Constructing the Voronoi diagram of a set of line segments in parallel

Preliminary version
  • Michael T. Goodrich
  • Colm Ó'Dúnlaing
  • Chee K. Yap
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 382)


In this paper we give a parallel algorithm for constructing the Voronoi diagram of a polygonal scene, i.e., a set of line segments in the plane such that no two segments intersect except possibly at their endpoints. Our algorithm runs in O(log2n) time using O(n) processors in the CREW PRAM model.


Line Segment Parallel Algorithm Voronoi Diagram Computational Geometry Medial Axis 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. [1]
    A. Aggarwal, B. Chazelle, L. Guibas, C. Ó'Dúnaling, and C. Yap, “Parallel Computational Geometry,” Algorithmica, Vol. 3, No. 3, 1988, 293–328.CrossRefGoogle Scholar
  2. [2]
    M.J. Atallah, R. Cole, and M.T. Goodrich, “Cascading Divide-and-Conquer: A Technique for Designing Parallel Algorithms,” SIAM J. Comput., in press.Google Scholar
  3. [3]
    M.J. Atallah and M.T. Goodrich, “Efficient Parallel Solutions to Some Geometric Problems,” J. of Par. and Dist. Comp., Vol. 3, 1986, 492–507.CrossRefGoogle Scholar
  4. [4]
    M.J. Atallah and M.T. Goodrich, “Parallel Algorithms for Some Functions of Two Convex Polygons,” Algorithmica, Vol. 3, 1988, 535–548.CrossRefGoogle Scholar
  5. [5]
    M.J. Atallah and U. Vishkin, “Finding Euler Tours in Parallel,” J. Comp. and System Sci., Vol. 29, 1985, 330–337.Google Scholar
  6. [6]
    G. Bilardi and A. Nicolau, “Adaptive Bitonic Sorting: An Optimal Parallel Algorithm for Shared Memory Machines,” TR 86-769, Dept. of Comp. Sci., Cornell Univ., August 1986.Google Scholar
  7. [7]
    H. Blum, “A Transformation for Extracting New Descriptors of Shape,” Proc. Symp. Models for Perception of Speech and Visual Form, W. Whaten-Dunn, ed., Cambridge, MA: M.I.T. Press, 1967, 362–380.Google Scholar
  8. [8]
    A. Chow, “Parallel Algorithms for Geometric Problems,” Ph.D. thesis, Comp. Sci. Dept., Univ. of Illinois at Urbana-Champaign, 1980.Google Scholar
  9. [9]
    R. Cole, “Parallel Merge Sort,” SIAM J. Computing, Vol. 17, No. 4, August 1988, 770–785.CrossRefGoogle Scholar
  10. [10]
    R. Cole and M.T. Goodrich, “Optimal Parallel Algorithms for Polygon and Point-Set Problems,” Algorithmica, in press.Google Scholar
  11. [11]
    R.L. Drysdale, III, “Generlaized Voronoi Diagrams and Geometric Searching,” Computer Science Report STAN-CS-79-705, Stanford Univ., Ph.D. thesis, 1979.Google Scholar
  12. [12]
    S. Fortune, “A Sweepline Algorithm for Voronoi Diagrams,” Proc. 2nd ACM Symp. on Computational Geometry, 1986, 313–322.Google Scholar
  13. [13]
    M.T. Goodrich, “Efficient Parallel Techniques for Computational Geometry,” Ph.D. thesis, Dept. of Computer Science, Purdue Univ., August 1987.Google Scholar
  14. [14]
    M.T. Goodrich, “Finding the Convex Hull of a Sorted Point Set in Parallel,” Info. Proc. Letters, Vol. 26, December 1987, 173–179.Google Scholar
  15. [15]
    L. Guibas, L. Ramshaw, and J. Stolfi, “A Kinetic Framework for Computational Geometry,” Proc. 24th IEEE Symp. on Found. of Comp. Sci., 1983, 100–111.Google Scholar
  16. [16]
    R.M. Karp and V. Ramachandran, “A Survey of Parallel Algorithms for Shared-Memory Machines,” to appear in Handbook of Theoretical Computer Science, North-Holland.Google Scholar
  17. [17]
    D.G. Kirkpatrick, “Efficient Computation of Continuous Skeletons,” Proc. 20th IEEE Symp. on Foundations of Computer Science, 1979, 18–27.Google Scholar
  18. [18]
    C.P. Kruskal, L. Rudolph, and M. Snir, “The Power of Parallel Prefix,” Proc. 1985 IEEE Int. Conf. on Parallel Processing, 180–185.Google Scholar
  19. [19]
    R.E. Ladner and M.J. Fischer, “Parallel Prefix Computation,” J. ACM, October 1980, 831–838.Google Scholar
  20. [20]
    D.T. Lee, “Medial Axis Transformation of a Planar Shape,” IEEE Trans. on Pattern Analysis and Machine Intelligence, Vol. PAMI-4, No. 4, July 1982, 363–369.Google Scholar
  21. [21]
    D.T. Lee and R.L. Drysdale, III,“Generalization of Voronoi Diagrams in the Plane,” SIAM Journal on Computing, Vol. 10, No. 1, Feb. 1981, 73–87.Google Scholar
  22. [22]
    J.S.B. Mitchell, “On Maximum Flows in Polyhedral Domains,” Proc. 4th ACM Symp. on Computational Geometry, 1988, 341–351.Google Scholar
  23. [23]
    C. Ó'Dúnlaing and C. Yap, “A ‘Retraction’ Method for Planning the Motion of a Disc, J. Algorithms, Vol. 6, 1985, 104–111.Google Scholar
  24. [24]
    F.P. Preparata, “The Medial Axis of a Simple Polygon,” Proc. 6th Symp. on Mathematical Foundations of Computer Science, 1977, 443–450.Google Scholar
  25. [25]
    M.I. Shamos, “Geometric Complexity,” Proc. 7th ACM Symp. on Theory of Computing, 1975, 224–233.Google Scholar
  26. [26]
    G.T. Toussaint, “Solving Geometric Problems with Rotating Calipers,” Proc. IEEE MELECON '83, Athens, Greece, May 1983.Google Scholar
  27. [27]
    H. Wagener, “Optimally Parallel Algorithms for Convex Hull Determination,” unpublished manuscript, September 1985.Google Scholar
  28. [28]
    J.C. Wyllie, “The Complexity of Parallel Computation,” Ph.D. thesis, Technical Report TR 79-387, Department of Computer Science, Cornell University, 1979.Google Scholar
  29. [29]
    C.K. Yap, “An O(n log n) Algorithm for the Voronoi Diagram of a Set of Simple Curve Segments,” Discrete and Computational Geometry, Vol. 2, 1987, 365–393.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1989

Authors and Affiliations

  • Michael T. Goodrich
    • 1
  • Colm Ó'Dúnlaing
    • 2
  • Chee K. Yap
    • 3
  1. 1.Dept. of Computer ScienceJohns Hopkins Univ.Baltimore
  2. 2.School of MathematicsUniv. of DublinDublin 2Irish Republic
  3. 3.Courant InstituteNew York Univ.New York

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