Techniques for solving geometric problems on mesh-connected computers

The contributions of this thesis are twofold: (i) we solve optimally some problems on conventional Mesh-Connected Computers, which were not previously solved optimally, and (ii) we present new algorithms for several geometric problems on more realistic models. On conventional Mesh-Connected Computers, in which the n processors are arranged as a (multidimensional) array, we present a new technique for optimally performing n searches on a class of hierarchical DAGs, which leads to the first optimal mesh algorithms for the three dimensional convex hull and convex polyhedra intersection problems, settling an open problem which was posed in (AW88) and in (MS88b). The previous algorithms were a log n factor away from optimality. On the more realistic models (RAM/ARRAY(d)), in which the d-dimensional Mesh-Connected Computer is of fixed-size p and is attached to a random access machine, we present new algorithms for several geometric problems, which achieve the same speedup for a problem of arbitrary size n $\geq$ p as for a problem of size p. The problems include that of computing the all nearest neighbors of a planar set of points, the measure and perimeter of a union of rectangles, visibility of a set of nonintersecting line segments from a point, and dominance counting between two planar sets of points. All of the problems have sequential time complexity $\Theta$(n log n) and have O($p\sp{1/d}$) solutions for a problem of size p on a d-dimensional Mesh-Connected Computers of p-processors. Hence, the RAM/ARRAY(d) achieves the speedup of $O(p\sp{1-1/d}$ log p) for a problem of size p. Thus our contribution is to show that the speedup of $O(p\sp{1-1/d}$ log p) can be achieved for arbitrarily large problem size.