## LSU Historical Dissertations and Theses

#### Title

Parallel Geometric Algorithms.

1992

Dissertation

#### Degree Name

Doctor of Philosophy (PhD)

Computer Science

Si-Qing Zheng

#### Abstract

Geometric algorithms have many important applications in science and technology. Some geometric problems require fast response time that could not be achieved by traditional sequential algorithms. However, the speed, power and versatility of parallel computers can be exploited to develop efficient geometric algorithms as shown in this dissertation. Our study focuses on designing efficient parallel geometric algorithms and analyzing their computational complexities. In this research, first we developed a parallel algorithm to find the maxima of a set of N points in the d-dimensional space, d $>$ 3, on a hypercube SIMD machine. Our algorithm is a parallel implementation from the sequential algorithm given by Kung, Luccio, and Preparata (KLP75). Although the time complexity, $O(N\sp{0.77}\log\sp{d-1}\ N),$ of our algorithm is not optimal, it is the first sublinear time algorithm for solving the high dimensional maxima problem. Next, we developed another parallel algorithm to construct the Voronoi diagram of a point set in the plane. Our algorithm is based on the sequential algorithm given by Brown (B79). We use an $N\times N$ mesh of trees (MOT) SIMD computer and get the optimal time complexity $O(log\sp2N).$. Finally, we developed another MOT algorithm to solve the congruent pattern problem. Given a simple polygon P with k edges and a planar graph G with N edges, $N>k.$ The problem is to find all the patterns (cycles) in G which are congruent to P. Our algorithm is based on the CREW PRAM algorithm given by Jeong, Kim, and Baek (JKB92). We also use an $N\times N$ MOT and get the optimal time complexity $O(k\log N).$.

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