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Parallel construction of quadtrees and quality triangulations

  • Marshall Bern
  • David Eppstein
  • Shang-Hua Teng
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 709)

Abstract

We describe efficient PRAM algorithms for constructing unbalanced quadtrees, balanced quadtrees, and quadtree-based finite element meshes. Our algorithms take time O(log n) for point set input and O(log n log k) time for planar straight-line graphs, using O(n+k/ log n) processors, where n measures input size and k output size.

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References

  1. [1]
    M. J. Atallah, R. Cole, and M.T. Goodrich. Cascading divide-and-conquer: a technique for designing parallel algorithms. SIAM J. Comput. 18 (1989) 499–532.CrossRefGoogle Scholar
  2. [2]
    B. Baker, E. Grosse, and C. Rafferty. Nonobtuse triangulation of polygons. Discrete Comput. Geom. 3 (1988) 147–168.Google Scholar
  3. [3]
    O. Berkman, D. Breslauer, Z. Galil, B. Schieber, and U. Vishkin. Highly parallelizable problems. 21st Symp. Theory of Computing (1989) 309–319.Google Scholar
  4. [4]
    M. Bern, D. Eppstein, and J.R. Gilbert. Provably good mesh generation. 31st Symp. Found. Comput. Sci. (1990) 231–241. To appear in J. Comp. Sys. Sci. Google Scholar
  5. [5]
    M. Bern and D. Eppstein. Mesh generation and optimal triangulation. In Euclidean Geometry and the Computer, World Scientific, 1992.Google Scholar
  6. [6]
    R.P. Brent. The parallel evaluation of general arithmetic expressions. J. ACM 21 (1974) 201–206.CrossRefGoogle Scholar
  7. [7]
    L.P. Chew. Guaranteed-quality triangular meshes. TR-89-983, Cornell, 1989.Google Scholar
  8. [8]
    R. Cole. Parallel merge sort. SIAM J. Comput. 17 (1988) 770–785.CrossRefGoogle Scholar
  9. [9]
    R. Cole and U. Vishkin. Optimal parallel algorithms for expression tree evaluation and list ranking. 3rd Aegean Workshop on Computing, Springer LNCS 319 (1988).Google Scholar
  10. [10]
    D. Eppstein. Approximating the minimum weight triangulation. 3rd Symp. Discrete Algorithms (1992) 48–57. To appear in Discrete Comput. Geom. Google Scholar
  11. [11]
    D. Eppstein and Z. Galil. Parallel algorithmic techniques for combinatorial computation. Ann. Rev. Comput. Sci. 3 (1988) 233–283.CrossRefGoogle Scholar
  12. [12]
    M.L. Fredman and D.E. Willard. Blasting through the information-theoretic barrier with fusion trees. 22nd Symp. Theory of Computing (1990) 1–7.Google Scholar
  13. [13]
    E.A. Melissaratos and D.L. Souvaine. Coping with inconsistencies: A new approach to produce quality triangulations of polygonal domains with holes. 8th Symp. Comput. Geom. (1992) 202–211.Google Scholar
  14. [14]
    S.A. Mitchell and S.A. Vavasis. Quality mesh generation in three dimensions. 8th Symp. Comput. Geom. (1992) 212–221.Google Scholar
  15. [15]
    J. Ruppert. A new and simple algorithm for quality 2-dimensional mesh generation. 4th Symp. Discrete Algorithms (1993) 83–92.Google Scholar
  16. [16]
    H. Samet. The quadtree and related hierarchical data structures. Computing Surveys 16 (1984) 188–260.CrossRefGoogle Scholar
  17. [17]
    D.E. Willard. Applications of the fusion tree method to computational geometry and searching. 3rd Symp. Discrete Algorithms (1992) 286–295.Google Scholar
  18. [18]
    R.D. Williams. Adaptive parallel meshes with complex geometry. Tech. Report CRPC-91-2, Center for Research on Parallel Computation, Cal. Tech.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1993

Authors and Affiliations

  • Marshall Bern
    • 1
  • David Eppstein
    • 2
  • Shang-Hua Teng
    • 3
  1. 1.Xerox Palo Alto Research CenterPalo Alto
  2. 2.Department of Information and Computer ScienceUniversity of CaliforniaIrvine
  3. 3.Department of MathematicsMassachusetts Inst. of TechnologyCambridge

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