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Efficient VLSI Parallel Algorithm for Delaunay Triangulation on Orthogonal Tree Network in Two and Three Dimensions
March 1990 (vol. 39 no. 3)
pp. 400-404

An algorithm with worst case time complexity O(log/sup 2/N) in two dimensions and O(m/sup 1/2/log N) in three dimensions with N input points and m as the number of tetrahedra in triangulation is given. Its AT/sup 2/ VLSI complexity on Thompson's logarithmic delay model, (1983) is O(N/sup 2/log/sup 6/N) in two dimensions and O(m/sup 2/Nlog/sup 4/ N) in three dimensions.

[1] A. Aggarwal, B. Chazelle, L. Guibas, C. O. Dunlaing, and C. Yap, "Parallel computational geometry," manuscript, 1987.
[2] M. J. Atallah, R. Cole, and M. T. Goodrich, "Cascading divide-and-conquer: A technique for designing parallel algorithms," manuscript, 1987.
[3] J.D. Boissonnat, "Geometric Structures for Three-Dimensional Shape Representation,"ACM Trans. Graphics, Jan. 1987, pp. 266-286.
[4] J. C. Cavendish, "Automatic triangulation of arbitrary planar domains for finite element method,"Int. J. Numer. Methods Eng., vol. 8, pp. 679-696, 1974.
[5] R. C. Chang and R. C. T. Lee, "AnO(nlogn) minimal spanning tree algorithm for n points in the plane,"BIT, vol. 26, pp. 7-16, 1986.
[6] A. L. Chow, "Parallel algorithms for geometric problems," Ph.D. dissertation, Dep. Comput. Sci., Univ. of Illinois, Urbana, IL, 1980.
[7] D. Dadoun and D. G. Kirkpatrick, "Parallel processing for efficient subdivision search," inProc. ACM Symp. Computational Geometry, 1987, pp. 205-214.
[8] H. Edelsbrunner,Algorithms in Combinatorial Geometry, Springer-Verlag, New York, 1987.
[9] D. A. Field, "Implementing Watson's algorithm in 3-dimensions," inProc. ACM Symp. Computational Geometry, 1986, pp. 246-259.
[10] L. Guibas and J. Stolfi, "Primitives for the Manipulation of General Subdivisions and the Computation of Voronoi Diagrams,"ACM Trans. Graphics, Vol. 4, No. 2, April 1985, pp. 74-123.
[11] D. J. Kuck,The Structure of Computers and Computations, vol. 1. New York: Wiley, 1978.
[12] M. Lu, "Constructing the Voronoi diagram on a mesh connected computer," inProc. Int. Conf. Parallel Processing, 1986, pp. 806-811.
[13] D. Nath, S. N. Maheshwari, and P. C. P. Bhatt, "Efficient VLSI networks for parallel processing based on orthogonal trees,"IEEE Trans. Comput., vol. C-32, pp. 569-581, June 1983.
[14] F. P. Preparata, "A mesh-connected area-time optimal VLSI multiplier of large integers,"IEEE Trans. Comput., vol. C-32, pp. 194-198, Feb. 1983.
[15] F. P. Preparata and M. I. Shamos,Computational Geometry, an Introduction. New York: Springer-Verlag, 1985.
[16] S. Saxena, P. C. P. Bhatt, and V. C. Prasad, "Parallel algorithms for Delaunay triangulation in two and three dimensions," Tech. Rep. JUN/1/87, Comput. Sci. Eng., Indian Institute of Technology, New Delhi.
[17] J. M. Smith, D. T. Lee, and J. S. Liebman, "AnO(nlogn) heuristic for Steiner minimal tree problem on the Euclidean metric,"Networks, vol. 11, pp. 23-29, Spring 1981.
[18] C. D. Thompson, "The VLSI complexity of sorting,"IEEE Trans. Comput., vol. C-32, pp. 1171-1184, Dec. 1983.
[19] C. A. Wang and Y. H. Tsin, "AnO(logN) time parallel algorithm for triangulating a set of points in the plane,"IPL, vol. 25, pp. 55-60, Apr. 6, 1987.

Index Terms:
VLSI parallel algorithm; Delaunay triangulation; orthogonal tree network; worst case time complexity; tetrahedra; logarithmic delay model; computational complexity; parallel algorithms; trees (mathematics).
F. Saxena, P.C.P. Bhatt, V.C. Prasad, "Efficient VLSI Parallel Algorithm for Delaunay Triangulation on Orthogonal Tree Network in Two and Three Dimensions," IEEE Transactions on Computers, vol. 39, no. 3, pp. 400-404, March 1990, doi:10.1109/12.48871
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