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13
Geometric Range Searching
, 1994
"... In geometric range searching, algorithmic problems of the following type are considered: Given an npoint set P in the plane, build a data structure so that, given a query triangle R, the number of points of P lying in R can be determined quickly. Problems of this type are of crucial importance in c ..."
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Cited by 57 (3 self)
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In geometric range searching, algorithmic problems of the following type are considered: Given an npoint set P in the plane, build a data structure so that, given a query triangle R, the number of points of P lying in R can be determined quickly. Problems of this type are of crucial importance in computational geometry, as they can be used as subroutines in many seemingly unrelated algorithms. We present a survey of results and main techniques in this area.
Fast Rendering of Irregular Grids
, 2007
"... We propose a fast algorithm for rendering general irregular grids. Our method uses a sweepplane approach to accelerate ray casting, and can handle disconnected and nonconvex (even with holes) unstructured irregular grids with a rendering cost that decreases as the “disconnectedness” decreases. The ..."
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Cited by 46 (11 self)
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We propose a fast algorithm for rendering general irregular grids. Our method uses a sweepplane approach to accelerate ray casting, and can handle disconnected and nonconvex (even with holes) unstructured irregular grids with a rendering cost that decreases as the “disconnectedness” decreases. The algorithm is carefully tailored to exploit spatial coherence even if the image resolution differs substantially from the object space resolution. In this paper, we establish the practicality of our method through experimental results based on our implementation, and we also provide theoretical results, both lower and upper bounds, on the complexity of ray casting of irregular grids.
The lazy sweep ray casting algorithm for rendering irregular grids
 IEEE Transactions on Visualization and Computer Graphics
, 1997
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Dynamic Ray Shooting and Shortest Paths in Planar Subdivisions via Balanced Geodesic Triangulations
 J. Algorithms
, 1997
"... We give new methods for maintaining a data structure that supports ray shooting and shortest path queries in a dynamicallychanging connected planar subdivision S. Our approach is based on a new dynamic method for maintaining a balanced decomposition of a simple polygon via geodesic triangles. We ma ..."
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Cited by 39 (3 self)
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We give new methods for maintaining a data structure that supports ray shooting and shortest path queries in a dynamicallychanging connected planar subdivision S. Our approach is based on a new dynamic method for maintaining a balanced decomposition of a simple polygon via geodesic triangles. We maintain such triangulations by viewing their dual trees as balanced trees. We show that rotations in these trees can be implemented via a simple "diagonal swapping" operation performed on the corresponding geodesic triangles, and that edge insertion and deletion can be implemented on these trees using operations akin to the standard split and splice operations. We also maintain a dynamic point location structure on the geodesic triangulation, so that we may implement ray shooting queries by first locating the ray's endpoint and then walking along the ray from geodesic triangle to geodesic triangle until we hit the boundary of some region of S. The shortest path between two points in the same ...
On Fat Partitioning, Fat Covering and the Union Size of Polygons
, 1993
"... The complexity of the contour of the union of simple polygons with n vertices in total can be O(n 2) in general. A notion of fatness for simple polygons is introduced, which extends most of the existing fatness definitions. It is proved that a set of fat polygons with n vertices in total has unio ..."
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Cited by 33 (3 self)
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The complexity of the contour of the union of simple polygons with n vertices in total can be O(n 2) in general. A notion of fatness for simple polygons is introduced, which extends most of the existing fatness definitions. It is proved that a set of fat polygons with n vertices in total has union complexity is O(nloglogn), which is a generalization of a similar result for fat triangles [19]. Applications to several basic problems in computational geometry are given, such as efficient hidden surface removal, motion planning, injection molding, etc. The result is based on a new method to partition a fat simple polygon P with n vertices into O(n) fat convex quadrilaterals, and a method to cover (but not partition) a fat convex quadrilateral with O(1) fat triangles. The maximum overlap of the triangles at any point is two, which is optimal for any coveting of a fat simple polygon by a linear number of fat triangles.
How Hard Is Halfspace Range Searching?
, 1993
"... We investigate the complexity of halfspace range searching: Given n points in d space, build a data structure that allows us to determine efficiently how many points lie in a query halfspace. We establish a tradeoff between the storage m and the worstcase query time t in the Fredman/Yao arithmetic ..."
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Cited by 22 (0 self)
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We investigate the complexity of halfspace range searching: Given n points in d space, build a data structure that allows us to determine efficiently how many points lie in a query halfspace. We establish a tradeoff between the storage m and the worstcase query time t in the Fredman/Yao arithmetic model of computation. We show that t must be at least on the order of (n= log n) 1\Gamma d\Gamma1 d(d+1) =m 1=d : Although the bound is unlikely to be optimal, it falls reasonably close to the recent upper bound of O \Gamma n=m 1=d \Delta upper bound established by Matousek. We also show that it is possible to devise a sequence of n inserts and halfspace range queries that require a total time of n 2\GammaO(1=d) . Our results imply the first nontrivial lower bounds for spherical range searching in any fixed dimension. For example they show that, with linear storage, circular range queries in the plane require\Omega \Gamma n 1=3 \Delta time (modulo a logarithmic factor).
