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Klee's Measure Problem
In computational geometry, Klee's measure problem is the problem of determining how efficiently the measure of a union of ( multidimensional) rectangular ranges can be computed. Here, a ''d''-dimensional rectangular range is defined to be a Cartesian product of ''d'' intervals of real numbers, which is a subset of R''d''. The problem is named after Victor Klee, who gave an algorithm for computing the length of a union of intervals (the case ''d'' = 1) which was later shown to be optimally efficient in the sense of computational complexity theory. The computational complexity of computing the area of a union of 2-dimensional rectangular ranges is now also known, but the case ''d'' ≥ 3 remains an open problem. History and algorithms In 1977, Victor Klee considered the following problem: given a collection of ''n'' intervals in the real line, compute the length of their union. He then presented an algorithm to solve this problem with computational complexity (or "running t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Set A Rectangles (Klee's Trevis)
Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electronics and computing *Set (abstract data type), a data type in computer science that is a collection of unique values ** Set (C++), a set implementation in the C++ Standard Library * Set (command), a command for setting values of environment variables in Unix and Microsoft operating-systems * Secure Electronic Transaction, a standard protocol for securing credit card transactions over insecure networks * Single-electron transistor, a device to amplify currents in nanoelectronics * Single-ended triode, a type of electronic amplifier * Set!, a programming syntax in the scheme programming language Biology and psychology * Set (psychology), a set of expectations which shapes perception or thought *Set or sett, a badger's den *Set, a small tuber ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Michael Fredman
Michael Lawrence Fredman is an emeritus professor at the Computer Science Department at Rutgers University, United States. He earned his Ph.D. degree from Stanford University in 1972 under the supervision of Donald Knuth. He was a member of the mathematics department at the Massachusetts Institute of Technology from 1974 to 1976. and of the Computer Science and Engineering department at the University of California, San Diego until 1992. . Among his contributions to computer science are the development of the in a joint work with , ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
American Mathematical Monthly
''The American Mathematical Monthly'' is a mathematical journal founded by Benjamin Finkel in 1894. It is published ten times each year by Taylor & Francis for the Mathematical Association of America. The ''American Mathematical Monthly'' is an expository journal intended for a wide audience of mathematicians, from undergraduate students to research professionals. Articles are chosen on the basis of their broad interest and reviewed and edited for quality of exposition as well as content. In this the ''American Mathematical Monthly'' fulfills a different role from that of typical mathematical research journals. The ''American Mathematical Monthly'' is the most widely read mathematics journal in the world according to records on JSTOR. Tables of contents with article abstracts from 1997–2010 are availablonline The MAA gives the Lester R. Ford Awards annually to "authors of articles of expository excellence" published in the ''American Mathematical Monthly''. Editors *2022– ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Convex Body
In mathematics, a convex body in n-dimensional Euclidean space \R^n is a compact convex set with non-empty interior. A convex body K is called symmetric if it is centrally symmetric with respect to the origin; that is to say, a point x lies in K if and only if its antipode, - x also lies in K. Symmetric convex bodies are in a one-to-one correspondence with the unit balls of norms on \R^n. Important examples of convex bodies are the Euclidean ball, the hypercube In geometry, a hypercube is an ''n''-dimensional analogue of a square () and a cube (). It is a closed, compact, convex figure whose 1- skeleton consists of groups of opposite parallel line segments aligned in each of the space's dimensions, ... and the cross-polytope. See also * * References * {{Authority control Multi-dimensional geometry ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Convex Volume Approximation
In the analysis of algorithms, several authors have studied the computation of the volume of high-dimensional convex bodies, a problem that can also be used to model many other problems in combinatorial enumeration. Often these works use a black box model of computation in which the input is given by a subroutine for testing whether a point is inside or outside of the convex body, rather than by an explicit listing of the vertices or faces of a convex polytope. It is known that, in this model, no deterministic algorithm can achieve an accurate approximation, and even for an explicit listing of faces or vertices the problem is #P-hard. However, a joint work by Martin Dyer, Alan M. Frieze and Ravindran Kannan provided a randomized polynomial time approximation scheme for the problem, providing a sharp contrast between the capabilities of randomized and deterministic algorithms. The main result of the paper is a randomized algorithm for finding an \varepsilon approximation to the vo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lower Bound
