Algorithmica

, Volume 3, Issue 1–4, pp 293–327

Parallel computational geometry

  • A. Aggarwal
  • B. Chazelle
  • L. Guibas
  • C. Ó'Dúnlaing
  • C. Yap
Article

Abstract

We present efficient parallel algorithms for several basic problems in computational geometry: convex hulls, Voronoi diagrams, detecting line segment intersections, triangulating simple polygons, minimizing a circumscribing triangle, and recursive data-structures for three-dimensional queries.

Key words

Parallel algorithms Computational geometry Data structures 

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References

  1. A. Aggarwal, J. S. Chang, and C. K. Yap (1985). Minimum area circumscribing polygons.Visual Comput.,1, pp. 112–117.MATHCrossRefGoogle Scholar
  2. A. Aggarwal, M. Klawe, S. Moran, P. Shor, and R. Wilber (1986). Geometric applications of a matrix searching algorithm.Proc. 2nd ACM Symposium on Computational Geometry, pp. 285–292.Google Scholar
  3. A. Aggarwal, B. Chazelle, L. Guibas, C. Ó'Dúnlaing, and C. K. Yap (1987). Parallel computational geometry. Robotics Report No. 115, Courant Institute, New York University. (Reported at the 26th IEEE FOCS Symposium, Portland, Oregon, 1985.)Google Scholar
  4. M. Ajtai, J. Komlós, and E. Szemerédi (1982). AnO(n log(n)) Sorting Network.Proc. 15th ACM Symposium on Theory of Computing, pp. 1–9. Also inCombinatorica,3(1) (1983), pp. 1–19.Google Scholar
  5. S. Akl (1983). Parallel algorithm for convex hulls. Manuscript, Department of Computer Science, Queen's University, Kingston, Ontario.Google Scholar
  6. M. J. Atallah and M. T. Goodrich (1985). Efficient parallel solutions to geometric problems.Proc. 1985 IEEE Conference on Parallel Processing, pp. 411–417.Google Scholar
  7. M. J. Atallah and M. T. Goodrich (1986). Efficient plane sweeping in parallel.Proc. 2nd Symposium on Computational Geometry, pp. 216–225.Google Scholar
  8. M. J. Atallah, R. Cole, and M. T. Goodrich (1987). Cascading divide- and-conquer: a technique for designing parallel algorithms.Proc. 28th IEEE FOCS Symposium, pp. 151–160.Google Scholar
  9. M. Ben-Or (1983). Lower bounds for algebraic computational trees.Proc. 15th ACM Symposium on Theory of Computing, pp. 80–86.Google Scholar
  10. M. Ben-Or, D. Kozen, and J. Reif (1984). The complexity of elementary algebra and geometry.Proc. 16th ACM Symposium on Theory of Computing, pp. 457–464.Google Scholar
  11. J. L. Bentley (1977). Algorithms for Klee's rectangle problems. Unpublished manuscript, CMU.Google Scholar
  12. J. L. Bentley and D. Wood (1980). An optimal worst-case algorithm for reporting intersections of rectangles.IEEE Trans. Comput.,29, pp. 571–577.CrossRefMathSciNetGoogle Scholar
  13. J. E. Boyce, D. P. Dobkin, R. L. Drysdale, and L. J. Guibas (1985). Finding extremal polygons.SIAM J. Comput.,14, pp. 134–147.MATHCrossRefMathSciNetGoogle Scholar
  14. R. P. Brent (1974). The parallel evaluation of general arithmetic expressions.J. Assoc. Comput. Mach.,21(2), pp. 201–206.MATHMathSciNetGoogle Scholar
  15. K. Q. Brown (1979). Voronoi diagrams from convex hulls.Inform. Process. Lett.,9(5), pp. 223–228.MATHCrossRefGoogle Scholar
  16. J. S. Chang (1986). Polygon optimization problems. Ph.D. Dissertation, Robotics Report No. 78, Department of Computer Science, Courant Institute, New York University.Google Scholar
  17. B. Chazelle (1982). A theorem on polygon cutting with applications.Proc. 23rd IEEE FOCS Symposium, pp. 339–349.Google Scholar
  18. B. M. Chazelle (1984). Computational geometry on a systolic chip.IEEE Trans. Comput.,33, pp. 774–785.CrossRefGoogle Scholar
  19. B. Chazelle (1986). Reporting and counting segment intersections.J. Comput. System Sci.,32(2), pp. 156–182.MATHCrossRefMathSciNetGoogle Scholar
  20. B. Chazelle and J. Incerpi (1984). Triangulation and shape-complexity.ACM Trans. Graphics,3(2), pp. 135–152.MATHCrossRefGoogle Scholar
  21. A. Chow (1980). Parallel algorithms for geometric problems. Ph.D. Dissertation, Computer Science Department, University of Illinois at Urbana-Champaign, 1980.Google Scholar
  22. A. Chow (1981). A parallel algorithm for determining convex hulls of sets of points in two dimensions.Proc. 19th Allerton Conference on Communication, Control and Computing, pp. 214–233.Google Scholar
  23. V. Chvátal (1975). A combinatorial theorem in plane geometry.J. Combin. Theory Ser. B,18, pp. 39–41.MATHCrossRefGoogle Scholar
  24. R. Cole (1986). Parallel merge sort.Proc. 27th IEEE FOCS Symposium, pp. 511–516.Google Scholar
