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Semi-Yao Graph
The ''k''-semi-Yao graph (''k''-SYG) of a set of ''n'' objects ''P'' is a geometric proximity graph, which was first described to present a kinetic data structure for maintenance of all the nearest neighbors on moving objects. It is named for its relation to the Yao graph, which is named after Andrew Yao. Construction The ''k''-SYG is constructed as follows. The space around each point ''p'' in ''P'' is partitioned into a set of polyhedral cones of opening angle \theta, meaning the angle of each pair of rays inside a polyhedral cone emanating from the apex is at most \theta, and then ''p'' connects to ''k'' points of ''P'' in each of the polyhedral cones whose projections on the cone axis is minimum. Properties * The ''k''-SYG, where ''k'' = 1, is known as the theta graph, and is the union of two Delaunay triangulations. * For a small \theta and an appropriate cone axis, the ''k''-SYG gives a supergraph of the ''k''-nearest neighbor graph (''k''-NNG). For example, in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kinetic Data Structure
A kinetic data structure is a data structure used to track an attribute of a geometric system that is moving continuously. For example, a kinetic convex hull data structure maintains the convex hull of a group of n moving points. The development of kinetic data structures was motivated by computational geometry problems involving physical objects in continuous motion, such as collision or visibility detection in robotics, animation or computer graphics. Overview Kinetic data structures are used on systems where there is a set of values that are changing as a function of time, in a known fashion. So the system has some values, and for each value v, it is known that v=f(t). Kinetic data structures allow queries on a system at the current virtual time t, and two additional operations: *\textrm(t): Advances the system to time t. *\textrm(v,f(t)): Alters the trajectory of value v to f(t), as of the current time. Additional operations may be supported. For example, kinetic data stru ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nearest Neighbor Graph
The nearest neighbor graph (NNG) is a directed graph defined for a set of points in a metric space, such as the Euclidean distance in the plane. The NNG has a vertex for each point, and a directed edge from ''p'' to ''q'' whenever ''q'' is a nearest neighbor of ''p'', a point whose distance from ''p'' is minimum among all the given points other than ''p'' itself. In many uses of these graphs, the directions of the edges are ignored and the NNG is defined instead as an undirected graph. However, the nearest neighbor relation is not a symmetric one, i.e., ''p'' from the definition is not necessarily a nearest neighbor for ''q''. In theoretical discussions of algorithms a kind of general position is often assumed, namely, the nearest (k-nearest) neighbor is unique for each object. In implementations of the algorithms it is necessary to bear in mind that this is not always the case. For situations in which it is necessary to make the nearest neighbor for each object unique, the set '' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Yao Graph
In computational geometry, the Yao graph, named after Andrew Yao, is a kind of geometric spanner, a weighted undirected graph connecting a set of geometric points with the property that, for every pair of points in the graph, their shortest path has a length that is within a constant factor of their Euclidean distance. The basic idea underlying the two-dimensional Yao graph is to surround each of the given points by equally spaced rays, partitioning the plane into sectors with equal angles, and to connect each point to its nearest neighbor in each of these sectors. Associated with a Yao graph is an integer parameter which is the number of rays and sectors described above; larger values of produce closer approximations to the Euclidean distance. The stretch factor is at most 1/(\cos \theta - \sin \theta), where \theta is the angle of the sectors. The same idea can be extended to point sets in more than two dimensions, but the number of sectors required grows exponentially wit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Andrew Yao
Andrew Chi-Chih Yao (; born December 24, 1946) is a Chinese computer scientist and computational theorist. He is currently a professor and the dean of Institute for Interdisciplinary Information Sciences (IIIS) at Tsinghua University. Yao used the minimax theorem to prove what is now known as Yao's Principle. Yao was a naturalized U.S. citizen, and worked for many years in the U.S. In 2015, together with Yang Chen-Ning, he renounced his U.S. citizenship and became an academician of the Chinese Academy of Sciences. Early life Yao was born in Shanghai, China. He completed his undergraduate education in physics at the National Taiwan University, before completing a Doctor of Philosophy in physics at Harvard University in 1972, and then a second PhD in computer science from the University of Illinois at Urbana–Champaign in 1975. Academic career Yao was an assistant professor at Massachusetts Institute of Technology (1975–1976), assistant professor at Stanford University ( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Theta Graph
