Finding a closest visible vertex pair between two polygons

AbstractGiven nonintersecting simple polygonsP andQ, two verticespεP andqε Q are said to be visible if $$\overline {pq}$$ does not properly intersectP orQ. We present a parallel algorithm for finding a closest pair among all visible pairs (p,q),pεP andqεQ. The algorithm runs in time O(logn) using O(n) processors on a CREW PRAM, wheren=¦P¦+¦Q¦. This algorithm can be implemented serially in Θ(n) time, which gives a new optimal sequential solution for this problem.

[1]  David G. Kirkpatrick,et al.  Determining the Separation of Preprocessed Polyhedra - A Unified Approach , 1990, ICALP.

[2]  Robert E. Tarjan,et al.  Triangulating a Simple Polygon , 1978, Inf. Process. Lett..

[3]  David G. Kirkpatrick,et al.  Optimal Parallel Algorithms for Convex Polygon Separation , 1989 .

[4]  Richard M. Karp,et al.  Parallel Algorithms for Shared-Memory Machines , 1991, Handbook of Theoretical Computer Science, Volume A: Algorithms and Complexity.

[5]  Nancy M. Amato.  An Optimal Algorithm for Finding the Separation of Simple Polygons , 1993, WADS.

[6]  Edward P. F. Chan,et al.  Finding the minimum visible vertex distance between two non-intersecting simple polygons , 1986, SCG '86.

[7]  Michael T. Goodrich,et al.  Parallel methods for visibility and shortest path problems in simple polygons (preliminary version) , 1990, SCG '90.

[8]  Francis Y. L. Chin,et al.  A unifying approach for a class of problems in the computational geometry of polygons , 1985, The Visual Computer.

[9]  Michael T. Goodrich,et al.  Triangulating a Polygon in Parallel , 1989, J. Algorithms.

[10]  Alok Aggarwal,et al.  Geometric applications of a matrix-searching algorithm , 1987, SCG '86.

[11]  Leonidas J. Guibas,et al.  Parallel computational geometry , 1988, 26th Annual Symposium on Foundations of Computer Science (sfcs 1985).

[12]  Bernard Chazelle.  Triangulating a simple polygon in linear time , 1991, Discret. Comput. Geom..

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

[14]  Mikhail J. Atallah,et al.  An efficient parallel algorithm for the row minima of a totally monotone matrix , 1991, SODA '91.

[15]  Godfried T. Toussaint,et al.  An optimal algorithm for computing the minimum vertex distance between two crossing convex polygons , 1983, Computing.

[16]  Mikhail J. Atallah,et al.  Efficient Parallel Solutions to Some Geometric Problems , 1986, J. Parallel Distributed Comput..

[17]  Alok Aggarwal,et al.  Geometric Applications of a Matrix Searching Algorithm , 1986, Symposium on Computational Geometry.

[18]  Nancy M. Amato.  Determining the separation of simple polygons , 1994, Int. J. Comput. Geom. Appl..

[19]  G. Toussaint,et al.  Finding the minimum vertex distance between two disjoint convex polygons in linear time , 1985 .

[20]  Alok Aggarwal,et al.  Computing the Minimum Visible Vertex Distance between Two Polygons (Preliminary Version) , 1989, WADS.

[21]  Hossam A. ElGindy,et al.  A new linear convex hull algorithm for simple polygons , 1984, IEEE Trans. Inf. Theory.

[22]  Chee-Keng Yap,et al.  AnO(n logn) algorithm for the voronoi diagram of a set of simple curve segments , 1987, Discret. Comput. Geom..

[23]  Michael T. Goodrich,et al.  Parallel methods for visibility and shortest-path problems in simple polygons , 1992, Algorithmica.

[24]  David G. Kirkpatrick,et al.  Efficient computation of continuous skeletons , 1979, 20th Annual Symposium on Foundations of Computer Science (sfcs 1979).

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

[26]  S. Suri.  Minimum link paths in polygons and related problems , 1987 .

[27]  Subhash Suri,et al.  Matrix searching with the shortest path metric , 1993, SIAM J. Comput..

[28]  Mikhail J. Atallah,et al.  Parallel algorithms for some functions of two convex polygons , 2005, Algorithmica.

[29]  Leonidas J. Guibas,et al.  Optimal shortest path queries in a simple polygon , 1987, SCG '87.

[30]  Michael T. Goodrich,et al.  Constructing the Voronoi diagram of a set of line segments in parallel , 1993, Algorithmica.

[31]  Fang-Rong Hsu,et al.  Parallel algorithms for computing the closest visible vertex pair between two polygons , 1992, Int. J. Comput. Geom. Appl..

[32]  Nancy M. Amato.  Computing the Minimum Visible Vertex Distance Between Two Nonintersecting Simple Polygons , 1992 .

[33]  Steven Fortune,et al.  A sweepline algorithm for Voronoi diagrams , 1986, SCG '86.