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13
Dynamic Trees and Dynamic Point Location
 In Proc. 23rd Annu. ACM Sympos. Theory Comput
, 1991
"... This paper describes new methods for maintaining a pointlocation data structure for a dynamicallychanging monotone subdivision S. The main approach is based on the maintenance of two interlaced spanning trees, one for S and one for the graphtheoretic planar dual of S. Queries are answered by using ..."
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This paper describes new methods for maintaining a pointlocation data structure for a dynamicallychanging monotone subdivision S. The main approach is based on the maintenance of two interlaced spanning trees, one for S and one for the graphtheoretic planar dual of S. Queries are answered by using a centroid decomposition of the dual tree to drive searches in the primal tree. These trees are maintained via the linkcut trees structure of Sleator and Tarjan, leading to a scheme that achieves vertex insertion/deletion in O(log n) time, insertion/deletion of kedge monotone chains in O(log n + k) time, and answers queries in O(log 2 n) time, with O(n) space, where n is the current size of subdivision S. The techniques described also allow for the dual operations expand and contract to be implemented in O(log n) time, leading to an improved method for spatial pointlocation in a 3dimensional convex subdivision. In addition, the interlacedtree approach is applied to online pointlo...
Computational Geometry
 in optimization 2.5D and 3D NC surface machining. Computers in Industry
, 1996
"... Introduction Computational geometry evolves from the classical discipline of design and analysis of algorithms, and has received a great deal of attention in the last two decades since its inception in 1975 by M. Shamos[108]. It is concerned with the computational complexity of geometric problems t ..."
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Cited by 12 (0 self)
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Introduction Computational geometry evolves from the classical discipline of design and analysis of algorithms, and has received a great deal of attention in the last two decades since its inception in 1975 by M. Shamos[108]. It is concerned with the computational complexity of geometric problems that arise in various disciplines such as pattern recognition, computer graphics, computer vision, robotics, VLSI layout, operations research, statistics, etc. In contrast with the classical approach to proving mathematical theorems about geometryrelated problems, this discipline emphasizes the computational aspect of these problems and attempts to exploit the underlying geometric properties possible, e.g., the metric space, to derive efficient algorithmic solutions. The classical theorem, for instance, that a set S is convex if and only if for any 0 ff 1 the convex combination ffp + (1 \Gamma<F
Randomized FullyScalable BSP Techniques for MultiSearching and Convex Hull Construction
 In ACMSIAM Symposium on Discrete Algorithms
, 1997
"... We study randomized techniques for designing efficient algorithms on a pprocessor bulksynchronous parallel (BSP) computer, which is a parallel multicomputer that allows for general processortoprocessor communication rounds provided each processor is guaranteed to send and receive at most h items ..."
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Cited by 10 (2 self)
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We study randomized techniques for designing efficient algorithms on a pprocessor bulksynchronous parallel (BSP) computer, which is a parallel multicomputer that allows for general processortoprocessor communication rounds provided each processor is guaranteed to send and receive at most h items in any round. The measure of efficiency we use is in terms of the internal computation time of the processors and the number of communication rounds needed to solve the problem at hand. We present techniques that achieve optimal efficiency in these bounds over all possible values for p, and we call such techniques fullyscalable for this reason. In particular, we address two fundamental problems: multisearching and convex hull construction. Our methods result in algorithms that use internal time that is O( n log n p ) and, for h = \Theta(n=p), a number of communication rounds that is O( log n log(h+1) ), with high probability. Both of these bounds are asymptotically optimal for the BSP ...
Parallelizing An Algorithm For Visibility On Polyhedral Terrain
, 1995
"... The best known outputsensitive sequential algorithm for computing the viewshed on a polyhedral terrain from a given viewpoint was proposed by Katz, Overmars, and Sharir 10 , and achieves time complexity O((k + nff(n)) log n) where n and k are the input and output sizes respectively, and ff() is t ..."
