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135
A unified approach to dynamic point location, ray shooting, and shortest paths in planar maps
 SIAM Journal on Computing
, 1996
"... Abstract. We describe a new technique for dynamically maintaining the trapezoidal decomposition of a connected planar map dX/ [ with n vertices and apply it to the development of a unified dynamic data structure that supports pointlocation, rayshooting, and shortestpath queries in A4. The space re ..."
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Cited by 24 (8 self)
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Abstract. We describe a new technique for dynamically maintaining the trapezoidal decomposition of a connected planar map dX/ [ with n vertices and apply it to the development of a unified dynamic data structure that supports pointlocation, rayshooting, and shortestpath queries in A4. The space requirement is O(n log n). Pointlocation queries take time O(log n). Rayshooting and shortestpath queries take time O(log n) (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(log n) time (amortized for vertex updates). This is the first polylogtime dynamic data structure for shortestpath and rayshooting queries. It is also the first dynamic pointlocation data structure for connected planar maps that achieves optimal query time. Key words, point location, ray shooting, shortest path, computational geometry, dynamic algorithm
A New Approach to Subdivision Simplification
, 1995
"... The line simplification problem is an old and wellstudied problem in cartography. Although there are several algorithms to compute a simplification, there seem to be no algorithms that perform line simplification in the context of other geographical objects. This paper presents a nearly quadratic t ..."
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Cited by 24 (0 self)
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The line simplification problem is an old and wellstudied problem in cartography. Although there are several algorithms to compute a simplification, there seem to be no algorithms that perform line simplification in the context of other geographical objects. This paper presents a nearly quadratic time algorithm for the following line simplification problem: Given a polygonal line, a set of extra points, and a real ffl ? 0, compute a simplification that guarantees (i) a maximum error ffl, (ii) that the extra points remain on the same side of the simplified chain as of the original chain, and (iii) that the simplified chain has no selfintersections. The algorithm is applied as the main subroutine for subdivision simplification. 1 Introduction The line simplification problem is a wellstudied problem in various disciplines including geographic information systems [Buttenfield '85, Cromley '88, Douglas & Peucker '73, Hershberger & Snoeyink '92, Li & Openshaw '92, McMaster '87], digital...
Efficient Visibility Queries in Simple Polygons
"... We present a method of decomposing a simple polygon that allows the preprocessing of the polygon to efficiently answer visibility queries of various forms in an output sensitive manner. Using O(n3 log n) preprocessing time and O(n3) space, we can, given a query point q inside or outside an n verte ..."
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Cited by 24 (2 self)
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We present a method of decomposing a simple polygon that allows the preprocessing of the polygon to efficiently answer visibility queries of various forms in an output sensitive manner. Using O(n3 log n) preprocessing time and O(n3) space, we can, given a query point q inside or outside an n vertex polygon, recover the visibility polygon of q in O(log n + k) time, where k is the size of the visibility polygon, and recover the number of vertices visible from q in O(log n) time. The key notion
More Planar TwoCenter Algorithms
 Comput. Geom. Theory Appl
, 1997
"... This paper considers the planar Euclidean twocenter problem: given a planar npoint set S, find two congruent circular disks of the smallest radius covering S. The main result is a deterministic algorithm with running time O(n log 2 n log 2 log n), improving the previous O(n log 9 n) bound ..."
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Cited by 22 (2 self)
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This paper considers the planar Euclidean twocenter problem: given a planar npoint set S, find two congruent circular disks of the smallest radius covering S. The main result is a deterministic algorithm with running time O(n log 2 n log 2 log n), improving the previous O(n log 9 n) bound of Sharir and almost matching the randomized O(n log 2 n) bound of Eppstein. If a point in the intersection of the two disks is given, then we can solve the problem in O(n log n) time with high probability. Keywords: twocenter, randomization, parametric search 1 Introduction Consider the following "facility location" problem: given a set S of n "demand" points in IR d and a number p, find a set T of p "supply"points in IR d minimizing max s2S min t2T d(s; t), where d(s; t) denotes the Euclidean distance between s and t. Geometrically, the problem is equivalent to finding p congruent disks of the smallest radius covering S and is referred to as the (Euclidean) pcenter problem. Th...
Methods for Achieving Fast Query Times in Point Location Data Structures
, 1997
"... Given a collection S of n line segments in the plane, the planar point location problem is to construct a data structure that can efficiently determine for a given query point p the first segment(s) in S intersected by vertical rays emanating out from p. It is well known that linearspace data struc ..."
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Cited by 20 (1 self)
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Given a collection S of n line segments in the plane, the planar point location problem is to construct a data structure that can efficiently determine for a given query point p the first segment(s) in S intersected by vertical rays emanating out from p. It is well known that linearspace data structures can be constructed so as to achieve O(log n) query times. But applications, such as those common in geographic information systems, motivate a reexamination of this problem with the goal of improving query times further while also simplifying the methods needed to achieve such query times. In this paper we perform such a reexamination, focusing on the issues that arise in three different classes of pointlocation query sequences: ffl sequences that are reasonably uniform spatially and temporally (in which case the constant factors in the query times become critical), ffl sequences that are nonuniform spatially or temporally (in which case one desires data structures that adapt to s...
Exponential structures for efficient cacheoblivious algorithms
 In Proceedings of the 29th International Colloquium on Automata, Languages and Programming
, 2002
"... Abstract. We present cacheoblivious data structures based upon exponential structures. These data structures perform well on a hierarchical memory but do not depend on any parameters of the hierarchy, including the block sizes and number of blocks at each level. The problems we consider are searchi ..."
