This Article 
   
 Share 
   
 Bibliographic References 
   
 Add to: 
 
Digg
Furl
Spurl
Blink
Simpy
Google
Del.icio.us
Y!MyWeb
 
 Search 
   
Mesh Computer Algorithms for Computational Geometry
March 1989 (vol. 38 no. 3)
pp. 321-340
Asymptotically optimal parallel algorithms are presented for use on a mesh computer to determine several fundamental geometric properties of figures. For example, given multiple figures represented by the Cartesian coordinates of n or fewer planar vertices, distributed one point per processor on a two-dimensional mesh computer with n simple processing elements, Theta (n/sup 1/2/≤-time algori

[1] A. Aggarwal, B. Chazelle, L. Guibas, C. O'Dunlaing, and C. Yap, "Parallel computational geometry," inProc. 1985 Symp. Foundations Comput. Sci., pp. 468-477.
[2] S. Akl, "Parallel algorithms for convex hulls," Dep. Comput. Sci., Queens Univ., Kingston Ont., Canada, 1983.
[3] M. J. Atallah and M. T. Goodrich, "Efficient parallel solutions to some geometric problems,"J. Parallel Distributed Comput., vol. 3, pp. 492-507, 1986.
[4] M. Atallah and R. Kosaraju, "Graph problems on a mesh connected processor array,"J. Assoc. Comp. Mach., vol. 31, pp. 649-667, 1983.
[5] M. J. Atallah and S. E. Hambrusch, "Solving tree problems on a mesh-connected processor array," inProc. 26th Symp. Foundations Comput. Sci., 1985, pp. 222-231.
[6] D. Avis, "On the complexity of finding the convex hull of a set of points," Tech. Rep. FOCS 79.2, School of Comput. Sci., McGill Univ., 1979.
[7] G. H. Barnes, R. M. Brown, M. Kato, D. J. Kuck, D. L. Slotnick, and R. A. Stokes, "The ILLIAC IV Computer,"IEEE Trans. Comput., vol. C-17, pp. 746-757, 1968.
[8] K. E. Batcher, "Design of a massively parallel processor,"IEEE Trans. Comput., vol. 29, pp. 836-840, 1981.
[9] J. L. Bentley, B. W. Weide, and A. C. Yao, "Optimal expected-time algorithms for closest point problems," inProc. Allerton Conf., 1978.
[10] J. L. Bentley and T. A. Ottman, "Algorithms for counting and reporting geometric intersections,"IEEE Trans. Comput., vol. 28, pp. 643-647, 1979.
[11] J. L. Bentley, "Multidimensional divide and conquer,"Commun. ACM, vol. 23, pp. 214-219, 1980.
[12] B. Chazelle, "Computational geometry on a systolic chip,"IEEE Trans. Comput., vol. C-33, pp. 774-785, 1984.
[13] A. Chow, "A parallel algorithm for determining convex hulls of sets of points in two dimensions," inProc. 19th Allerton Conf. Commun., Contr., Comput., 1981, pp. 214-233.
[14] F. Dehne, "O(n1/2)algorithms for the maximal elements and ECDF searching problem on a mesh-connected parallel computer,"Inform. Proc. Lett., vol. 22, pp. 303-306, 1986.
[15] M. L. B. Duff and D. M. Watson, "The cellular logic array image processor,"Comput. J., vol. 20, pp. 68-72, 1977.
[16] C. R. Dyer and A. Rosenfeld, "Parallel image processing by memory augmented cellular automata,"IEEE Trans. Pattern Anal. Machine Intell., vol. PAMI-3, pp. 29-41, 1981.
[17] H. Freeman and R. Shapira, "Determining the minimal-area enclosing rectangle for an arbitrary closed curve,"Commun. Assoc. Comput. Mach., vol. 18, pp. 409-413, 1975.
[18] S. I. Gass,Linear Programming. New York: McGraw-Hill, 1969.
[19] F. C. A. Groen, P. W. Verbeek, N. d. Jong, and J. W. Klumper, "The smallest box around a package," Tech. Rep. Instit. Appl. Phy., Delft Univ. of Technology.
[20] K. Hwang and K-s. Fu, "Integrated computer architectures for image processing and database management,"IEEE Computer, vol. 15, pp. 51-60, 1982.
[21] C. S. Jeong and D. T. Lee, "Parallel geometric algorithms on a mesh connected computer," Tech. Rep. 87-02-FC-01 (revised), Dep. EECS, Northwestern Univ.
