An efficient algorithm for the three-dimensional diameter problem
- S. Bespamyatnikh
- … show all 1 hide
Purchase on Springer.com
$39.95 / €34.95 / £29.95*
Rent the article at a discountRent now
* Final gross prices may vary according to local VAT.
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.
- 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. CrossRef
- 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. CrossRef
- F. Aurenhammer, Power diagrams: properties, algorithms and applications, SIAM J. Comput., 16 (1987), 78–96. CrossRef
- F. Aurenhammer, Improved algorithms for discs and balls using power diagrams, J. Algorithms, 9 (1988), 151–161. CrossRef
- F. Aurenhammer, Linear combinations from power domains, Geom. Dedicata, 28 (1988), 45–52. CrossRef
- H. Brönninman, B. Chazelle, and J. Matoušek, Product range spaces, sensitive sampling, and derandomization, SIAM J. Comput., 28(5) (1999), 1552–1575. CrossRef
- J. L. Bentley and T. A. Ottmann, Algorithm for reporting and counting geometric intersections, IEEE Trans. Comput., 28 1979, 643–647. CrossRef
- 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. CrossRef
- K. L. Clarkson and P. W. Shor, Applications of random sampling in computational geometry, II. Discrete Comput. Geom., 4 (1989), 387–421. CrossRef
- H. Davenport and A. Schinzel, A combinatorial problem connected with differential equations, Amer. J. Math., 87 (1965), 684–689. CrossRef
- D. P. Dobkin and D. G. Kirkpatrick, Fast detection of polyhedral intersection, Theoret. Comput. Sci., 27 (1983), 241–253. CrossRef
- H. Edelsbrunner, Computing the extreme distances between two convex polygons, J. Algorithms, 6 (1985), 213–224. CrossRef
- 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. CrossRef
- H. Imai, M. Iri, and K. Murota, Voronoi diagrams in the Laguerre geometry and its applications, SIAM J. Comput., 14 (1985), 93–105. CrossRef
- 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. CrossRef
- 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. CrossRef
- 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. CrossRef
- A. C. Yao, On constructing minimum spanning trees in k-dimensional spaces and related problems, SIAM J. Comput., 11 (1982), 721–736. CrossRef
- An efficient algorithm for the three-dimensional diameter problem
Discrete & Computational Geometry
Volume 25, Issue 2 , pp 235-255
- Cover Date
- Print ISSN
- Online ISSN
- Additional Links
- Industry Sectors
- S. Bespamyatnikh (1)
- Author Affiliations
- 1. Department of Computer Science, University of British Columbia, V6T 1Z2, Vancouver, British Columbia, Canada