An Optimal Algorithm for Finding the Separation of Simple Polygons

Given simple polygons P and Q, their separation, denoted by σ(P,Q), is defined to be the minimum distance between their boundaries. We present an optimal Θ(N) algorithm for determining the separation of two simple polygons P and Q, where ¦P¦+¦Q¦=¦N¦. The best previous algorithm for this problem is due to Kirkpatrick and has complexity O(N log N). In addition, a parallel version of our algorithm can be implemented in O(log N) time using O(N) processors on a CREW PRAM. Our results are obtained by providing a unified treatment of the separation and the closest visible vertex problems for simple polygons.

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

[2]  Herbert Edelsbrunner,et al.  Computing the Extreme Distances Between Two Convex Polygons , 1985, J. Algorithms.

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

[4]  Francis Y. L. Chin,et al.  Optimal Algorithms for the Intersection and the Minimum Distance Problems Between Planar Polygons , 1983, IEEE Transactions on Computers.

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

[6]  Michael T. Goodrich,et al.  Constructing the Voronoi Diagram of a Set of Line Segments in Parallel (Preliminary Version) , 1989, WADS.

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

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

[9]  David M. Mount.  Intersection detection and separators for simple polygons , 1992, SCG '92.

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

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

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

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

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