We describe e#cient PRAM algorithms for constructing unbalanced quadtrees, balanced quadtrees, and quadtree-based finite element meshes. Our algorithms take time O(log n) for point set input and O(log n log k) time for planar straight-line graphs, using O(n + k/ log n) processors, where n measures input size and k output size. 1. Introduction A crucial preprocessing step for the finite element method is mesh generation, and the most general and versatile type of two-dimensional mesh is an unstructured triangular mesh. Such a mesh is simply a triangulation of the input domain (e.g., a polygon), along with some extra vertices, called Steiner points. Not all triangulations, however, serve equally well; numerical and discretization error depend on the quality of the triangulation, meaning the shapes and sizes of triangles. A typical quality guarantee gives a lower bound on the minimum angle in the triangulation. Baker et al. 1 first proved the existence of quality triangulations fo...

this paper and supplied many helpful comments. This research was supported in part by NSF grants DCR-85-11713, CCR-86-05353, and CCR-88-14977, and by DARPA contract N00039-84-C-0165.

Abstract. Currently, there is a significant gap between the best sequential and parallel complexities of many fundamental problems related to digraph reachability. This complexity bottleneck essentially reflects a seemingly unavoidable reliance on transitive closure techniques in parallel algorithms for digraph reachability. To pinpoint the nature of the bottleneck, we de* velop a collection of polylog-time reductions among reachability problems. These reductions use only linear processors and work for general graphs. Furthermore, for planar digraphs, we give polylog-time algorithms for the following problems: (1) directed ear decomposition, (2) topological ordering, (3) digraph reachability, (4) descendent counting, and (5) depth-first search. These algorithms use only linear processors and therefore reduce the complexity to within a polylog factor of optimal.

This paper considers the computation of matrix chain products of the form M1 \Theta M2 \Theta \Delta \Delta \Delta \Theta Mn\Gamma1 . The order in which the matrices are multiplied affects the number of operations. The best sequential algorithm for computing an optimal order of matrix multiplication runs in O(n log n) time while the best known parallel NC algorithm runs in O(log 2 n) time using n 6 = log 6 n processors. This paper presents the first approximating optimal parallel algorithm for this problem and for the problem of finding a near-optimal triangulation of a convex polygon. The algorithm runs in O(log n) time using n= log n processors on a CREW PRAM, and in O(log log n) time using n= log log n processors on a weak CRCW PRAM. It produces an order of matrix multiplications and a partition of polygon which differ from the optimal ones at most 0.1547 times. 1 Introduction The problem of computing an optimal order of matrix multiplication (the matrix chain product proble...

Computing shortest paths in a directed graph has received considerable attention in the sequential RAM model of computation. However, developing a polylog-time parallel algorithm that is close to the sequential optimal in terms of the total work done remains an elusive goal. We present a first step in this direction by giving efficient parallel algorithms for shortest paths in planar layered digraphs. We show that these graphs admit special kinds of separators called one-way separators which allow the paths in the graph to cross it only once. We use these separators to give divide and conquer solutions to the problem of finding the shortest paths between any two vertices. We first give a simple algorithm that works in the CREW model and computes the shortest path between any two vertices in an n-node planar layered digraph in time O(log³ n) using n/log n processors. A CRCW version of this algorithm runs in time O(log² n log log n) and uses n/log log n processors. We then use re...

Let q be a nonnegative integer. For a graph G, the maximum q-dependent set problem is to find a maximum cardinality subset of the set of vertices of G such that none of the vertices in the subset has more than q neighbours in the subset. We observe that the decision version of this natural extension of the maximum independent set problem is NP-complete for every fixed q. Then, we present a simple linear-time algorithm for the maximum q-dependent set problem restricted to trees. For an input tree on n nodes, the algorithm runs in time bounded by cn where c is a constant independent of q: Using the algorithm, we derive the d q+1 q+2 \Theta ne lower bound on the size of maximum q-dependent set in a tree and prove it to be tight. In addition, we present an O(logn)-time parallel implementation of the algorithm which uses an EREW PRAM with only O(n= log n) processors. We also observe that the maximum q-dependent set problem can be solved in linear time when restricted to graphs ...

Graphs are the most widely used of all mathematical structures. There is uncountable number of interesting computational problems defined in terms of graphs. A graph can be seen as a collection of vertices (V ), and a collection of edges (E) joining all or some of the vertices. One is very often interested in finding subsets, either from the set V of vertices or from the set E of edges, which possess some predefined property. A subset S of vertices or edges in a graph G is said to be maximum with respect to a property if, among all the subsets of G having this property, S is one having the largest cardinality. Set S is said to be maximal with respect to a property, if the set has the property, and it is not a proper subset of another set having the property. A subset of vertices in a graph, is said to be independent if no two of them are adjacent. The problems of finding maximum and maximal independent sets in a graph are well known problems. One of the more natural generalizations of...

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ere capable of anything and instilling in us a desire to be the best in whatever we did. I would also like to thank my high school teachers Mr. Jaypal Chandra and Ms. Bhuvaneshvari for showing me that education could be fun, and Professors. M.V. Tamhankar, and H. Subramanian for some truly inspiring courses in mathematics. At Brown, I would like to thank Professors Philip Klein, Roberto Tamassia, and Jeff Vitter for advising this thesis and for teaching me much of what I know. I would like to thank Prof. Vitter for introducing me to research and for his confidence in my abilities. His constant encouragement kept me motivated during times when the going was tough. I would like to thank Prof. Tamassia for encouraging my interest in dynamic graph algorithms and for suggesting the problem solved in Chapter 5. A large portion of the results in this thesis were obtained in joint work with Prof. Phil Klein. I would like to thank him for his boundless enthusiasm for research and for the innume

Computing shortest paths in a directed graph has received considerable attention in the sequential RAM model of computation. However, developing a polylog-time parallel algorithm that is close to the sequential optimal in terms of the total work done remains an elusive goal. We present a first step in this direction by showing that for an n-node planar layered digraph with nonnegative edge-weights the shortest path between any two vertices can be computed in O(log³ n) time with n processors in a CREW PRAM. A CRCW version of our algorithm runs in time O(log² n log log n) and uses n log n/log log n processors. Our results make use of the existence of special kinds of separators in planar layered digraphs, called one-way separators, to implement a divide and conquer solution.