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Constrained Delaunay triangulations
 Algorithmica
, 1989
"... Given a set of n vertices in the plane together with a set of noncrossing edges, the constrained Delaunay triangulation (CDT) is the triangulation of the vertices with the following properties: (1) the prespecified edges are included in the triangulation, and (2) it is as close as possible to the De ..."
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Cited by 217 (5 self)
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Given a set of n vertices in the plane together with a set of noncrossing edges, the constrained Delaunay triangulation (CDT) is the triangulation of the vertices with the following properties: (1) the prespecified edges are included in the triangulation, and (2) it is as close as possible to the Delaunay triangulation. We show that the CDT can be built in optimal O(n log n) time using a divideandconquer technique. This matches the time required to build an arbitrary (unconstrained) Delaunay triangulation and the time required to build an arbitrary constrained (nonDelaunay) triangulation. CDTs, because of their relationship with Delaunay triangulations, have a number of properties that should make them useful for the finiteelement method. Applications also include motion planning in the presence of polygonal obstacles in the plane and constrained Euclidean minimum spanning trees, spanning trees subject to the restriction that some edges are prespecified. I’wnishi0tt to copy without tix all or part of thk material is granlcd provided thal IIIC wpics arc not nude or distributed li)r direct commercial advanlagc, the ACM copyright wficc and the title of lhc publication and its date appear. and notice is given that copying is hy permission ol the Association Car Computing Machinery. ‘To copy otherwise. or to republish. requires a fee and/or specific permission.
Spanning Trees and Spanners
, 1996
"... We survey results in geometric network design theory, including algorithms for constructing minimum spanning trees and lowdilation graphs. ..."
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Cited by 149 (2 self)
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We survey results in geometric network design theory, including algorithms for constructing minimum spanning trees and lowdilation graphs.
An Optimal Algorithm for Euclidean Shortest Paths in the Plane
 SIAM J. Comput
, 1997
"... We propose an optimaltime algorithm for a classical problem in plane computational geometry: computing a shortest path between two points in the presence of polygonal obstacles. Our algorithm runs in worstcase time O(n log n) and requires O(n log n) space, where n is the total number of vertice ..."
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Cited by 116 (2 self)
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We propose an optimaltime algorithm for a classical problem in plane computational geometry: computing a shortest path between two points in the presence of polygonal obstacles. Our algorithm runs in worstcase time O(n log n) and requires O(n log n) space, where n is the total number of vertices in the obstacle polygons. The algorithm is based on an efficient implementation of wavefront propagation among polygonal obstacles, and it actually computes a planar map encoding shortest paths from a fixed source point to all other points of the plane; the map can be used to answer singlesource shortest path queries in O(logn) time. The time complexity of our algorithm is a significant improvement over all previously published results on the shortest path problem. Finally, we also discuss extensions to more general shortest path problems, involving nonpoint and multiple sources. 1 Introduction 1.1 The Background and Our Result The Euclidean shortest path problem is one of the o...
Planar spanners and approximate shortest path queries among obstacles
 in the plane, Proc. 4th European Sympos. Algorithms
, 1996
"... Abstract. We consider the problem of finding an obstacleavoiding path between two points s and t in the plane, amidst a set of disjoint polygonal obstacles with a total of n vertices. The length of this path should be within a small constant factor c of the length of the shortest possible obstacle ..."
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Cited by 49 (15 self)
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Abstract. We consider the problem of finding an obstacleavoiding path between two points s and t in the plane, amidst a set of disjoint polygonal obstacles with a total of n vertices. The length of this path should be within a small constant factor c of the length of the shortest possible obstacleavoiding st path measured in the Lvmetric. Such an approximate shortest path is called a cshort path, or a short path with stretch]actor c. The goal is to preprocess the obstaclescattered plane by creating an efficient data structure that enables fast reporting of a cshort path (or its length). In this paper, we give a family of algorithms for the above problem that achieve an interesting tradeoff between the stretch factor, the query time and the preprocessing bounds. Our main results are algorithms that achieve logarithmic length query time, after subquadratic time and space preprocessing. 1
RAY SHOOTING AND OTHER APPLICATIONS OF SPANNING TREES WITH LOW STABBING NUMBER
, 1992
"... This paper considers the following problem: Given a set G of n (possibly intersecting) line segments in the plane, prcproccss it so that, given a query ray p emanating from a point p, one can quickly compute the intersection point &(G, p) of p with a segment of G that lies nearest to p. The pape ..."
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Cited by 31 (11 self)
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This paper considers the following problem: Given a set G of n (possibly intersecting) line segments in the plane, prcproccss it so that, given a query ray p emanating from a point p, one can quickly compute the intersection point &(G, p) of p with a segment of G that lies nearest to p. The paper presents an algorithm that preproccsses G, in time 0 ( 3/2 log n), into a data structure of size O(nc(n) log4 n), so that for a query ray p, /,(, p) can be computed in time O(v/nc(ni log2 n), where w is a constant < 4.33 and a(n) is a functional inverse of Ackermann’s function. If the given segments are nonintersecting, the storage goes down to O(n log3 n) and the query time becomes O(v/ log2 n). The main tool used is spanning trees (on the set of segment endpoints) with low stabbing number, i.e., with the property that no line intersects more than O(x/) edges of the tree. Such trees make it possible to obtain faster algorithms for several other problems, including implicit point location, polygon containment, and implicit hidden surface removal.
