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55
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 169 (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.
Voronoi Diagrams
 Handbook of Computational Geometry
"... Voronoi diagrams can also be thought of as lower envelopes, in the sense mentioned at the beginning of this subsection. Namely, for each point x not situated on a bisecting curve, the relation p x q defines a total ordering on S. If we construct a set of surfaces H p , p S,in3space such t ..."
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Cited by 143 (19 self)
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Voronoi diagrams can also be thought of as lower envelopes, in the sense mentioned at the beginning of this subsection. Namely, for each point x not situated on a bisecting curve, the relation p x q defines a total ordering on S. If we construct a set of surfaces H p , p S,in3space such that H p is below H q i# p x q holds, then the projection of their lower envelope equals the abstract Voronoi diagram.
Spanning Trees and Spanners
, 1996
"... We survey results in geometric network design theory, including algorithms for constructing minimum spanning trees and lowdilation graphs. 1 Introduction This survey covers topics in geometric network design theory. The problem is easy to state: connect a collection of sites by a "good" network. ..."
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Cited by 143 (2 self)
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We survey results in geometric network design theory, including algorithms for constructing minimum spanning trees and lowdilation graphs. 1 Introduction This survey covers topics in geometric network design theory. The problem is easy to state: connect a collection of sites by a "good" network. For instance, one may wish to connect components of a VLSI circuit by networks of wires, in a way that uses little surface area on the chip, draws little power, and propagates signals quickly. Similar problems come up in other applications such as telecommunications, road network design, and medical imaging [1]. One network design problem, the Traveling Salesman problem, is sufficiently important to have whole books devoted to it [79]. Problems involving some form of geometric minimum or maximum spanning tree also arise in the solution of other geometric problems such as clustering [12], mesh generation [56], and robot motion planning [93]. One can vary the network design problem in many w...
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 86 (1 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 40 (14 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 paper pr ..."
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Cited by 32 (12 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.
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 27 (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
Approximating Generalized Voronoi Diagrams in Any Dimension
, 1995
"... Generalized Voronoi diagrams of objects are difficult to compute in a robust way, especially in higher dimensions. For a number of applications an approximation of the real diagram within some predetermined precision is sufficient. In this paper we study the computation of such approximate Voronoi d ..."
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Cited by 23 (0 self)
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Generalized Voronoi diagrams of objects are difficult to compute in a robust way, especially in higher dimensions. For a number of applications an approximation of the real diagram within some predetermined precision is sufficient. In this paper we study the computation of such approximate Voronoi diagrams. The emphasis is on practical applicability, therefore we are mainly concerned with fast (in terms of running time) computation, generality, robustness, and easy implementation, rather than optimal combinatorial and computational complexity. Given a set of disjoint convex sites in any dimension we describe a general algorithm that approximates their Voronoi diagram with arbitrary precision; the only primitive operation that is required is the computation of the distance from a point to a site. The method is illustrated by its application to motion planning using retraction. To justify our claims on practical applicability, we provide experimental results obtained with implementations...
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 21 (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
Finding the Largest Area AxisParallel Rectangle in a Polygon
 Fifth Canadian Conference on Computational Geometry
, 1997
"... This paper considers the geometric optimization problem of finding the Largest area axisparallel Rectangle (LR) in an nvertex general polygon. We characterize the LR for general polygons by considering different cases based on the types of contacts between the rectangle and the polygon. A general ..."
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Cited by 15 (0 self)
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This paper considers the geometric optimization problem of finding the Largest area axisparallel Rectangle (LR) in an nvertex general polygon. We characterize the LR for general polygons by considering different cases based on the types of contacts between the rectangle and the polygon. A general framework is presented for solving a key subproblem of the LR problem which dominates the running time for a variety of polygon types. This framework permits us to transform an algorithm for orthogonal polygons into an algorithm for nonorthogonal polygons. Using this framework, we show that the LR in a general polygon (allowing holes) can be found in O(n log 2 n) time. This matches the running time of the best known algorithm for orthogonal polygons. References are given for the application of the framework to other types of polygons. For each type, the running time of the resulting algorithm matches the running time of the best known algorithm for orthogonal polygons of that type. A lower...