphere in geography, and the celestial sphere is an important concept in astronomy. Manufactured items including pressure vessels and most curved mirrors and lenses are based on spheres. Spheres roll smoothly in any direction, so most balls used in sports and toys are spherical, as are ball bearings. Basic terminology As mentioned earlier is th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fractional Cascading
In computer science, fractional cascading is a technique to speed up a sequence of binary searches for the same value in a sequence of related data structures. The first binary search in the sequence takes a logarithmic amount of time, as is standard for binary searches, but successive searches in the sequence are faster. The original version of fractional cascading, introduced in two papers by Chazelle and Guibas in 1986 (; ), combined the idea of cascading, originating in range searching data structures of and , with the idea of fractional sampling, which originated in . Later authors introduced more complex forms of fractional cascading that allow the data structure to be maintained as the data changes by a sequence of discrete insertion and deletion events. Example As a simple example of fractional cascading, consider the following problem. We are given as input a collection of ''k'' ordered lists ''Li'' of numbers, such that the total length Σ, ''Li'', of all lists is ''n' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dan Willard
Dan Edward Willard is an American computer scientist and logician, and is a professor of computer science at the University at Albany. Education and career Willard did his undergraduate studies in mathematics at Stony Brook University, graduating in 1970. He went on to graduate studies in mathematics at Harvard University, earning a master's degree in 1972 and a doctorate in 1978. After leaving Harvard, he worked at Bell Labs for four years before joining the Albany faculty in 1983.Curriculum vitae accessed 2013-06-04. Contributions Although trained as a mathematician and employed as a computer scientist, Willard's most highly cited publication is in . In 1973, wi ...[...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Range Tree
In computer science, a range tree is an ordered tree data structure to hold a list of points. It allows all points within a given range to be reported efficiently, and is typically used in two or higher dimensions. Range trees were introduced by Jon Louis Bentley in 1979. Similar data structures were discovered independently by Lueker, Lee and Wong, and Willard. The range tree is an alternative to the ''k''-d tree. Compared to ''k''-d trees, range trees offer faster query times of (in Big O notation) O(\log^dn+k) but worse storage of O(n\log^ n), where ''n'' is the number of points stored in the tree, ''d'' is the dimension of each point and ''k'' is the number of points reported by a given query. Bernard Chazelle improved this to query time O(\log^ n + k) and space complexity O\left(n\left(\frac\right)^\right). Data structure A range tree on a set of 1-dimensional points is a balanced binary search tree on those points. The points stored in the tree are stored in the le ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Communications Of The ACM
''Communications of the ACM'' is the monthly journal of the Association for Computing Machinery (ACM). It was established in 1958, with Saul Rosen as its first managing editor. It is sent to all ACM members. Articles are intended for readers with backgrounds in all areas of computer science and information systems. The focus is on the practical implications of advances in information technology and associated management issues; ACM also publishes a variety of more theoretical journals. The magazine straddles the boundary of a science magazine, trade magazine, and a scientific journal. While the content is subject to peer review, the articles published are often summaries of research that may also be published elsewhere. Material published must be accessible and relevant to a broad readership. From 1960 onward, ''CACM'' also published algorithms, expressed in ALGOL. The collection of algorithms later became known as the Collected Algorithms of the ACM. See also * ''Journal of th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Big O Notation
Big ''O'' notation is a mathematical notation that describes the limiting behavior of a function when the argument tends towards a particular value or infinity. Big O is a member of a family of notations invented by Paul Bachmann, Edmund Landau, and others, collectively called Bachmann–Landau notation or asymptotic notation. The letter O was chosen by Bachmann to stand for '' Ordnung'', meaning the order of approximation. In computer science, big O notation is used to classify algorithms according to how their run time or space requirements grow as the input size grows. In analytic number theory, big O notation is often used to express a bound on the difference between an arithmetical function and a better understood approximation; a famous example of such a difference is the remainder term in the prime number theorem. Big O notation is also used in many other fields to provide similar estimates. Big O notation characterizes functions according to their growth rate ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