In mathematics, particularly in order theory, an upper bound or majorant of a subset of some preordered set is an element of that is greater than or equal to every element of . Dually, a lower bound or minorant of is defined to be an element of that is less than or equal to every element of . A set with an upper (respectively, lower) bound is said to be bounded from above or majorized (respectively bounded from below or minorized) by that bound. The terms bounded above (bounded below) are also used in the mathematical literature for sets that have upper (respectively lower) bounds. Examples For example, is a lower bound for the set (as a subset of the integers or of the real numbers, etc.), and so is . On the other hand, is not a lower bound for since it is not smaller than every element in . The set has as both an upper bound and a lower bound; all other numbers are either an upper bound or a lower bound for that . Every subset of the natural numbers has a lowe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Trellis (graph)
A trellis is a graph whose nodes are ordered into vertical slices (''time'') with each node at each time connected to at least one node at an earlier and at least one node at a later time. The earliest and latest times in the trellis have only one node. Trellises are used in encoders and decoders for communication theory and encryption. They are also the central datatype used in Baum–Welch algorithm or the Viterbi AlgorithmRyan, M. S., & Nudd, G. R. (1993). The viterbi algorithm. University of Warwick, Department of Computer Science. for Hidden Markov Models. The trellis graph is named for its similar appearance to an architectural trellis. References See also * Trellis modulation * Trellis quantization Trellis quantization is an algorithm that can improve data compression in DCT-based encoding methods. It is used to optimize residual DCT coefficients after motion estimation in lossy video compression encoders such as Xvid and x264. Trellis qu ... Application-specif ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kd-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 the hyper ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mark Overmars
Markus Hendrik Overmars (; born 29 September 1958 in Zeist, Netherlands) is a Dutch computer scientist and teacher of game programming known for his game development application GameMaker. GameMaker lets people create computer games using a drag-and-drop interface. He is the former head of the ''Center for Geometry, Imaging, and Virtual Environments'' at Utrecht University, in the Netherlands. This research center concentrates on computational geometry and its application in areas like computer graphics, robotics, geographic information systems, imaging, multimedia, virtual environments, and games. Overmars received his Ph.D. in 1983 from Utrecht University under the supervision of Jan van Leeuwen, and continued to be a member of the faculty of the same university until September 2013. Overmars has published over 100 journal papers, largely on computational geometry, and is the co-author of several books including a widely used computational geometry text. Overmars has also wo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quadtree
A quadtree is a tree data structure in which each internal node has exactly four children. Quadtrees are the two-dimensional analog of octrees and are most often used to partition a two-dimensional space by recursively subdividing it into four quadrants or regions. The data associated with a leaf cell varies by application, but the leaf cell represents a "unit of interesting spatial information". The subdivided regions may be square or rectangular, or may have arbitrary shapes. This data structure was named a quadtree by Raphael Finkel and J.L. Bentley in 1974. A similar partitioning is also known as a ''Q-tree''. All forms of quadtrees share some common features: * They decompose space into adaptable cells * Each cell (or bucket) has a maximum capacity. When maximum capacity is reached, the bucket splits * The tree directory follows the spatial decomposition of the quadtree. A tree-pyramid (T-pyramid) is a "complete" tree; every node of the T-pyramid has four child nodes excep ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jan Van Leeuwen
Jan van Leeuwen (born December 17, 1946, in Waddinxveen) is a Dutch computer scientist and Emeritus professor of computer science at the Department of Information and Computing Sciences at Utrecht University.Curriculum vitae retrieved 2011-03-27. Education and career Van Leeuwen completed his undergraduate studies in mathematics at in 1967 and received a PhD in mathematics in 1972 from the same institution under the supervision of .. After postdoctoral studies at the |
Computer Graphics
Computer graphics deals with generating images with the aid of computers. Today, computer graphics is a core technology in digital photography, film, video games, cell phone and computer displays, and many specialized applications. A great deal of specialized hardware and software has been developed, with the displays of most devices being driven by computer graphics hardware. It is a vast and recently developed area of computer science. The phrase was coined in 1960 by computer graphics researchers Verne Hudson and William Fetter of Boeing. It is often abbreviated as CG, or typically in the context of film as computer generated imagery (CGI). The non-artistic aspects of computer graphics are the subject of computer science research. Some topics in computer graphics include user interface design, sprite graphics, rendering, ray tracing, geometry processing, computer animation, vector graphics, 3D modeling, shaders, GPU design, implicit surfaces, visualization, scientific c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