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Cited by 4 (0 self)
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The best known outputsensitive sequential algorithm for computing the viewshed on a polyhedral terrain from a given viewpoint was proposed by Katz, Overmars, and Sharir 10 , and achieves time complexity O((k + nff(n)) log n) where n and k are the input and output sizes respectively, and ff() is the inverse Ackermann's function. In this paper, we present a parallel algorithm that is based on the work mentioned above, and achieves O(log 2 n) time complexity, with work complexity O((k + nff(n)) log n) in a CREW PRAM model. This improves on previous parallel complexity while maintaining work efficiency with respect to the best sequential complexity known. Keywords: Terrain visibility, parallel algorithms, CREW PRAM, profile tree 1. Introduction Visibility analysis on terrain is a fundamental problem in computer graphics, navigation, and engineering applications. Several aspects of this problem have been Permanent address: Jet Propulsion Lab, 4800 Oak Grove Dr. MS 168522, Pasade...
PARALLEL ALGORITHMS IN GEOMETRY
"... The goal of parallel algorithm design is to develop parallel computational methods that run very fast with as few processors as possible, and there is an extensive literature of such algorithms for computational geometry problems. There are several different parallel computing models, and in order t ..."
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Cited by 3 (0 self)
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The goal of parallel algorithm design is to develop parallel computational methods that run very fast with as few processors as possible, and there is an extensive literature of such algorithms for computational geometry problems. There are several different parallel computing models, and in order to maintain a focus in this
Parallel Algorithms for Batched Range Searching on CoarseGrained Multicomputers
, 1997
"... We define the batched rangesearching problem as follows: given a set S of n points and a set Q of m hyperrectangles, report for each hyperrectangle which points it contains. This problem has applications in, for example, computeraided design and engineering. We present several parallel algorithms ..."
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We define the batched rangesearching problem as follows: given a set S of n points and a set Q of m hyperrectangles, report for each hyperrectangle which points it contains. This problem has applications in, for example, computeraided design and engineering. We present several parallel algorithms for this problem on coarsegrained multicomputers. Our algorithms are based on wellknown average and worstcase efficient sequential algorithms. One of our algorithms solves the ddimensional batched rangesearching problem in O(T s (n log d\Gamma1 p; p)+T s (m log d\Gamma1 p; p)+ ((m + n) log d\Gamma1 (n=p) + m log d\Gamma1 p log(n=p) + k)=p) time on a p  processor coarsegrained multicomputer. (T s (n; p) denotes the time globally to sort n numbers on a p processor multicomputer, and k is the total number of reported points.)
Parallel Algorithms for Searching Monotone Matrices on Coarse Grained Multicomputers
"... This work has been submitted for publication elsewhere. Copyright may then be transferred, and the present version of the article may be superseded by a revised one. The WWW page at the URL stated below will contain uptodate information about the current version and copyright status of this articl ..."
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This work has been submitted for publication elsewhere. Copyright may then be transferred, and the present version of the article may be superseded by a revised one. The WWW page at the URL stated below will contain uptodate information about the current version and copyright status of this article. Additional copyright information is found on the next page of this document.
Computational Geometry II
"... Introduction This is a follow up on the previous Chapter dealing with geometric problems and their efficient solutions. The classes of problems that we address in this Chapter include proximity, optimization, intersection, searching, point location, and some discussions of geometric software that i ..."
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Introduction This is a follow up on the previous Chapter dealing with geometric problems and their efficient solutions. The classes of problems that we address in this Chapter include proximity, optimization, intersection, searching, point location, and some discussions of geometric software that is under development. 2 Proximity Geometric problems pertaining to the questions of how close two geometric entities are among a collection of objects or how similar two geometric patterns match each other abound. For example, in pattern classification and clustering, features that are similar according to some metric, are to be clustered in a group. The two aircrafts that are closest at any time instant in the air space will have the largest likelihood of collision with each other. In some cases one may be interested in how far apart or how dissimilar the objects are. Some of