In computational geometry, the Theta graph, or \Theta-graph, is a type of geometric spanner similar to a Yao graph. The basic method of construction involves partitioning the space around each vertex into a set of ''cones'', which themselves partition the remaining vertices of the graph. Like Yao Graphs, a \Theta-graph contains at most one edge per cone; where they differ is how that edge is selected. Whereas Yao Graphs will select the nearest vertex according to the metric space of the graph, the \Theta-graph defines a fixed ray contained within each cone (conventionally the bisector of the cone) and selects the nearest neighbor with respect to orthogonal projections to that ray. The resulting graph exhibits several good spanner properties.. \Theta-graphs were first described by ClarksonK. Clarkson. 1987. Approximation algorithms for shortest path motion planning. In Proceedings of the nineteenth annual ACM symposium on Theory of computing (STOC '87), Alfred V. Aho (Ed.). ACM, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Delaunay Triangulation
In mathematics and computational geometry, a Delaunay triangulation (also known as a Delone triangulation) for a given set P of discrete points in a general position is a triangulation DT(P) such that no point in P is inside the circumcircle of any triangle in DT(P). Delaunay triangulations maximize the minimum of all the angles of the triangles in the triangulation; they tend to avoid sliver triangles. The triangulation is named after Boris Delaunay for his work on this topic from 1934. For a set of points on the same line there is no Delaunay triangulation (the notion of triangulation is degenerate for this case). For four or more points on the same circle (e.g., the vertices of a rectangle) the Delaunay triangulation is not unique: each of the two possible triangulations that split the quadrangle into two triangles satisfies the "Delaunay condition", i.e., the requirement that the circumcircles of all triangles have empty interiors. By considering circumscribed spheres, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Computational Geometry (journal)
''Computational Geometry'', also known as ''Computational Geometry: Theory and Applications'', is a peer-reviewed mathematics journal for research in theoretical and applied computational geometry, its applications, techniques, and design and analysis of geometric algorithms. All aspects of computational geometry are covered, including the numerical, graph theoretical and combinatorial aspects, as well as fundamental problems in various areas of application of computational geometry: in computer graphics, pattern recognition, image processing, robotics, electronic design automation, CAD/CAM, and geographical information systems. The journal was founded in 1991 by Jörg-Rüdiger Sack and Jorge Urrutia.. It is indexed by ''Mathematical Reviews'', Zentralblatt MATH, Science Citation Index, and Current Contents ''Current Contents'' is a rapid alerting service database from Clarivate Analytics, formerly the Institute for Scientific Information and Thomson Reuters. It is publis ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometric Spanner
A geometric spanner or a -spanner graph or a -spanner was initially introduced as a weighted graph over a set of points as its vertices for which there is a -path between any pair of vertices for a fixed parameter . A -path is defined as a path through the graph with weight at most times the spatial distance between its endpoints. The parameter is called the stretch factor or dilation factor of the spanner. In computational geometry, the concept was first discussed by L.P. Chew in 1986, although the term "spanner" was not used in the original paper. The notion of graph spanners has been known in graph theory: -spanners are spanning subgraphs of graphs with similar dilation property, where distances between graph vertices are defined in graph-theoretical terms. Therefore geometric spanners are graph spanners of complete graphs embedded in the plane with edge weights equal to the distances between the embedded vertices in the corresponding metric. Spanners may be used in co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Computational Geometry
Computational geometry is a branch of computer science devoted to the study of algorithms which can be stated in terms of geometry. Some purely geometrical problems arise out of the study of computational geometric algorithms, and such problems are also considered to be part of computational geometry. While modern computational geometry is a recent development, it is one of the oldest fields of computing with a history stretching back to antiquity. Analysis of algorithms, Computational complexity is central to computational geometry, with great practical significance if algorithms are used on very large datasets containing tens or hundreds of millions of points. For such sets, the difference between O(''n''2) and O(''n'' log ''n'') may be the difference between days and seconds of computation. The main impetus for the development of computational geometry as a discipline was progress in computer graphics and computer-aided design and manufacturing (Computer-aided design, CAD/Compu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
