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)


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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  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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