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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 21 (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...
Lineartime reconstruction of Delaunay triangulations with applications
 In Proc. Annu. European Sympos. Algorithms, number 1284 in Lecture Notes Comput. Sci
, 1997
"... Many of the computational geometers' favorite data structures are planar graphs, canonically determined by a set of geometric data, that take \Theta(n log n) time to compute. Examples include 2d Delaunay triangulation, trapezoidations of segments, and constrained Voronoi diagrams, and 3d convex hu ..."
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Cited by 20 (3 self)
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Many of the computational geometers' favorite data structures are planar graphs, canonically determined by a set of geometric data, that take \Theta(n log n) time to compute. Examples include 2d Delaunay triangulation, trapezoidations of segments, and constrained Voronoi diagrams, and 3d convex hulls. Given such a structure, one can determine a permutation of the data in O(n) time such that the data structure can be reconstructed from the permuted data in O(n) time by a simple incremental algorithm. As a consequence, one can permute a data file to "hide" a geometric structure, such as a terrian model based on the Delaunay triangulation of a set of sampled points, without disrupting other applications. One can even include "importance" in the ordering so the incremental reconstruction produces approximate terrain models as the data is read or received. For the Delaunay triangulation, we can also handle input in degenerate position, even though the data structures may no longer be cano...
Two and ThreeDimensional Point Location in Rectangular Subdivisions
 Journal of Algorithms
, 1995
"... We apply van Emde Boastype stratified trees to point location problems in rectangular subdivisions in 2 and 3 dimensions. In a subdivision with n rectangles having integer coordinates from [0; U \Gamma 1], we locate an integer query point in O((log log U ) d ) query time using O(n) space when d ..."
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Cited by 20 (1 self)
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We apply van Emde Boastype stratified trees to point location problems in rectangular subdivisions in 2 and 3 dimensions. In a subdivision with n rectangles having integer coordinates from [0; U \Gamma 1], we locate an integer query point in O((log log U ) d ) query time using O(n) space when d 2 or O(n log log U ) space when d = 3. Applications and extensions of this "fixed universe" approach include spatial point location using logarithmic time and linear space in rectilinear subdivisions having arbitrary coordinates, point location in coriented polygons or fat triangles in the plane, point location in subdivisions of space into "fat prisms," and vertical ray shooting among horizontal "fat objects." Like other results on stratified trees, our algorithms run on a RAM model and make use of perfect hashing. 1 Introduction The point location problemwhich seeks to preprocess a set of disjoint geometric objects to be able to determine quickly which object contains a query point...
An Experimental Analysis of Change Propagation in Dynamic Trees
, 2005
"... Change propagation is a technique for automatically adjusting the output of an algorithm to changes in the input. The idea behind change propagation is to track the dependences between data and function calls, so that, when the input changes, functions affected by that change can be reexecuted to u ..."
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Cited by 20 (12 self)
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Change propagation is a technique for automatically adjusting the output of an algorithm to changes in the input. The idea behind change propagation is to track the dependences between data and function calls, so that, when the input changes, functions affected by that change can be reexecuted to update the computation and the output. Change propagation makes it possible for a compiler to dynamize static algorithms. The practical effectiveness of change propagation, however, is not known. In particular, the cost of dependence tracking and change propagation may seem significant. The contributions of the paper are twofold. First, we present some experimental evidence that change propagation performs well when compared to direct implementations of dynamic algorithms. We implement change propagation on treecontraction as a solution to the dynamic trees problem and present an experimental evaluation of the approach. As a second contribution, we present a library for dynamictrees that support a general interface and present an experimental evaluation by considering a broad set of applications. The dynamictrees library relies on change propagation to handle edge insertions/deletions. The applications that we consider include path queries, subtree queries, leastcommonancestor queries, maintenance of centers and medians of trees, nearestmarkedvertex queries, semidynamic minimum spanning trees, and the maxflow algorithm of Sleator and Tarjan.
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...
Improved Algorithms for Dynamic Shortest Paths
, 2000
"... We describe algorithms for finding shortest paths and distances in outerplanar and planar digraphs that exploit the particular topology of the input graph. An important feature of our algorithms is that they can work in a dynamic environment, where the cost of any edge can be changed or the edge ca ..."
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Cited by 15 (3 self)
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We describe algorithms for finding shortest paths and distances in outerplanar and planar digraphs that exploit the particular topology of the input graph. An important feature of our algorithms is that they can work in a dynamic environment, where the cost of any edge can be changed or the edge can be deleted. In the case of outerplanar digraphs, our data structures can be updated after any such change in only logarithmic time. A distance query is also answered in logarithmic time. In the case of planar digraphs, we give an interesting tradeoff between preprocessing, query, and update times depending on the value of a certain topological parameter of the graph. Our results can be extended to nvertex digraphs of genus O(n1−ε) for any ε>0.
Dynamization of the Trapezoid Method for Planar Point Location
, 1991
"... We present a fully dynamic data structure for point location in a monotone subdivision, based on the trapezoid method. The operations supported are insertion and deletion of vertices and edges, and horizontal translation of vertices. Let n be the current number of vertices of the subdivision. Point ..."
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Cited by 14 (4 self)
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We present a fully dynamic data structure for point location in a monotone subdivision, based on the trapezoid method. The operations supported are insertion and deletion of vertices and edges, and horizontal translation of vertices. Let n be the current number of vertices of the subdivision. Point location queries take O(log n) time, while updates take O(log2 n) time. The space requirement is O(n log n). This is the first fully dynamic point location data structure for monotone subdivisions that achieves optimal query time.
Average case analysis of dynamic geometric optimization
, 1995
"... We maintain the maximum spanning tree of a planar point set, as points are inserted or deleted, in O(log³ n) expected time per update in Mulmuley’s averagecase model of dynamic geometric computation. We use as subroutines dynamic algorithms for two other geometric graphs: the farthest neighbor fore ..."
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Cited by 13 (3 self)
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We maintain the maximum spanning tree of a planar point set, as points are inserted or deleted, in O(log³ n) expected time per update in Mulmuley’s averagecase model of dynamic geometric computation. We use as subroutines dynamic algorithms for two other geometric graphs: the farthest neighbor forest and the rotating caliper graph related to an algorithm for static computation of point set widths and diameters. We maintain the former graph in expected time O(log² n) per update and the latter in expected time O(log n) per update. We also use the rotating caliper graph to maintain the diameter, width, and minimum enclosing rectangle of a point set in expected time O(log n) per update. A subproblem uses a technique for averagecase orthogonal range search that may also be of interest.
Simplification Culling of Static and Dynamic Scene Graphs
, 1998
"... We present a new approach for simplifying large polygonal environments composed of hundreds or thousands of objects. Our algorithm represents the environment using a scene graph and automatically computes levels of detail (LOD) for each node in the graph. For drastic simplification, the algorithm us ..."
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Cited by 11 (5 self)
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We present a new approach for simplifying large polygonal environments composed of hundreds or thousands of objects. Our algorithm represents the environment using a scene graph and automatically computes levels of detail (LOD) for each node in the graph. For drastic simplification, the algorithm uses hierarchical levels of detail (HLOD) to represent the simplified geometry of whole portions of the scene graph. When HLOD are rendered, the algorithm can ignore these portions, thereby performing simplification culling. For dynamic environments, HLOD are incrementally computed on the fly. The algorithm is applicable to all models and involves no user intervention. It generates high quality and drastic simplifications and has been applied to CAD models composed of hundreds of thousands of polygons. In practice, it achieves significant speedups in rendering large static and dynamic environments with little loss in image quality.