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Efficient Geometric Algorithms on the EREW PRAM
January 1995 (vol. 6 no. 1)
pp. 41-47

Abstract—We present a technique that can be used to obtain efficient parallel geometric algorithms in the EREW PRAM computational model. This technique enables us to solve optimally a number of geometric problems in $O(\log n)$ time using $O(n/\log n)$ EREW PRAM processors, where $n$ is the input size of a problem. These problems include: computing the convex hull of a set of points in the plane that are given sorted, computing the convex hull of a simple polygon, computing the common intersection of half-planes whose slopes are given sorted, finding the kernel of a simple polygon, triangulating a set of points in the plane that are given sorted, triangulating monotone polygons and star-shaped polygons, and computing the all dominating neighbors of a sequence of values. PRAM algorithms for these problems were previously known to be optimal (i.e., in $O(\log n)$ time and using $O(n/\log n)$ processors) only on the CREW PRAM, which is a stronger model than the EREW PRAM.

Index Terms—Computational geometry, convex hulls, kernel, parallel algorithms, parallel random access machines, read conflicts, simple polygons, triangulation, visibility.

[1] A. Aggarwal, B. Chazelle, L. Guibas, C. O'Dunlang, and C. Yap,“Parallel computational geometry,”Algorithmica, vol. 3, no. 3, pp. 293–327, 1988.
[2] M.J. Atallah,D.Z. Chen,, and H. Wagener,“An optimal parallel algorithm for the visibility of a simple polygon from apoint,” J. ACM, vol. 38, pp. 516-533, 1991.
[3] M.J. Atallah,R. Cole,, and M.T. Goodrich,“Cascading divide-and-conquer: A technique for designing parallelalgorithms,” SIAM J. Computing, vol. 18, no. 3, pp. 499-532, 1989.
[4] M. J. Atallah and M. T. Goodrich,“Efficient parallel solutions to some geometric problems,”J. Parallel&Distrib. Comput., vol. 3, pp. 492–507, 1986.
[5] M. J. Atallah and M. T. Goodrich,“Parallel algorithms for some functions of two convex polygons,”Algorithmica, vol. 3, pp. 535–548, 1988.
[6] O. Berkman,D. Breslauer,Z. Galil,B. Schieber,, and U. Vishkin,“Highly parallelizable problems,” Proc. Ann. Symp. Theory of Computing, pp. 770-780, 1989.
[7] G. Bilardi and A. Nicolau,“Adaptive bitonic sorting: An optimal parallel algorithm for shared-memory machines,”SIAM J. Comput., vol. 18, pp. 216–228, 1989.
[8] D.Z. Chen and S. Guha,“Testing a simple polygon for monotonicity optimally n parallel,” Information Processing Letters, vol. 47, no. 6, pp. 325-331, 1993.
[9] R. Cole, "Parallel Merge Sort," SIAM J. Computing, vol. 17, pp. 770-785, 1988.
[10] R. Cole and M.T. Goodrich,“Optimal parallel algorithms for polygon and point-set problems,” Proc. Fourth Ann. ACM Symp. Computational Geometry, pp. 211-220, 1988; also Algorithmica, vol. 7, no. 1, pp. 3-23, 1992.
[11] R. Cole and U. Vishkin, "Approximate Parallel Scheduling. Part 1: The Basic Technique with Applications to Optimal Parallel List Ranking in Logarithmic Time," SIAM J. Computing, vol. 18, pp. 128-142, 1988.
[12] A. Fournier and D. Y. Montuno,“Triangulating simple polygons and equivalent problems,”ACM Trans. Graphics, vol. 3, no. 2, pp. 153–174, 1984.