  25. G. E. Collins (1975). Quantifier elimination for real closed fields by cylindrical algebraic decomposition.2nd GI Conference on Automata Theory and Formal Languages. Lecture Notes in Computer Science, Vol. 33, Springer-Verlag, Berlin, pp. 134–183.Google Scholar
  26. S. Cook and C. Dwork (1982). Bounds on the time for parallel RAMS to compute simple functions.Proc. 14th ACM Symposium on Theory of Computing, pp. 231–233.Google Scholar
  27. N. Dadoun and D. G. Kirkpatrick (1987). Parallel processing for efficient subdivision search.Proc. 3rd Annual Symposium on Computational Geometry, 1987, 205–214.Google Scholar
  28. D. P. Dobkin and D. G. Kirkpatrick (1983). Fast detection of polyhedral intersections.Theoret. Comput. Science,27, pp. 241–253.MATHCrossRefMathSciNetGoogle Scholar
  29. H. Freeman and R. Shapira (1975). Determining the minimum-area encasing rectangle for an arbitrary closed curve.Comm. ACM,18(7), 409–413.MATHCrossRefMathSciNetGoogle Scholar
  30. L. J. Guibas and J. Stolfi (1983). Primitives for the manipulation of general subdivisions and the computation of Voronoi diagrams. Proc.15th ACM Symposium on Theory of Computing, pp. 221–234. Also to appear inACM Trans. Graphics.Google Scholar
  31. V. Klee and M. C. Laskowski (1985). Finding the smallest triangles containing a given polygon.J. Algorithms,6, pp. 457–464.CrossRefMathSciNetGoogle Scholar
  32. D. Kozen and C. K. Yap (1985). Algebraic cell decomposition in NC.Proc. 26th IEEE FOCS Symposium, pp. 515–521.Google Scholar
  33. T. Leighton (1984). Tight bounds on the complexity of parallel sorting.Proc. 16th ACM Symposium on Theory of Computing, pp. 71–80.Google Scholar
  34. W. Lipski, Jr., and F. P. Preparata (1981). Segments, rectangles, contours.J. Algorithms,2, pp. 63–76.MATHCrossRefMathSciNetGoogle Scholar
  35. D. Nath, S. N. Maheshwari, and P. C. P. Bhatt (1981). Parallel algorithms for the convex hull in two dimensions.Proc. Conference on Analysis Problem Classes and Programming for Parallel Computing, pp. 358–372.Google Scholar
  36. J. O'Rourke, A. Aggarwal, S. Madilla, and M. Baldwin (1986). An optimal algorithm for finding minimal enclosing triangles.J. Algorithms,7(2), 258–269.MATHCrossRefMathSciNetGoogle Scholar
  37. M. H. Overmars and J. Van Leeuwen (1981). Maintenance of configurations in the plane.J. Comput. Systems Sci.,23, pp. 166–204.MATHCrossRefGoogle Scholar
  38. F. Preparata and S. J. Hong (1977). Convex hulls of finite sets of points in two and three dimensions.Comm. ACM,20, pp. 87–93.MATHCrossRefMathSciNetGoogle Scholar
  39. F. Preparata and M. I. Shamos (1985).Computational Geometry: An Introduction. Springer-Verlag, Berlin.Google Scholar
  40. M. I. Shamos (1977). Computational geometry. Ph.D. Dissertation, Yale University.Google Scholar
  41. M.I. Shamos and D. Hoey (1975). Closest point problems.Proc. 16th IEEE Symposium on Foundations of Computing, pp. 151–162.Google Scholar
  42. Y. Shiloach and U. Vishkin (1982). AnO(log(n)) parallel connectivity algorithm.J. Algorithms,3, pp. 57–67.MATHCrossRefMathSciNetGoogle Scholar
  43. R. E. Tarjan and U. Vishkin (1985). An efficient parallel biconnectivity algorithm.SIAM J. Comput., 14(4), pp. 862–874.MATHCrossRefMathSciNetGoogle Scholar
  44. G. T. Toussaint (1983). Solving geometric problems using “rotating callipers.”Proc. IEEE Melecon '83.Google Scholar
  45. L. G. Valiant (1975). Parallelism in comparison problems.SIAM J. Comput.,4(3), pp. 348–355.MATHCrossRefMathSciNetGoogle Scholar
  46. H. Wagener (1985). Parallel computational geometry using polygon ordering. Ph.D. Thesis, Technical University of Berlin.Google Scholar
  47. H. Wagener (1987). Optimally parallel algorithms for convex hull determination (submitted).Google Scholar
  48. C. K. Yap (1987). What can be parallelized in computational geometry? International Workshop on Parallel Algorithms and Architecture, Humboldt University, Berlin, DDR (invited talk). Proceedings to appear.Google Scholar

Copyright information

© Springer-Verlag New York Inc. 1988

Authors and Affiliations

  • A. Aggarwal
    • 1
  • B. Chazelle
    • 2
  • L. Guibas
    • 3
    • 4
  • C. Ó'Dúnlaing
    • 5
    • 6
  • C. Yap
    • 5
  1. 1.IBM Thomas J. Watson Research CenterYorktown HeightsUSA
  2. 2.Department of Computer SciencePrinceton UniversityPrincetonUSA
  3. 3.Digital Equipment Corporation Systems Research LaboratoriesPalo AltoUSA
  4. 4.Computer Science DepartmentStanford UniversityStanfordUSA
  5. 5.Courant Institute of Mathematical SciencesNew York UniversityNew YorkUSA
  6. 6.School of MathematicsTrinity CollegeRepublic of Ireland

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