[22] R. J. Lipton and R. E. Tarjan, "Applications of a planar separator theorem," inProc. 18th Annu. IEEE Symp. Foundations Comput. Sci., 1977, pp. 162-170.
[23] M. Lu, "Constructing the Voronoi diagram on a mesh-connected computer," inProc. 1985 Int. Conf. Parallel Processing, pp. 806- 811.
[24] M. Lu and P. Varman, "Solving geometric proximity problems on mesh-connected computers," inProc. 1985 Workshop Comput. Architecture Pattern Anal. Image Database Management, pp. 248- 255.
[25] E. M. McCreight, "Priority search trees," Tech. Rep. Xerox PARCL CSL-81-5, 1981.
[26] C. Mead and L. Conway,Introduction to VLSI Systems. Reading, MA: Addison-Wesley, 1980, pp. 150-152.
[27] R. Miller and Q. F. Stout, "Computational geometry on a mesh-connected computer," inProc. 1984 Int. Conf. Parallel Processing, pp. 66-73.
[28] R. Miller and Q. F. Stout, "Geometric algorithms for digitized pictures on a mesh-connected computer,"IEEE Trans. Pattern Anal. Machine Intell., vol. PAMI-7, pp. 216-228, 1985.
[29] R. Miller and Q. F. Stout, "Varying diameter and problem size in mesh-connected computers," inProc. 1985 Int. Conf. Parallel Processing, pp. 697-699.
[30] R. Miller and Q. F. Stout,Parallel Algorithms for Regular Architectures. Cambridge, MA: MIT Press, 1988, to be published.
[31] D. Nassimi and S. Sahni, "Data broadcasting in SIMD computers,"IEEE Trans. Comput., vol. C-30, pp. 101-107, 1981.
[32] D. Nath, S. N. Maheshwari, and P. C. P. Bhatt, "Parallel algorithms for the convex hull in two dimensions," inProc. Conf. Anal. Problem Classes Programming Parallel Comput., 1981, pp. 358- 372.
[33] J. O'Rourke, C.-B. Chen, T. Olson, and D. Naddor, "A new linear algorithm for intersecting convex polygons,"Comput. Graph. Image Processing, vol. 19, pp. 384-391, 1982.
[34] F. P. Preparata and D. T. Lee, "Computational geometry-A survey,"IEEE Trans. Comput., C-33, pp. 1072-1100, 1984.
[35] F. P. Preparata and M. I. Shamos,Computational Geometry, an Introduction. New York: Springer-Verlag, 1985.
[36] J. Reif and Q. F. Stout, "Optimal component labeling algorithms for mesh computers and VLSI," to be published.
[37] M. I. Shamos, "Computational geometry," Ph.D. dissertation, Dep. Comput. Sci., Yale Univ., 1978.
[38] M. I. Shamos and D. Hoey, "Geometric intersection problems," inProc. Seventh Annu. IEEE Symp. Foundations Comput. Sci., 1975, pp. 151-162.
[39] Q. F. Stout, "Broadcasting in mesh-connected computers," inProc. 1982 Conf. Inform. Sci. Sys., Princeton Univ., pp. 85-90.
[40] Q. F. Stout, "Topological matching," inProc. 15th ACM Symp. Theory of Computing, 1983, pp. 24-31.
[41] Q. F. Stout, Tree-based graph algorithms for some parallel computers," inProc. 1985 Int. Conf. Parallel Processing, pp. 727-730.
[42] C. D. Thompson and H. T. Kung, "Sorting on a mesh connected processor array,"Commun. ACM, pp. 263-271, 1972.
[43] G. T. Toussaint, "Pattern recognition and geometrical complexity," inProc. 5th Int. Conf. Pattern Recognition, 1980, pp. 1324-1347.
[44] A. Yao, "A lower bound to finding convex hulls," Dep. Computer Sci., Stanford Univ., 1979.

Index Terms:
mesh computer algorithms; asymptotically optimal parallel algorithms; computational geometry; Cartesian coordinates; two-dimensional mesh computer; convex hull; all-nearest neighbor problems; intersection problems; computational geometry; parallel algorithms.
Citation:
R. Miller, Q.F. Stout, "Mesh Computer Algorithms for Computational Geometry," IEEE Transactions on Computers, vol. 38, no. 3, pp. 321-340, March 1989, doi:10.1109/12.21120
Usage of this product signifies your acceptance of the Terms of Use.