[13] M. R. Garey, D. S. Johnson, F. P. Preparata, and R. E. Tarjan,“Triangulating a simple polygon,”Inform. Process. Lett., vol. 7, no. 4, pp. 175–179, 1978.
[14] M. T. Goodrich,“Finding the convex hull of a sorted point set in parallel,”Inform. Process. Lett., vol. 26, pp. 173–179, 1987/1988.
[15] ——,“Triangulating a polygon in parallel,”J. Algorithms, vol. 10, pp. 327–351, 1989.
[16] ——,“Planar separators and parallel polygon triangulation,”inProc. 24th Annu. ACM Symp. Theory Comput., Victoria, B.C., Canada, 1992.
[17] R. L. Graham,“An efficient algorithm for determining the convex hull of a finite planar set,”Inform. Process. Lett., vol. 1, pp. 132–133, 1972.
[18] R. L. Graham and F. F. Yao,“Finding the convex hull of a simple polygon,”J. Algorithms, vol. 4, no. 4, pp. 324–331, 1983.
[19] T. Hagerup and C. Rub,"Optimal merging and sorting on the EREW PRAM," Information Processing Letters, vol. 33, pp. 181-185, 1989.
[20] D. G. Kirkpatrick and T. Przytycka,“An optimal parallel algorithm for all dominating neighbors problem and its applications,”Manuscript, 1990.
[21] C. P. Kruskal, L. Rudolph, and M. Snir,“The power of parallel prefix,”IEEE Trans. Comput., vol. C-34, pp. 965–968, 1985.
[22] R.E. Ladner and M.J. Fischer, "Parallel Prefix Computation," J. ACM, vol. 27, no. 4, pp. 831-838, Oct. 1980.
[23] D. T. Lee,“On finding the convex hull of a simple polygon,”Int. J. Comput. Inform. Sci., vol. 12, no. 2, pp. 87–98, 1983.
[24] D.T. Lee and F.P. Preparata,“An optimal algorithm for finding the kernel of a polygon,” J. ACM, vol. 26, pp. 415-421, 1979.
[25] D. T. Lee and F. P. Preparata,“Computational geometry—A survey,”IEEE Trans. Comput., vol. C-33, pp. 1072–1101, Dec. 1984.
[26] E. Merks,“An optimal parallel algorithm for triangulating a set of points in the plane,”inInt. J. Parallel Programming, vol. 15, no. 5, pp. 399–411, 1986.
[27] R. Miller and Q. F. Stout,“Efficient parallel convex hull algorithms,”IEEE Trans. Comput., vol. 37, pp. 1605–1618, Dec. 1988.
[28] W. Paul, U. Vishkin, and H. Wagener,“Parallel dictionaries on 2-3 trees,”inProc. 10th Coll. Autom., Lang., Prog. (ICALP),LNCS 154, Springer, Berlin, 1983, pp. 597–609.
[29] F.P. Preparata and M.I. Shamos, Computational Geometry. Springer-Verlag, 1985.
[30] H. Wagener,“Parallel computational geometry using polygonal order,”Ph.D. dissertation, Technical Univ. of Berlin, FRG, 1985.
[31] H. Wagener,“Triangulating a monotone polygon in parallel,”inComputational Geometry and Its Applications; also inProc. Int. Workshop Computational Geometry (CG'88), Wurzburg, FRG, 1988, pp. 136–147.
[32] C.A. Wang and Y.H. Tsin, "An O(log n) Time Parallel Algorithm for Triangulating a Set of Points in the Plane," Information Processing Letters, vol. 25, pp. 55-60, Apr. 1987.
[33] T. C. Woo and S. Y. Shin,“A linear time algorithm for triangulating a point-visible polygon,”ACM Trans. Graphics, vol. 4, no. 1, pp. 60–70, 1985.
[34] C. K. Yap.“Parallel triangulation of a polygon in two calls to the trapezoidal map,”Algorithmica, vol. 3, pp. 279–288, 1988.

Citation:
Danny Z. Chen, "Efficient Geometric Algorithms on the EREW PRAM," IEEE Transactions on Parallel and Distributed Systems, vol. 6, no. 1, pp. 41-47, Jan. 1995, doi:10.1109/71.363412
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