Abstract
We present a plane-sweep algorithm that solves the all — nearest — neighbors problem with respect to an arbitrary Minkowski-metric d t (1 ≤ t ≤ ∞) for a set of non-intersecting planar compact convex objects, such as points, line segments, circular arcs and convex polygons. The algorithm also applies if we replace the condition of disjointness by the weaker condition that the objects in the configuration are diagonal-disjoint. For configurations of points, line segments or disks the algorithm runs in asymptotically optimal tune O(n log n). For a configuration of n convex polygons with a total of N edges it finds nearest neighbors with respect to the Euclidean L 2-metric in time O(n log N) if each polygon is given by its vertices in cyclic order.
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F. Bartling, Th. Graf, K. Hinrichs: A plane sweep algorithm for finding a closest pair among convex planar objects, Preprints Angewandte Mathematik und Informatik, Universität Münster, Bericht Nr. 1/93-1.
F. Bartling, K. Hinrichs: A plane-sweep algorithm for finding a closest pair among convex planar objects, A. Finkel, M. Jantzen (eds.), STACS 92, 9th Annual Symposium on Theoretical Aspects of Computer Science, Lecture Notes in Computer Science 577, 221–232, Springer-Verlag, Berlin, 1992.
J.L. Bentley and T. Ottmann, Algorithms for reporting and counting intersections, IEEE Transactions on Computers C28, 643–647 (1979).
B. Chazelle, D. P. Dobkin: Intersection of convex objects in two and three dimensions, Journal of the ACM, 34(1), 1–27 (1987).
F. Chin, C. A. Wang: Optimal algorithms for the intersection and the minimum distance problems between planar polygons, IEEE Trans. Comput. 32(12), 1203–1207 (1983).
H. Edelsbrunner: Computing the extreme distances between two convex polygons, Journal of Algorithms 6, 213–224 (1985).
S. Fortune: A sweepline algorithm for Voronoi diagrams, Algorithmica 2, 153–174 (1987).
P.-O. Fjällström. J.Katajainen, J. Petersson: Algorithms for the all-nearest-neighbors problem, Report 92/2, Dept. of Computer Science, University of Copenhagen, Denmark, 1992.
K.Hinrichs, J.Nievergelt, P.Schorn: An all-round algorithm for 2-dimensional nearest-neighbor problems, Acta Informatics, 29(4), 383–394 (1992).
P. Schorn: Robust algorithms in a program library for geometric computation, PhD Dissertation No. 9519, ETH Zürich, Switzerland, 1991
M.Shamos, D.Hoey: Closest-point problems, Proceedings of the 16th Annual IEEE Symposium on Foundations of Computer Science, 151–162 (1975).
P.Vaidya: An O(n log n) algorithm for the all-nearest-neighbours-problem, Discrete & Computational Geometry, 4, 101–115 (1989).
C.K. Yap: An O(n log n) algorithm for the Voronoi diagram of a set of simple curve segments, Discrete Comput. Geometry 2, 365–393 (1987).
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© 1993 Springer-Verlag Berlin Heidelberg
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Graf, T., Hinrichs, K. (1993). A plane-sweep algorithm for the all-nearest-neighbors problem for a set of convex planar objects. In: Dehne, F., Sack, JR., Santoro, N., Whitesides, S. (eds) Algorithms and Data Structures. WADS 1993. Lecture Notes in Computer Science, vol 709. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-57155-8_261
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DOI: https://doi.org/10.1007/3-540-57155-8_261
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