In IEEE Transactions on Computers C-38 (1989), pp. 321-340.

Mesh Computer Algorithms for Computational Geometry

Russ Miller
Dept of Comp Sci & Eng, State University of New York at Buffalo

Quentin F. Stout
EECS Department, University of Michigan

Abstract: We present asymptotically optimal parallel algorithms for using 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 2-dimensional mesh computer with n simple processing elements, we give Theta(sqrt(n)) algorithms for identifying the convex hull and smallest enclosing box of each figure. Given two such figures, we give a Theta(sqrt(n)) algorithm to decide if the two figures are linearly separable. Given n or fewer planar points, we give Theta(sqrt(n)) time algorithms to solve the all-nearest neighbor problem for points and for sets of points. Given n or fewer circles, convex figures, hyperplanes, simple polygons, orthogonal polygons, or iso-oriented rectangles, we give Theta(sqrt(n)) time algorithms to solve a variety of area and intersection problems. Since any serial computer has worst-case time of Omega(n) when processing n points, our algorithms show that the mesh computer provides significantly better solutions to these problems.

Keywords: parallel algorithm, mesh computer, computational geometry, planar point data, convexity, proximity, area, intersection, parallel computer, discrete mathematics, computer science.

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Much of the content of this paper has been incorporated into the book Parallel Algorithms for Regular Architectures: Meshes and Pyramids, by R. Miller and Q.F. Stout. More information about the book is available.

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