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Topologically Sweeping Visibility Complexes via Pseudotriangulations
, 1996
"... This paper describes a new algorithm for constructing the set of free bitangents of a collection of n disjoint convex obstacles of constant complexity. The algorithm runs in time O(n log n + k), where k is the output size, and uses O(n) space. While earlier algorithms achieve the same optimal run ..."
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Cited by 85 (9 self)
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This paper describes a new algorithm for constructing the set of free bitangents of a collection of n disjoint convex obstacles of constant complexity. The algorithm runs in time O(n log n + k), where k is the output size, and uses O(n) space. While earlier algorithms achieve the same optimal running time, this is the first optimal algorithm that uses only linear space. The visibility graph or the visibility complex can be computed in the same time and space. The only complicated data structure used by the algorithm is a splittable queue, which can be implemented easily using redblack trees. The algorithm is conceptually very simple, and should therefore be easy to implement and quite fast in practice. The algorithm relies on greedy pseudotriangulations, which are subgraphs of the visibility graph with many nice combinatorial properties. These properties, and thus the correctness of the algorithm, are partially derived from properties of a certain partial order on the faces of th...
Computing the Visibility Graph via Pseudotriangulations
 In Proc. 11th Annu. ACM Sympos. Comput. Geom
, 1995
"... We show that the k free bitangents of a collection of n pairwise disjoint convex plane sets can be computed in time O(k+n log n) and O(n) working space. The algorithm uses only one advanced data structure, namely a splittable queue. We introduce (weakly) greedy pseudotriangulations, whose combinat ..."
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Cited by 31 (2 self)
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We show that the k free bitangents of a collection of n pairwise disjoint convex plane sets can be computed in time O(k+n log n) and O(n) working space. The algorithm uses only one advanced data structure, namely a splittable queue. We introduce (weakly) greedy pseudotriangulations, whose combinatorial properties are crucial for our method. 1 Introduction Consider a collection O of pairwise disjoint convex objects in the plane. We are interested in problems in which these objects arise as obstacles, either in connection with visibility problems where they can block the view from an other geometric object, or in motion planning, where these objects may prevent a moving object from moving along a straight line path. The visibility graph is a central object in such contexts. For polygonal obstacles the vertices of these polygons are the nodes of the visibility graph, and two nodes are connected by an arc if the corresponding vertices can see each other. [9] describes the first nontriv...
Hierarchical Motion Planning Using a Spatial Index
, 1996
"... We investigate the problem of constructing a shortest path of a pointlike robot between two configurations in the euclidean plane cluttered with (intersecting) convex polygonal obstacles. One common approach is to construct the visibility graph and search within this graph in a total time of O(n ..."
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Cited by 16 (0 self)
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We investigate the problem of constructing a shortest path of a pointlike robot between two configurations in the euclidean plane cluttered with (intersecting) convex polygonal obstacles. One common approach is to construct the visibility graph and search within this graph in a total time of O(n 2 ). We show that in general it is not necessary to construct the entire visibility graph. In contrast, we develop two hierarchical motionplanning techniques based on the monotonous bisector tree and the visibility graph, which are shown to be more efficient in scenes of low object density. We show that in our setting the visibility graph can be incrementally constructed in time O(n 2 log n), where n is the complexity of the scene. A shortest path can then be constructed in time O(l 4 log l+n log l), if a shortest path of length l exists. The dependency on n can be further reduced by the use of a spatial data structure complying a query assumption to O(l 4 log l + l 2 log n). If ...
Minimal Tangent Visibility Graphs
, 1995
"... We prove the tight lower bound 4n \Gamma 4 on the size of tangent visibility graphs on n pairwise disjoint bounded obstacles in the euclidean plane, and we give a simple description of the configurations of convex obstacles which realize this lower bound. 1980 Mathematics Subject Classification: 68U ..."
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We prove the tight lower bound 4n \Gamma 4 on the size of tangent visibility graphs on n pairwise disjoint bounded obstacles in the euclidean plane, and we give a simple description of the configurations of convex obstacles which realize this lower bound. 1980 Mathematics Subject Classification: 68U05,52A10,52C10,52B10,05C90. Key Words and Phrases: Visibility graphs, triangulations, pseudotriangles, pseudotriangulations, convex hulls, relative convex hulls, plane trees, maps, DavenportSchinzel sequences. To appear in the special issue of Computational Geometry: Theory and Applications devoted to the 6th Canadian Conference on Computational Geometry held in Saskatoon, August 1994. y D'epartement de Math'ematiques et d'Informatique, Ecole normale sup'erieure, ura 1327 du Cnrs, 45 rue d'Ulm 75230 Paris Cedex 05, France (pocchiola@dmi.ens.fr). This work was partially supported by PRC "Math'ematiques et Informatique". z Dept. of Math. and Comp. Sc, University of Groningen P.O.Box 8...