What can be parallelized in computational geometry?

  • Chee-Keng Yap
Invited Papers
Part of the Lecture Notes in Computer Science book series (LNCS, volume 269)


This paper has two goals. First, we point out that most problems in computational geometry in fact have fast parallel algorithms (that is, in NC*) by reduction to the cell decomposition result of Kozen and Yap. We illustrate this using a new notion of generalized Voronoi diagrams that subsumes all known instances. While the existence of NC* algorithms for computational geometry is theoretically significant, it leaves much to be desired for specific problems. Therefore, the second part of the paper surveys some recent results in a fast growing list of parallel algorithms for computational geometry.


Convex Hull Parallel Algorithm Voronoi Diagram Computational Geometry Delaunay Triangulation 
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, J. S. Chang, and C. K. Yap. Minimum area circumscribing polygons. The Visual Computer, 1:112–117, 1985.CrossRefGoogle Scholar
  2. [2]
    A. Aggarwal, B. Chazelle, L. Guibas, C. Ó'Dúnlaing, and C. Yap. Parallel computational geometry. In IEEE Foundations of Computer Science, pages 468–477, 1985. To appear, special issue Algorithmica.Google Scholar
  3. [3]
    M. Ajtai, J. Komlós, and E. Szemerédi. Sorting in c log n parallel steps. Combinatorica, 3:1–19, 1983.Google Scholar
  4. [4]
    Dennis S. Arnon. A Cellular Decomposition Algorithm for Semi-Algebraic Sets. Tech. Rep. No. 353, Dept. of Comp. Sci., Univ. of Wisconsin, 1979. PhD thesis.Google Scholar
  5. [5]
    M. J. Atallah, R. Cole, and M. T. Goodrich. Private communication.Google Scholar
  6. [6]
    Mikhail J. Atallah and Michael T. Goodrich. Efficient parallel solutions to geometric problems. In Proc. 1985 Int. Conf. on Parallel Processing, pages 411–417, 1985. To appear, J. of Parallel and Distributed Computing.Google Scholar
  7. [7]
    Mikhail J. Atallah and Michael T. Goodrich. Efficient plane sweeping in parallel. In 2nd ACM Symp. on Computational Geometry, pages 216–225, 1986.Google Scholar
  8. [8]
    M. Ben-Or, D. Kozen, and J. Reif. The complexity of elementary algebra and geometry. In ACM Symposium on Theory of Computing, pages 457–464, 1984.Google Scholar
  9. [9]
    Alan Borodin and John Hopcroft. Routing, merging and sorting on parallel models of computation. In 14th STOC, pages 338–344, 1982.Google Scholar
  10. [10]
    J. E. Boyce, D. P. Dobkin, R. L. Drysdale, and L. J. Guibas. Finding extremal polygons. SIAM J. Computing, 14:134–147, 1985.CrossRefGoogle Scholar
  11. [11]
    J. S. Chang and C. K. Yap. A polynomial solution for the potato-peeling problem. Discrete and Comput. Geo., 1:155–182, 1986.Google Scholar
  12. [12]
    Anita Chow. Parallel algorithsm for geometric problems. PhD thesis, Department of Computer Sci., University of Illinois, Urbana-Champaign, 1980.Google Scholar
  13. [13]
    Anita L. Chow. A practical parallel algorithm for reporting intersection of rectangles. In IEEE Symp. Parallel Processing, pages 304–305, 1981.Google Scholar
  14. [14]
    Richard Cole. Parallel Merge Sort. Technical Report 278, Courant Institute, NYU, 1987. Earlier version in 27th FOCS, submitted for publication.Google Scholar
  15. [15]
    Richard Cole and Chee Yap. Geometric retrieval problems. Information and Control, 63:39–57, 1984.Google Scholar
  16. [16]
    George E. Collins. Quantifier elimination for real closed fields by cylindrical algebraic decomposition. In 2nd GI Conf. on Automata Theory and Formal Languages, Lecture Notes in Computer Science, pages 134–183, Springer-Verlag, 1975.Google Scholar
  17. [17]
    Stephen A. Cook. Towards a Complexity Theory of Synchronous Parallel Computation. Technical Report 141/80, Dept. of Computer Science, University of Toronto, 1980.Google Scholar
  18. [18]
    N. Dadoun and D. Kirkpatrick. Parallel processing for efficient subdivision search. In 3nd ACM Symp. Comput. Geo., 1987. to appear.Google Scholar
  19. [19]
    H. Edelsbrunner and R. Seidel. Voronoi diagrams and arrangements. In ACM Symposium on Computational Geometry, pages 251–262, 1985.Google Scholar