K-d Tree
In computer science, a ''k''-d tree (short for ''k-dimensional tree'') is a space-partitioning data structure for organizing points in a ''k''-dimensional space. ''k''-d trees are a useful data structure for several applications, such as searches involving a multidimensional search key (e.g. range searches and nearest neighbor searches) and creating point clouds. ''k''-d trees are a special case of binary space partitioning trees. Description The ''k''-d tree is a binary tree in which ''every'' node is a ''k''-dimensional point. Every non-leaf node can be thought of as implicitly generating a splitting hyperplane that divides the space into two parts, known as half-spaces. Points to the left of this hyperplane are represented by the left subtree of that node and points to the right of the hyperplane are represented by the right subtree. The hyperplane direction is chosen in the following way: every node in the tree is associated with one of the ''k'' dimensions, with ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jon Bentley (computer Scientist)
Jon Louis Bentley (born February 20, 1953) is an American computer scientist who is credited with the heuristic-based partitioning algorithm ''k''-d tree. Education and career Bentley received a B.S. in mathematical sciences from Stanford University in 1974, and M.S. and PhD in 1976 from the University of North Carolina at Chapel Hill; while a student, he also held internships at the Xerox Palo Alto Research Center and Stanford Linear Accelerator Center. After receiving his Ph.D., he joined the faculty at Carnegie Mellon University as an assistant professor of computer science and mathematics. At CMU, his students included Brian Reid, John Ousterhout, Jeff Eppinger, Joshua Bloch, and James Gosling, and he was one of Charles Leiserson's advisors. Later, Bentley moved to Bell Laboratories, where he co-authored an optimized Quicksort algorithm with Doug McIlroy. He found an optimal solution for the two dimensional case of Klee's measure problem: given a set of ''n'' rectang ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Orthogonal Range Query
In mathematics, orthogonality is the generalization of the geometric notion of '' perpendicularity''. By extension, orthogonality is also used to refer to the separation of specific features of a system. The term also has specialized meanings in other fields including art and chemistry. Etymology The word comes from the Ancient Greek ('), meaning "upright", and ('), meaning "angle". The Ancient Greek (') and Classical Latin ' originally denoted a rectangle. Later, they came to mean a right triangle. In the 12th century, the post-classical Latin word ''orthogonalis'' came to mean a right angle or something related to a right angle. Mathematics Physics * In optics, polarization states are said to be orthogonal when they propagate independently of each other, as in vertical and horizontal linear polarization or right- and left-handed circular polarization. * In special relativity, a time axis determined by a rapidity of motion is hyperbolic-orthogonal to a space axis of sim ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dynamization
In computer science, dynamization is the process of transforming a static data structure into a dynamic one. Although static data structures may provide very good functionality and fast queries, their utility is limited because of their inability to grow/shrink quickly, thus making them inapplicable for the solution of dynamic problems, where the amount of the input data changes. Dynamization techniques provide uniform ways of creating dynamic data structures. Decomposable search problems We define problem P of searching for the predicate M match in the set S as P(M,S). Problem P is ''decomposable'' if the set S can be decomposed into subsets S_i and there exists an operation + of result unification such that P(M,S) = P(M,S_0) + P(M,S_1) + \dots + P(M,S_n). Decomposition Decomposition is a term used in computer science to break static data structures into smaller units of unequal size. The basic principle is the idea that any decimal number can be translated into a representation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Semigroup
In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative internal binary operation on it. The binary operation of a semigroup is most often denoted multiplicatively: ''x''·''y'', or simply ''xy'', denotes the result of applying the semigroup operation to the ordered pair . Associativity is formally expressed as that for all ''x'', ''y'' and ''z'' in the semigroup. Semigroups may be considered a special case of magmas, where the operation is associative, or as a generalization of groups, without requiring the existence of an identity element or inverses. The closure axiom is implied by the definition of a binary operation on a set. Some authors thus omit it and specify three axioms for a group and only one axiom (associativity) for a semigroup. As in the case of groups or magmas, the semigroup operation need not be commutative, so ''x''·''y'' is not necessarily equal to ''y''·''x''; a well-known example of an operation that is as ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