Connections between thetagraphs, Delaunay triangulations, and orthogonal surfaces
 In Proceedings of the 36th International Conference on Graph Theoretic Concepts in Computer Science (WG 2010
, 2010
"... Abstract. Θkgraphs are geometric graphs that appear in the context of graph navigation. The shortestpath metric of these graphs is known to approximate the Euclidean complete graph up to a factor depending on the cone number k and the dimension of the space. TDDelaunay graphs, a.k.a. triangulard ..."
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Cited by 29 (2 self)
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Abstract. Θkgraphs are geometric graphs that appear in the context of graph navigation. The shortestpath metric of these graphs is known to approximate the Euclidean complete graph up to a factor depending on the cone number k and the dimension of the space. TDDelaunay graphs, a.k.a. triangulardistance Delaunay triangulations, introduced by Chew, have been shown to be plane 2spanners of the 2D Euclidean complete graph, i.e., the distance in the TDDelaunay graph between any two points is no more than twice the distance in the plane. Orthogonal surfaces are geometric objects defined from independent sets of points of the Euclidean space. Orthogonal surfaces are well studied in combinatorics (orders, integer programming) and in algebra. From orthogonal surfaces, geometric graphs, called geodesic embeddings can be built. In this paper, we introduce a specific subgraph of the Θ6graph defined in the 2D Euclidean space, namely the halfΘ6graph, composed of the evencone edges of the Θ6graph. Our main contribution is to show that these graphs are exactly the TDDelaunay graphs, and are strongly connected to the geodesic embeddings of orthogonal surfaces of coplanar points in the 3D Euclidean space. Using these new bridges between these three fields, we establish: – Every Θ6graph is the union of two spanning TDDelaunay graphs. In particular, Θ6graphs are 2spanners of the Euclidean graph, and the bound of 2 on the stretch factor is the best possible. It was not known that Θ6graphs are tspanners for some constant t, and Θ7graphs were only known to be tspanners for t ≈ 7.562. – Every plane triangulation is TDDelaunay realizable, i.e., every combinatorial plane graph for which all its interior faces are triangles is the TDDelaunay graph of some point set in the plane. Such realizability property does not hold for classical Delaunay triangulations.
On the Construction of Abstract Voronoi Diagrams
 DISCRETE COMPUT GEOM 6:211224 (1991)
, 1991
"... We show that the abstract Voronoi diagram of n sites in the plane can be constructed in time O(n log n) by a randomized algorithm. This yields an alternative, but simpler, O(n log n) algorithm in many previously considered cases and the first O(n log n) algorithm in some cases, e.g., disjoint conv ..."
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Cited by 25 (2 self)
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We show that the abstract Voronoi diagram of n sites in the plane can be constructed in time O(n log n) by a randomized algorithm. This yields an alternative, but simpler, O(n log n) algorithm in many previously considered cases and the first O(n log n) algorithm in some cases, e.g., disjoint convex sites with the Euclidean distance function. Abstract Voronoi diagrams are given by a family of bisecting curves and were recently introduced by Klein [13]. Our algorithm is based on Clarkson and Shor's randomized incremental construction technique [7].
A Practical Evaluation of Kinetic Data Structures
 In Proc. 13th Annu. ACM Sympos. Comput. Geom
, 1997
"... this paper is to study and validate the use of the kinetic data structures proposed in [BGH97] in practice. The implementation and experimental evaluation of such structures brings to light several important issues which were not addressed in the original paper. For instance, KDSs implement the cont ..."
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Cited by 25 (6 self)
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this paper is to study and validate the use of the kinetic data structures proposed in [BGH97] in practice. The implementation and experimental evaluation of such structures brings to light several important issues which were not addressed in the original paper. For instance, KDSs implement the continuous maintenance of the function of interest through a discreteevent simulation. The calculation of the event times poses numerical difficulties which raise questions about the integrity of the structure if nearby events should happen out of order, or if the time of an event changes when recalculated. Also, the algorithmic measures for evaluating the quality of KDSs in [BGH97] are all worstcase measures, which may not reflect behavior under ordinary motions which occur in practice. It may well be that simple data structures do better in such situations. In order to study these issues, we have compared the KDSs of [BGH97] with simpler variants under various common data/motion distributions, each meant to generate a different amount of
A compact piecewiselinear Voronoi diagram for convex sites in the plane
 Discrete Comput. Geom
, 1996
"... In the plane, the postofice problem, which asks for the closest site to a query site, and retraction motion planning, which asks for a onedimensional retract of the free space of a robot, are both classtcally solved by computing a Voronoi diagram. When the sites are k disjoint convex sets, we giv ..."
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Cited by 25 (4 self)
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In the plane, the postofice problem, which asks for the closest site to a query site, and retraction motion planning, which asks for a onedimensional retract of the free space of a robot, are both classtcally solved by computing a Voronoi diagram. When the sites are k disjoint convex sets, we give a compact representation of the Voronoi diagram, using O(k) line segments, that is suficient for logarithmic time postofice location queries and motion planning. If these sets are polygons with n total vertices, we compute this diagram optimally in O ( k log n) deterministic time for the Euclidean metric and in O(k logn logm) deterministic time for the convex distance function defined by a convex mgon. 1