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Cited by 20 (3 self)
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Abstract. We present cacheoblivious data structures based upon exponential structures. These data structures perform well on a hierarchical memory but do not depend on any parameters of the hierarchy, including the block sizes and number of blocks at each level. The problems we consider are searching, partial persistence and planar point location. On a hierarchical memory where data is transferred in blocks of size B, some of the results we achieve are: – We give a linearspace data structure for dynamic searching that supports searches and updates in optimal O(log B N) worstcase I/Os, eliminating amortization from the result of Bender, Demaine, and FarachColton (FOCS ’00). We also consider finger searches and updates and batched searches. – We support partiallypersistent operations on an ordered set, namely, we allow searches in any previous version of the set and updates to the latest version of the set (an update creates a new version of the set). All operations take an optimal O(log B (m + N)) amortized I/Os, where N is the size of the version being searched/updated, and m is the number of versions. – We solve the planar point location problem in linear space, taking optimal O(log B N) I/Os for point location queries, where N is the number of line segments specifying the partition of the plane. The preprocessing requires O((N/B) log M/B N) I/Os, where M is the size of the ‘inner ’ memory. 1
I/OEfficient Dynamic Point Location in Monotone Planar Subdivisions (Extended Abstract)
"... We present an efficient externalmemory dynamic data structure for point location in monotone planar subdivisions. Our data structure uses O(N=B) disk blocks to store a monotone subdivision of size N, where B is the size of a disk block. It supports queries in O(log2B N) I/Os (worstcase) and upda ..."
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Cited by 20 (15 self)
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We present an efficient externalmemory dynamic data structure for point location in monotone planar subdivisions. Our data structure uses O(N=B) disk blocks to store a monotone subdivision of size N, where B is the size of a disk block. It supports queries in O(log2B N) I/Os (worstcase) and updates in O(log2B N) I/Os (amortized). We also
Randomized incremental construction of threedimensional convex hulls and planar Voronoi diagrams, and approximate range counting
 in Proceedings of the Seventeenth Annual ACMSIAM Symposium on Discrete Algorithms
, 2006
"... We present new algorithms for approximate range counting, where, for a specified ε> 0, we want to count the number of data points in a query range, up to relative error of ε. We first describe a general framework, adapted from Cohen [12], for this task, and then specialize it to two important instan ..."
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Cited by 19 (7 self)
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We present new algorithms for approximate range counting, where, for a specified ε> 0, we want to count the number of data points in a query range, up to relative error of ε. We first describe a general framework, adapted from Cohen [12], for this task, and then specialize it to two important instances of range counting: halfspaces in R 3 and disks in the plane. The technique reduces the approximate range counting problem to that of finding the minimum rank of a data object in the range, with respect to a random permutation of the input. A major technical step in our analysis, which we believe to be of independent interest, is a bound of O(n log n) on the expected complexity of the overlay of all the Voronoi faces that are generated during a randomized incremental construction of the Voronoi diagram of n points in the plane. The same bound holds for the expected complexity of the overlay of all the faces of the minimization diagram of the lower envelope of n planes in R 3, or for the expected complexity of the overlay of all the normal (or Gaussian) diagram faces of the convex hull of n points in R 3, that are generated during a randomized incremental construction of the lower envelope or of the hull, respectively. All these bounds are tight in the worst case.
TwoPoint Euclidean Shortest Path Queries in the Plane (Extended Abstract)
, 1999
"... ) To appear in Proc. Tenth Annual ACMSIAM Symposium on Discrete Algorithms (SODA '99), January 1719, 1999 YiJen Chiang Joseph S. B. Mitchell y Abstract We consider the twopoint query version of the fundamental geometric shortest path problem: Given a set h of polygonal obstacles in the pla ..."
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Cited by 18 (2 self)
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) To appear in Proc. Tenth Annual ACMSIAM Symposium on Discrete Algorithms (SODA '99), January 1719, 1999 YiJen Chiang Joseph S. B. Mitchell y Abstract We consider the twopoint query version of the fundamental geometric shortest path problem: Given a set h of polygonal obstacles in the plane, having a total of n vertices, build a data structure such that for any two query points s and t we can efficiently determine the length, d(s; t), of an Euclidean shortest obstacleavoiding path, ß(s; t), from s to t. Additionally, our data structure should allow one to report the path ß(s; t), in time proportional to its (combinatorial) size. We present various methods for solving this twopoint query problem, including algorithms with o(n), O(log n+h), O(h log n), O(log 2 n) or optimal O(log n) query times, using polynomialspace data structures, with various tradeoffs between space and query time. While several results have been known for approximate twopoint Euclidean shortest p...
Geometric And Computational Aspects Of Manufacturing Processes
 Comput. & Graphics
, 1994
"... Two of the fundamental questions that arise in the manufacturing industry concerning every type of manufacturing process are: 1. Given an object, can it be built using a particular process? 2. Given that an object can be built using a particular process, what is the best way to construct the objec ..."
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Cited by 18 (7 self)
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Two of the fundamental questions that arise in the manufacturing industry concerning every type of manufacturing process are: 1. Given an object, can it be built using a particular process? 2. Given that an object can be built using a particular process, what is the best way to construct the object? The latter question gives rise to many different problems depending on how best is qualified. We address these problems for two complimentary categories of manufacturing processes: rapid prototyping systems and casting processes. The method we use to address these problems is to first define a geometric model of the process in question and then answer the questions on that model. In the category of rapid prototyping systems, we concentrate on stereolithography, which is emerging as one of the most popular rapid prototyping systems. We model stereolithography geometrically and then study the class of objects that admit a construction in this model. For the objects that admit a constructio...