In the past 20 years there has been treftlendous progress in developing and analyzing parallel algorithftls. Researchers have developed efficient parallel algorithms to solve most problems for which efficient sequential solutions are known. Although some ofthese algorithms are efficient only in a theoretical framework, many are quite efficient in practice or have key ideas that have been used in efficient implementations. This research on parallel algorithms has not only improved our general understanding ofparallelism but in several cases has led to improvements in sequential algorithms. Unf:ortunately there has been less success in developing good languages f:or prograftlftling parallel algorithftls, particularly languages that are well suited for teaching and prototyping algorithms. There has been a large gap between languages

In this paper we provide a theoretical foundation for the problem of network localization in which some nodes know their locations and other nodes determine their locations by measuring the distances to their neighbors. We construct grounded graphs to model network localization and apply graph rigidity theory to test the conditions for unique localizability and to construct uniquely localizable networks. We further study the computational complexity of network localization and investigate a subclass of grounded graphs where localization can be computed efficiently. We conclude with a discussion of localization in sensor networks where the sensors are placed randomly.

In this paper we provide a theoretical foundation for the problem of network localization in which some nodes know their locations and other nodes determine their locations by measuring the distances to their neighbors. We construct grounded graphs to model network localization and apply graph rigidity theory to test the conditions for unique localizability and to construct uniquely localizable networks. We further study the computational complexity of network localization and investigate a subclass of grounded graphs where localization can be computed efficiently. We conclude with a discussion of localization in sensor networks where the sensors are placed randomly.

This report deals with a parallel algorithmic technique that has proved to be very useful in the design of efficient parallel algorithms for several problems on undirected graphs. We describe this method for searching undirected graphs, called "open ear decomposition", and we relate this decomposition to graph biconnectivity. We present an efficient parallel algorithm for finding this decomposition and we relate it to a sequential algorithm based on depth-first search. We then apply open ear decomposition to obtain an efficient parallel algorithm for testing graph triconnectivity and for finding the triconnnected components of a graph.

We present a new algorithm based on open ear decomposition for testing vertex four-connectivity and for finding all separating triplets in a triconnected graph. A sequential implementation of our algorithm runs in O(n 2) time and a parallel implementation runs in O(log 2 n) time using O(n 2) processors on an ARBITRARY CRCW PRAM, where n is the number of vertices in the graph. This improves previous bounds for the problem for both the sequential and parallel cases. The sequential time bound is the best possible, to within a constant factor, if the input is specified in adjacency matrix form, or if the input graph is dense. 1.

The first half of the paper is a general introduction which emphasizes the central role that the PRAM model of parallel computation plays in algorithmic studies for parallel computers. Some of the collective knowledge-base on non-numerical parallel algorithms can be characterized in a structural way. Each structure relates a few problems and technique to one another from the basic to the more involved. The second half of the paper provides a bird's-eye view of such structures for: (1) list, tree and graph parallel algorithms; (2) very fast deterministic parallel algorithms; and (3) very fast randomized parallel algorithms. 1 Introduction Parallelism is a concern that is missing from "traditional" algorithmic design. Unfortunately, it turns out that most efficient serial algorithms become rather inefficient parallel algorithms. The experience is that the design of parallel algorithms requires new paradigms and techniques, offering an exciting intellectual challenge. We note that it had...

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.

It is demonstrated that a given value of average genus is shared by at most finitely many 2-connected simplicial graphs and by at most finitely many 3-connected graphs. Moreover, the distribution of values of average genus is sparse, in the following sense: within any finite real interval, there are at most finitely many different numbers that are values of average genus for 2-connected simplicial graphs or for 3-connected graphs. Thus, there are no limit points for the values of average genus of graphs in these classes. The potential applicability of these results to graph isomorphism testing is considered. April 27, 1992 1 Supported by Engineering Excellence Award from Texas A&M University. 2 Supported by ONR Contract N00014-85-0768. CUCS-020-92 1 Introduction In this paper, one of a series, a topological invariant of recent interest, the average genus of graphs, is investigated. By the average genus of a graph G, we mean the average value of the genus of the imbedding surfa...

Graphs of small average genus are characterized. In particular, a Kuratowski-type theorem is obtained: except for finitely many graphs, a cutedge-free graph has average genus less than or equal to 1 if and only if it is a necklace. We provide a complete list of those exceptions. A Kuratowski-type theorem for graphs of maximum genus 1 is also given. Some of the methods used in obtaining these results involve variations of a classical result of Whitney. April 27, 1992 1 Supported by Engineering Excellence Award from Texas A&M University, and by the National Science Foundation under Grant CCR-9110824. 2 Supported by ONR Contract N00014-85-0768. CUCS-019-92 1 Introduction By the average genus of a graph G, we mean the average value of the genus of the imbedding surface, taken over all orientable imbeddings of G. This value is evidently a rational number, and it is clearly an invariant of the homeomorphism type of a graph. The average genus of individual graphs is in the Gross-Furst ...

We present a parallel randomized algorithm running on a CRCW PRAM , to determine whether two planar graphs are isomorphic, and if so to find the isomorphism. We assume that we have a tree of separators for each planar graph (which can be computed by known algorithms in O(log 2 n) time with n 1+ffl processors, for any ffl ? 0). If n is the number of vertices, our algorithm takes O(log(n)) time with P = O i n 1:5 \Delta p log(n) j processors and with probability of failure at most 1 n . The algorithm needs 2 \Delta log(m) \Delta log(n) +O(log(n)) random bits. The number of random bits can be decreased to O(log(n)) by increasing the number of processors to n 3 2 +ffl ; for any ffl ? 0: Our parallel algorithm has significantly improved processor efficiency, compared to the previous logorithmic time parallel algorithm of Miller and Reif [MR 91], which requires n 4 randomized processors or n 5 deterministic processors. Keywords: planar graphs, graph isomorph...