Abstract
We explore a new approach for computing the diameter of n points in ℝ3 that is based on the restriction of the furthest-point Voronoi diagram to the convex hull. We show that the restricted Voronoi diagram has linear complexity. We present a deterministic algorithm with O(n log2 n) running time. The algorithm is quite simple and is a good candidate to be implemented in practice. Using our approach the chromatic diameter and all-furthest neighbors in ℝ3 can be found in the same running time.
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P. K. Agarwal and M. Sharir, Davenport-Schinzel Sequences and Their Geometric Applications, Cambridge University Press, Cambridge, 1995.
A. Aggarwal and D. Kravets, A linear time algorithm for finding all farthest neighbors in a convex polygon, Inform. Process. Lett., 31(1) (1989), 17–20.
N. M. Amato, M. T. Goodrich, and E. A. Ramos, Parallel algorithms for higher-dimensional convex hulls, Proc. 35th Annual IEEE Symposium on Foundations of Computer Science, pp. 340–347, 1994.
F. Aurenhammer, A criterion for the affine equality of cell complexes in ℝd and convex polyhedra in ℝd+1, Discrete Comput. Geom., 2 (1987), 49–64.
F. Aurenhammer, Power diagrams: properties, algorithms and applications, SIAM J. Comput., 16 (1987), 78–96.
F. Aurenhammer, Improved algorithms for discs and balls using power diagrams, J. Algorithms, 9 (1988), 151–161.
F. Aurenhammer, Linear combinations from power domains, Geom. Dedicata, 28 (1988), 45–52.
H. Brönninman, B. Chazelle, and J. Matoušek, Product range spaces, sensitive sampling, and derandomization, SIAM J. Comput., 28(5) (1999), 1552–1575.
J. L. Bentley and T. A. Ottmann, Algorithm for reporting and counting geometric intersections, IEEE Trans. Comput., 28 1979, 643–647.
B. Chazelle, A theorem on polygon cutting with applications, Proc. 23rd Annual IEEE Symposium on Foundations of Computer Science, pp. 339–349, 1982.
B. Chazelle, H. Edelsbrunner, L. Guibas, and M. Sharir, Diameter, width, closest line pair, and parametric search, Discrete Comput. Geom., 10 (1993), 145–158.
K. L. Clarkson and P. W. Shor, Applications of random sampling in computational geometry, II. Discrete Comput. Geom., 4 (1989), 387–421.
H. Davenport and A. Schinzel, A combinatorial problem connected with differential equations, Amer. J. Math., 87 (1965), 684–689.
D. P. Dobkin and D. G. Kirkpatrick, Fast detection of polyhedral intersection, Theoret. Comput. Sci., 27 (1983), 241–253.
H. Edelsbrunner, Computing the extreme distances between two convex polygons, J. Algorithms, 6 (1985), 213–224.
L. Guibas, J. Hershberger, D. Leven, M. Sharir and R. E. Tarjan, Linear time algorithms for visibility and shortest path problems inside triangulated simple polygons, Algorithmica, 2 (1987), 209–233.
H. Imai, M. Iri, and K. Murota, Voronoi diagrams in the Laguerre geometry and its applications, SIAM J. Comput., 14 (1985), 93–105.
D. T. Lee, Proximity and Reachability in the Plane, Technical Report No. R-831, Coordinate Science Laboratory, University of Illinois at Urbana, IL, 1978.
D. T. Lee, Two-dimensional Voronoi diagrams in the Lp-metric, J. Assoc. Comput. Mach., 27 (1980), 604–618.
J. Matoušek and O. Schwarzkopf, On ray shooting in convex polytopes, Discrete Comput. Geom., 10 (1993), 215–232.
J. Matoušek and O. Schwarzkopf, A deterministic algorithm for the three-dimensional diameter problem, Comput. Geom. Theory Appl., 6 (1996), 253–262.
N. Megiddo, Applying parallel computation algorithms in the design of serial algorithms, J. Assoc. Comput. Mach., 30 (1983), 852–865.
F. P. Preparata and S. J. Hong, Convex hulls of finite sets of points in two and three dimensions, Comm. ACM, 20(2) (1977), 87–93.
F. P. Preparata and M. I. Shamos, Computational Geometry, an Introduction, 3rd edn., Springer-Verlag, New York, 1990.
E. A. Ramos, Intersection of unit-balls and diameter of a point set in ℝ3, Comput. Geom. Theory Appl., 6 (1996), 57–65.
E. A. Ramos, Construction of 1-d lower envelopes and applications, Proc. 13th Annual ACM Symposium on Computational Geometry, pp. 57–66, 1997.
E. A. Ramos, Deterministic algorithms for 3-D diameter and some 2-D lower envelopes, Proc. 16th Annual ACM Symposium on Computational Geometry, pp. 290–299, 2000.
M. I. Shamos and D. Hoey, Closest-point problems, Proc. 16th Annual IEEE Symposium on Foundations of Computer Science, pp. 151–162, 1975.
G. T. Toussaint, The symmetric all-furthest neighbor problem, J. Comput. Math. Appl., 9(6) (1983), 747–754.
A. C. Yao, On constructing minimum spanning trees in k-dimensional spaces and related problems, SIAM J. Comput., 11 (1982), 721–736.
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Bespamyatnikh, S. An efficient algorithm for the three-dimensional diameter problem. Discrete Comput Geom 25, 235–255 (2001). https://doi.org/10.1007/s004540010086
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DOI: https://doi.org/10.1007/s004540010086