Abstract
We consider the following four problems for a set S of k points on a plane, equipped with the rectilinear metric and containing a set R of n disjoint rectangular obstacles (so that distance is measured by a shortest rectilinear path avoiding obstacles in R): (a) find a closest pair of points in S, (b) find a nearest neighbor for each point in S, (c) compute the rectilinear Voronoi diagram of S, and (d) compute a rectilinear minimal spanning tree of S. We describe O((n+k)log(n+k)) time sequential algorithms for (a) and (b) based on plane-sweep, and the consideration of geometrically special types of shortest paths, so-called z-first paths. For (c) we present an O((n+k)log(n+k) log n) time sequential algorithm that implements a sophisticated divide-and-conquer scheme with an added extension phase. In the extension phase of this scheme we introduce novel geometric structures, in particular so-called z-diagrams, and techniques associated with the Voronoi diagram. Problem (d) can be reduced to (c) and solved in O((n+k) log(n+k) log n) time as well. All our algorithms are near-optimal, as well as easy to implement.
Supported in part by a UW-Milwaukee Graduate School Research Committee Award.
Supported in part by the National Science Foundation under grants CCR-9004346 and IRI-9307506.
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References
B. Aronov, On the geodesic Voronoi diagram of point sites in a simple polygon, Algorithmica 4 (1989), 109–140.
M. J. Atallah, D. Chen, Parallel rectilinear shortest paths with rectangular obstacles, Proc. ACM Symp. on Parallel Algorithms and Architectures, 1990, 270–279; Computational Geometry: Theory and Applications 1 (1991), 79–113.
J. L. Bentley, M. I. Shamos, Divide-and-conquer in multidimensional space, Proc. ACM Symp. on Theory of Computing, 1976, 220–230.
P. J. de Rezende, D. T. Lee, Y. F. Wu, Rectilinear shortest paths with rectangular barriers, Proc. ACM Symp. on Computational Geometry, 1985, 204–213; Discrete and Computational Geometry 4 (1989), 41–53.
D. R. Fowler, J. M. Keil, The rectilinear Voronoi diagram with barriers, Proc. Allerton Conf. on Communication, Control, and Computing, 1987, 889–897.
M. I. Shamos, D. Hoey, Closest-point problems, Proc. IEEE Symp. on Foundations of Computer Science, 1975, 151–162.
Y. F. Wu, P. Widmayer, M. D. F. Schlag, C. K. Wong, Rectilinear shortest paths and minimum spanning trees in the presence of rectilinear obstacles, IEEE Trans. Computers C-36 (1987), 321–331.
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© 1993 Springer-Verlag Berlin Heidelberg
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Guha, S., Suzuki, I. (1993). Proximity problems and the Voronoi diagram on a rectilinear plane with rectangular obstacles. In: Shyamasundar, R.K. (eds) Foundations of Software Technology and Theoretical Computer Science. FSTTCS 1993. Lecture Notes in Computer Science, vol 761. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-57529-4_55
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DOI: https://doi.org/10.1007/3-540-57529-4_55
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