A Unified Approach to Dynamic Point Location, Ray Shooting, and Shortest Paths in Planar Maps
, 1992
"... We describe a new technique for dynamically maintaining the trapezoidal decomposition of a connected planar map M with 7 ~ vertices, and apply it to the development of a unified dynamic data structure that supports pointlocation, rayshooting, and shortestpath queries in M. The space requirement i ..."
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Cited by 20 (6 self)
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We describe a new technique for dynamically maintaining the trapezoidal decomposition of a connected planar map M with 7 ~ vertices, and apply it to the development of a unified dynamic data structure that supports pointlocation, rayshooting, and shortestpath queries in M. The space requirement is O(nlog n). Pointlocation queries take time O(log 7~). Rayshooting and shortestpath queries take time O(log3 TZ) (plus O(k) time if the k edges of the shortest path are reported in addition to its length). Updates consist of insertions and deletions of vertices and edges, and take O(log3 n) time (amortized for vertex updates).
Dynamic and I/OEfficient Algorithms for Computational Geometry and Graph Problems: Theoretical and Experimental Results
, 1995
"... As most important applications today are largescale in nature, highperformance methods are becoming indispensable. Two promising computational paradigms for largescale applications are dynamic and I/Oefficient computations. We give efficient dynamic data structures for several fundamental proble ..."
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Cited by 18 (4 self)
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As most important applications today are largescale in nature, highperformance methods are becoming indispensable. Two promising computational paradigms for largescale applications are dynamic and I/Oefficient computations. We give efficient dynamic data structures for several fundamental problems in computational geometry, including point location, ray shooting, shortest path, and minimumlink path. We also develop a collection of new techniques for designing and analyzing I/Oefficient algorithms for graph problems, and illustrate how these techniques can be applied to a wide variety of specific problems, including list ranking, Euler tour, expressiontree evaluation, leastcommon ancestors, connected and biconnected components, minimum spanning forest, ear decomposition, topological sorting, reachability, graph drawing, and visibility representation. Finally, we present an extensive experimental study comparing the practical I/O efficiency of four algorithms for the orthogonal s...
Trace Size vs Parallelism in TraceandReplay Debugging of SharedMemory Programs
 LANGUAGES AND COMPILERS FOR PARALLEL COMPUTING, LNCS
, 1993
"... Execution replay is a debugging strategy where a program is run over and over on an input that manifests bugs. For explicitly parallel sharedmemory programs, execution replay requires support of special tools  because these programs can be nondeterministic, their executions can differ from ru ..."
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Cited by 4 (0 self)
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Execution replay is a debugging strategy where a program is run over and over on an input that manifests bugs. For explicitly parallel sharedmemory programs, execution replay requires support of special tools  because these programs can be nondeterministic, their executions can differ from run to run on the same input. For such programs, executions must be traced before they can be replayed for debugging. We present improvements over our past work on an adaptive tracing strategy that records only a fraction of the execution'ssharedmemory references. Our past approach makes runtime tracing decisions by detecting and tracing exactly the nontransitive dynamic data dependences among the execution'sshared data. Tracing the nontransitive dependences provides sufficient information for a replay.Inthis paper we show that tracing exactly these dependences is not necessary.Instead, we present two algorithms that introduce and trace artificial dependences among some events that are actually independent. These artificial dependences reduce trace size, but introduce additional event orderings that can reduce the amount of parallelism achievable during replay.Wepresent one algorithm that always adds dependences guaranteed not to be on the critical path and thus do not slow replay.Another algorithm adds as many dependences as possible, slowing replay but reducing trace size further.Experiments show that we can improve the already high trace reduction of our past technique by up to two more orders of magnitude, without slowing replay.Our new techniques usually trace only 0.000250.2% of the sharedmemory references, a 36 order of magnitude reduction over past techniques which trace every access.
Trends and Developments in Computational Geometry
 COMPUTER GRAPHICS FORUM
, 1995
"... This report discusses some trends and achievements in computational geometry during the past five years, with emphasis on problems related to computer graphics. Furthermore, a direction of research in computational geometry is discussed, which could help in bringing the fields of computational ge ..."
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Cited by 2 (0 self)
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This report discusses some trends and achievements in computational geometry during the past five years, with emphasis on problems related to computer graphics. Furthermore, a direction of research in computational geometry is discussed, which could help in bringing the fields of computational geometry and computer graphics closer together.