Optimal parallel hypercube algorithms for polygon problems

We present parallel techniques on hypercubes for solving optimally a class of polygon problems. We thus obtain optimal O(log n)-time, n-processor hypercube algorithms for the problems of computing the portions of an n-vertex simple polygonal chain C that are visible from a given source point, computing the convex hull of C, testing an n-vertex simple polygon P for monotonicity, and other related problems as well. Previously it was not known how to achieve these complexity bounds on hypercubes, one of the main difficulties being that there is no known optimal sorting hypercube algorithm that achieves these bounds. In fact these are the first optimal geometric hypercube algorithms that do not assume that the input is given already sorted by x or y coordinates. The hypercube model we use is the standard one, with O(1) local memory per processor, and with one-port communication.<<ETX>>

[1]  Richard Cole,et al.  Cascading divide-and-conquer: A technique for designing parallel algorithms , 1989, 28th Annual Symposium on Foundations of Computer Science (sfcs 1987).

[2]  Bernard Chazelle,et al.  Intersection of convex objects in two and three dimensions , 1987, JACM.

[3]  Quentin F. Stout,et al.  Asymptotically efficient hypercube algorithms for computational geometry , 1990, [1990 Proceedings] The Third Symposium on the Frontiers of Massively Parallel Computation.

[4]  D. T. Lee,et al.  An Optimal Algorithm for Finding the Kernel of a Polygon , 1979, JACM.

[5]  Danny Ziyi Chen,et al.  Testing a Simple Polygon for Monotonicity Optimally in Parallel , 1993, Inf. Process. Lett..

[6]  C. Greg Plaxton,et al.  Deterministic sorting in nearly logarithmic time on the hypercube and related computers , 1990, STOC '90.

[7]  Danny Ziyi Chen,et al.  Efficient Geometric Algorithms on the EREW PRAM , 1995, IEEE Trans. Parallel Distributed Syst..

[8]  D. T. Lee,et al.  Visibility of a simple polygon , 1983, Comput. Vis. Graph. Image Process..

[9]  Andrew Rau-Chaplin,et al.  Implementing Data Structures on a Hypercube Multiprocessor, and Applications in Parallel Computational Geometry , 1989, J. Parallel Distributed Comput..

[10]  Kenneth E. Batcher,et al.  Sorting networks and their applications , 1968, AFIPS Spring Joint Computing Conference.

[11]  Richard Cole,et al.  Optimal parallel algorithms for polygon and point-set problems , 1988, SCG '88.

[12]  Afonso Ferreira,et al.  Parallel Fractional Cascading on Hypercube Multiprocessors , 1992, Comput. Geom..

[13]  Russ Miller,et al.  Efficient Parallel Convex Hull Algorithms , 1988, IEEE Trans. Computers.

[14]  Quentin F. Stout,et al.  Practical hypercube algorithms for computational geometry , 1990, [1990 Proceedings] The Third Symposium on the Frontiers of Massively Parallel Computation.

[15]  David Avis,et al.  A Linear Algorithm for Computing the Visibility Polygon from a Point , 1981, J. Algorithms.

[16]  Andrew Rau-Chaplin,et al.  Implementing Data Structures on a Hypercube Multiprocessor, and Applications in Parallel Computational Geometry , 1989, WG.

[17]  Sartaj Sahni,et al.  Parallel permutation and sorting algorithms and a new generalized connection network , 1982, JACM.

[18]  Franco P. Preparata,et al.  Testing a Simple Polygon for Monotonicity , 1981, Inf. Process. Lett..

[19]  F. Frances Yao,et al.  Finding the Convex Hull of a Simple Polygon , 1983, J. Algorithms.