  20. [20]
    Hossam ElGindy. A parallel algorithm for the shortest path problem in monotone polygons. may 1986. Dept of Comp. and Inf. Sci., Univ. of Penn.Google Scholar
  21. [21]
    Hossam ElGindy. A parallel algorithm for triangulating simplicial point sets in space with optimal speed-up. April 1986. Manuscript.Google Scholar
  22. [22]
    Steven Fortune. A sweepline algorithm for Voronoi diagrams. In 2nd ACM Symp. Comput. Geo., pages 313–322, 1986.Google Scholar
  23. [23]
    M. Goodrich, C. Ó'Dúnlaing, and C. Yap. Fast Parallel Algorithms for Voronoi Diagrams. Technical Report CSD-TR-538, Dept. of Comp. Sci., Purdue Univ., 1985.Google Scholar
  24. [24]
    Michael T. Goodrich. An Optimal parallel algorithm for the all nearest-neighbor problem for a convex polygon. Technical Report CSD-TR-533, Dept. of Comp. Sci., Purdue Univ., 1985.Google Scholar
  25. [25]
    David G. Kirkpatrick. Optimal search in planar subdivision. SIAM J. Computing, 12:28–35, 1983.CrossRefGoogle Scholar
  26. [26]
    Dexter Kozen and Chee Yap. Algebraic cell decomposition in NC. In IEEE Foundations of Computer Science, pages 515–521, 1985.Google Scholar
  27. [27]
    Russ Miller and Quentin F. Stout. Mesh Computer Algorithms for Computational Geometry. Technical Report 86-18, Dept. of Computer Science, SUNY at Buffalo, 1986.Google Scholar
  28. [28]
    D. Nath, S. N. Maheshwari, and P. C. P. Bhatt. Parallel algorithms for the convex hull in two dimensions. In Conf. on Analysis Problem Classes and Programming for Parallel Computing, pages 358–372, 1981.Google Scholar
  29. [29]
    C. Ó'Dúnlaing, M. Sharir, and C. Yap. Generalized Voronoi diagrams for moving a ladder: I. Topological analysis. Comm. Pure and Applied Math., 1984.Google Scholar
  30. [30]
    Franco Preparata. New parallel-sorting schemes. IEEE Trans. on Computers, C-27:669–673, 1978.Google Scholar
  31. [31]
    David Prill. On approximations and incidence in cylindrical algebraic decompositions. SIAM J. Computing, 15:972–993, 1986.Google Scholar
  32. [32]
    S. Saxena, P. C. P. Bhatt, and V. C. Prasad. Fast parallel algorithms for Delaunay triangulation. 1986. Extended abstract.Google Scholar
  33. [33]
    Jacob T. Schwartz. Ultracomputers. ACM Trans. on Programming Languages and Systems, 2:484–521, 1980.Google Scholar
  34. [34]
    Jacob T. Schwartz and Micha Sharir. On the piano movers' problem: II. General techniques for computing topological properties of real algebraic manifolds. Advances in Math., 4:298–351, 1983.Google Scholar
  35. [35]
    Uzi Vishkin. Synchronous Parallel Computation — a Survey. Technical Report 71, Courant Institute, NYU, 1983.Google Scholar
  36. [36]
    Hubert Wagener. Optimally parallel algorithms for convex hull determination. Algorithmica, 1987. (submitted).Google Scholar
  37. [37]
    Hubert Wagener. Parallel computational geometry using polygonal order. PhD thesis, Technical University of Berlin, 1985. West Germany.Google Scholar
  38. [38]
    Chee K. Yap. Parallel subresultant Sturm sequences. Technical Report, Robotics Lab. Report, Courant Institute, NYU, 1987. To appear.Google Scholar
  39. [39]
    Chee-Keng Yap. An O(n log n) algorithm for the Voronoi diagram of a set of simple curve segments. Technical Report 43, Robotics Lab, Courant Institute, NYU, 1985. Submitted for publication.Google Scholar
  40. [40]
    Chee-Keng Yap. Parallel triangulation of a simple polygon in three calls to the trapezoidal map. Technical Report, Robotics Lab, Courant Institute, NYU, March 1987.Google Scholar
  41. [41]
    Chee-Keng Yap. Space-time tradeoffs and first order problems in a model of programs. In 12th ACM Symp. Theory of Computing, 1980.Google Scholar
  42. [42]
    Chee-Keng Yap. Three studies on Computational Problems. PhD thesis, Yale University, 1980.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1987

Authors and Affiliations

  • Chee-Keng Yap
    • 1
  1. 1.Courant Institute of Mathematical SciencesNew York UniversityNew York

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