Proximity problems for points on a rectilinear plane with rectangular obstacles |
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Authors: | S Guha I Suzuki |
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Affiliation: | (1) Electrical Engineering and Computer Science Department, University of Wisconsin-Milwaukee, 53201 Milwaukee, WI, USA |
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Abstract: | We consider the following four problems for a setS ofk points on a plane, equipped with the rectilinear metric and containing a setR ofn disjoint rectangular obstacles (so that distance is measured by a shortest rectilinear path avoiding obstacles inR): (a) find aclosest pair of points inS, (b) find anearest neighbor for each point inS, (c) compute the rectilinearVoronoi diagram ofS, and (d) compute a rectilinearminimal spanning tree ofS. We describeO ((n+k) log(n+k))-time sequential algorithms for (a) and (b) based onplane-sweep, and the consideration of geometrically special types of shortest paths, so-calledz-first paths. For (c) we present anO ((n+k) log(n+k) logn)-time sequential algorithm that implements a sophisticateddivide-and-conquer scheme with an addedextension phase. In the extension phase of this scheme we introduce novel geometric structures, in particular so-calledz-diagrams, and techniques associated with the Voronoi diagram. Problem (d) can be reduced to (c) and solved inO ((n+k) log(n+k) logn) time as well. All our algorithms arenear-optimal, as well as easy to implement.
An extended abstract appeared inProc. 13th Conf. on the Foundations of Software Technology and Theoretical Computer Science, Bombay, 1993, Springer-Verlag, pp. 218–227. Sumanta Guha was supported in part by a UW-Milwaukee Graduate School Research
Committee Award. Ichiro Suzuki was supported in part by the National Science Foundation under Grants CCR-9004346 and IRI-9307506,
the Office of Naval Research under Grant N00014-94-1-0284, and an endowed chair supported by Hitachi Ltd. at the Faculty of
Engineering Science, Osaka University. |
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Keywords: | Computational geometry Rectilinear metric Obstacles Nearest neighbors Voronoi diagram Minimal spanning tree |
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