We discuss the following problem: given n points in the plane (the "sites"), and an arbitrary query point q, find the site that is closest to q. This problem can be solved by constructing the Voronoi diagram of the given sites, and then locating the query point in one of its regions. We give two algorithms, one that constructs the Voronoi diagram in O(n lg n) time, and another that inserts a new site in O(n) time. Both are based on the use of the Voronoi dual, the Delaunay triangulation, and are simple enough to be of practical value. The simplicity of both algorithms can be attributed to the separation of the geometrical and topological aspects of the problem, and to the use of two simple but powerful primitives, a geometric predicate and an operator for manipulating the tgpology of the diagram. The topology is represented by a new data structure for generalized diagrams, that is embeddings of graphs in two-dimensional manifolds. This structure represents simultaneously an embedding, its dual, and its mirror-image. Furthermore, just two operators are sufficient for building and modifying arbitrary diagrams. O. IntroductionOne of the ft,ndamental data structures of computational geometry is the Voronoi diagram. This diagram arises from consideration of the following natural problem. Let n points in the plane be given, called sites. We wish to preprocess them into a data structure, so that given a new query point q, we can efficiently locate the nearest neighbor of q among the sites. The n sites in fact partition the plane into a collection of n regions, each associated with one of the sites. If region P is associated with site p, then P is the locus of all points in the plane closer to p than to any of the other n --1 sites.This partition is known as the Voronoi diagram (or the Dirichlet, or Thiessen, tesselation) determined by the given sites.The closest site problem is can therefore be solved by constructing the Voronoi diagram, and then locating the query point in it. Using the currently best available algorithms, the Voronoi diagram of n points can be computed in O(nlgn) time and stored in O(n) space; these bounds have been shown to be optimal in the worst case [Sh]. Once we have the Voronoi diagram, we can construct in linear further time a structure with which we can do point location in a planar subdivision in O(lg n) time [Kil].The work of Jorge Stolfi. who is on leave from the University of Sit Paulo (Sit Paulo, Brazil) was partially s'upponed by a grant from Conselho Nacional de Desenvolvimento Cienlffioo e Tecnol6gico (CNPq).Permission to copy without fee all or part of this material is granted provided that the copies are not made or distributed for direct commercial advantage, the ACM copyright notice and the title of the publication and its date appear, and notice is given that copying is by permission of the Association for Computing Machinery. To copy otherwise, or to republish, requires a fee and/or specific permission.© 1983 ACM 0-89791-099-0/83/004/0221 $00.75Shamos [Sh] first pointed out that ...
Abstract-The image foresting transform (IFT) is a graph-based approach to the design of image processing operators based on connectivity. It naturally leads to correct and efficient implementations and to a better understanding of how different operators relate to each other. We give here a precise definition of the IFT, and a procedure to compute it-a generalization of Dijkstra's algorithm-with a proof of correctness. We also discuss implementation issues and illustrate the use of the IFT in a few applications.
Point location, often known in graphics as "hit detection," is one of the fundamental problems of computational geometry. In a point location query we want to identify which of a given collection of geometric objects contains a particular point. Let denote a subdivision of the Euclidean plane into monotone regions by a straight-line graph of m edges. In this paper we exhibit a substantial refinement of the technique of Lee and Preparata [SIAM J. Comput., 6 (1977), pp. 594-606] for locating a point in 5e based on separating chains. The new data structure, called a layered dag, can be built in O(m) time, uses O(m) storage, and makes possible point location in O(log rn) time. Unlike previous structures that attain these optimal bounds, the layered dag can be implemented in a simple and practical way, and is extensible to subdivisions with edges more general than straight-line segments.
Phase transitions and critical behavior are crucial issues both in theoretical and experimental neuroscience. We report analytic and computational results about phase transitions and self-organized criticality (SOC) in networks with general stochastic neurons. The stochastic neuron has a firing probability given by a smooth monotonic function Φ(V) of the membrane potential V, rather than a sharp firing threshold. We find that such networks can operate in several dynamic regimes (phases) depending on the average synaptic weight and the shape of the firing function Φ. In particular, we encounter both continuous and discontinuous phase transitions to absorbing states. At the continuous transition critical boundary, neuronal avalanches occur whose distributions of size and duration are given by power laws, as observed in biological neural networks. We also propose and test a new mechanism to produce SOC: the use of dynamic neuronal gains – a form of short-term plasticity probably located at the axon initial segment (AIS) – instead of depressing synapses at the dendrites (as previously studied in the literature). The new self-organization mechanism produces a slightly supercritical state, that we called SOSC, in accord to some intuitions of Alan Turing.
The following problem is discussed: given n points in the plane (the sites) and an arbitrary query point q , find the site that is closest to q . This problem can be solved by constructing the Voronoi diagram of the griven sites and then locating the query point inone of its regions. Two algorithms are given, one that constructs the Voronoi diagram in O ( n log n ) time, and another that inserts a new sit on O(n) time. Both are based on the use of the Voronoi dual, or Delaunay triangulation, and are simple enough to be of practical value. the simplicity of both algorithms can be attributed to the separation of the geometrical and topological aspects of the problem and to the use of two simple but powerful primitives, a geometric predicate and an operator for manipulating the topology of the diagram. The topology is represented by a new data structure for generalized diagrams, that is, embeddings of graphs in two-dimensional manifolds. This structure represents simultaneously an embedding, its dual, and its mirror image. Furthermore, just two operators are sufficients for building and modifying arbitrary diagrams.
Abstract. Affine arithmetic (AA) is a model for self-validated computation which, like standard interval arithmetic (IA), produces guaranteed enclosures for computed quantities, taking into account any uncertainties in the input data as well as all internal truncation and roundoff errors. Unlike standard IA, the quantity representations used by AA are first-order approximations, whose error is generally quadratic in the width of input intervals. In many practical applications, the higher asymptotic accuracy of AA more than compensates for the increased cost of its operations.
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