Abstract
We present an efficient algorithm for solving the “smallest k-enclosing circle” (kSC) problem: Given a set of n points in the plane and an integer k ≤ n, find the smallest disk containing k of the points. We resent several algorithms that run in O(nk logc n) time, where the constant c depends on the storage that the algorithm is allowed. When using O(nk) storage, the problem can be solved in time O(nk log2 n). When only O(n log n) storage is allowed, the running time is O(nk log2 n log n/k.). The method we describe can be easily extended to obtain efficient solutions of several related problems (with similar time and storage bounds). These related problems include: finding the smallest homothetic copy of a given convex polygon P, which contains k points from a given planar set, and finding the smallest hypodrome of a given length and orientation (formally defined in Section 4) containing k points from a given planar set.
Work on this paper by the second author has been supported by NSF Grant CCR-91-22103, and by grants from the U.S.-Israeli Binational Science Foundation, the G.I.F., the German-Israeli Foundation for Scientific Research and Development, and the Fund for Basic Research administered by the Israeli Academy of Sciences.
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References
P. Agarwal, M. Sharir, and S. Toledo. New applications of parametric searching in computational geometry, to appear in J. Algorithms, 1993.
P. K. Agarwal and J. Matoušek. Dynamic half-space range reporting and its applications, manuscript, 1992.
A. Aggarwal, H. Imai, N. Katoh, and S. Suri. Finding k points with minimum diameter and related problems. In Proc. 5th Annu. ACM Sympos. Comput. Geom., pages 283–291, 1989.
M. Ajtai, J. Komlós, and E. Szemerédi. Sorting in c log n parallel steps. Combinatorica, 3:1–19, 1983.
L. P. Chew and K. Kedem. Improvements on geometric pattern matching problems. In Proc. 3rd Scand. Workshop Algorithm Theory, volume 621 of Lecture Notes in Computer Science, pages 318–325. Springer-Verlag, 1992.
R. Cole. Slowing down sorting networks to obtain faster sorting algorithms. J. ACM 31:200–208, 1984.
H. Edelsbrunner, L. J. Guibas, and J. Stolfi. Optimal point location in a monotone subdivision. SIAM J. Comput., 15:317–340, 1986.
D. Eppstein. New algorithms for minimum area k-gons. In Proc. 3rd ACM-SIAM Sympos. Discrete Algorithms, pages 83–88, 1992.
T. C. Kao and D. M. Mount. An algorithm for computing compacted Voronoi diagrams defined by convex distance functions. In Proc. 3rd Canad. Conf. Comput. Geom., pages 104–109, 1991.
K. Kedem, R. Livne, J. Pach, and M. Sharir. On the union of Jordan regions and collision-free translational motion amidst polygonal obstacles. Discrete Comput. Geom., 1:59–71, 1986.
H.-P. Lenhof and M. Smid. Enumerating the k closest pairs optimally. In Proc. 33rd Annu. IEEE Sympos. Found. Comput. Sci., pages 380–386, 1992.
D. Leven and M. Sharir. Planning a purely translational motion for a convex object in two-dimensional space using generalized Voronoi diagrams. Discrete Comput. Geom., 2:9–31, 1987.
N. Megiddo. Applying parallel computation algorithms in the design of serial algorithms. J. ACM, 30:852–865, 1983.
N. Megiddo. Linear-time algorithms for linear programming in R 3 and related problems. SIAM J. Comput., 12:759–776, 1983.
F. P. Preparata and M. I. Shamos. Computational Geometry: an Introduction. Springer-Verlag, New York, NY, 1985.
R. Seidel. A simple and fast incremental randomized algorithm for computing trapezoidal decompositions and for triangulating polygons. Comput. Geom. Theory Appl., 1:51–64, 1991.
M. Sharir. On k-sets in arrangements of curves and surfaces. Discrete Comput. Geom., 6:593–613, 1991.
C. K. Yap. An O(n log n) algorithm for the Voronoi diagram of a set of simple curve segments. Discrete Comput. Geom., 2:365–393, 1987.
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© 1993 Springer-Verlag Berlin Heidelberg
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Efrat, A., Sharir, M., Ziv, A. (1993). Computing the smallest k-enclosing circle and related problems. 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_259
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DOI: https://doi.org/10.1007/3-540-57155-8_259
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