We consider the problem of orienting the edges of a planar graph in such a way that the out-degree of each vertex is minimized. If, for each vertex v, the out-degree is at most d, then we say that such an orientation is d-bounded. We prove the following results: ffl Each planar graph has a 5-bounded acyclic orientation, which can be constructed in linear time. ffl Each planar graph has a 3-bounded orientation, which can be constructed in linear time. ffl A 6-bounded acyclic orientation, and a 3-bounded orientation, of each planar graph can each be constructed in parallel time O(log n log n) on an EREW PRAM, using O(n= log n log n) processors. As an application of these results, we present a data structure such that each entry in the adjacency matrix of a planar graph can be looked up in constant time. The data structure uses linear storage, and can be constructed in linear time. Department of Mathematics and Computer Science, University of California, Riverside, CA 92521. On...

Tree contraction is a powerful technique for solving a large number of graph problems on families of recursively definable graphs. The method is based on processing the parse tree associated with a member of such a family of graphs in a bottom-up fashion, such that the solution to the problem is obtained at the root of the tree. Sequentially, this can be done in linear time with respect to the size of the input graph. In parallel, efficient and even cost optimal tree contraction algorithms have also been developed. In this paper we show how the method can be applied to compute the cardinality of the minimum vertex cover of a two-terminal series-parallel graph. We then construct a real-time paradigm for this problem and show that in the new computational environment, a parallel algorithm is superior to the best possible sequential algorithm, in terms of the accuracy of the solution computed. Specifically, there are cases in which the solution produced by a parallel algorithm ...

In this paper we develop a new linear-time algorithm for the recognition of series parallel graphs. The algorithm is based on a succinct representation of series parallel graphs for which the presence of an arc can be tested in constant time; space utilization is linear in the number of vertices. We show how to compute such a representation in linear time from a breadth-first spanning tree. Furthermore, we present a precise condition for the existence of such succinct representations in general, which is, for instance, satisfied by planar graphs.

In this paper, a parallel algorithm is given that, given a graph G = (V; E), decides whether G is a series parallel graph, and if so, builds a decomposition tree for G of series and parallel composition rules. The algorithm uses O(log |E| log |E|) time and O(|E|) operations on an EREW PRAM, and O(log |E|) time and O(|E|) operations on a CRCW PRAM (note that if G is a simple series parallel graph, then |E| = O(|V|)). With the same time and processor resources, a tree-decomposition of width at most two can be built of a given series parallel graph, and hence, very efficient parallel algorithms can be found for a large number of graph problems on series parallel graphs, including many well known problems, e.g., all problems that can be stated in monadic second order logic. The results hold for undirected series parallel graphs graphs, as well as for directed series parallel graphs.

In this paper, we present parallel algorithms for the coarse grained multicomputer (CGM) and bulk synchronous parallel computer (BSP) for solving two well known graph problems: (1) determining whether a graph G is bipartite, and (2) determining whether a bipartite graph G is convex. Our algorithms require O(...

Fast algorithms can be created for many graph problems when instances are confined to classes of graphs that are recursively constructed. This paper first describes some basic conceptual notions regarding the design of such fast algorithms, and then the coverage proceeds through several recursive graph classes. Specific classes include k-terminal graphs, trees, series-parallel graphs, k-trees, partial k-trees, Halin graphs, bandwidth-k graphs, pathwidth-k graphs, treewidthk graphs, branchwidth-k graphs, cographs, cliquewidth-k graphs, k-NLC graphs, k-HB graphs, and rankwidth-k graphs. The definition of each class is provided, after which some typical algorithms are applied to solve problems on instances of each class.

We present a parallel algorithm for solving the minimum weighted completion time scheduling problem for transitive series parallel graphs. The algorithm takes O(log 2 n) time with O(n 3 ) processors on a CREW PRAM, where n is the number of vertices of the input graph. This is the first NC algorithm for solving the problem. 1 Introduction Because of their practical applications and connection to other combinatorial and optimization problems, scheduling problems have played an important role in the field of algorithm design. Sequential algorithms for solving a variety of scheduling problems have been studied and many important results are known. (See [6] for a survey.) On the other hand, very little is known about the parallel complexity of scheduling problems. Dekel and Sahni [4] presented a parallel algorithm for a special case of the single processor scheduling problem: each job has a release time, a deadline and unit processing time (but no precedence constraints) . The algorith...

Computing short regular expressions equivalent to a given finite automaton is a hard task. In this work we present a class of acyclic automata for which it is possible to obtain in time O(n² log n) an equivalent regular expression of size O(n). A characterisation of this class is made using properties of the underlying digraphs that correspond to the series-parallel digraphs class. Using this characterisation we present an algorithm for the generation of automata of this class and an enumerative formula for the underlying digraphs with a given number of vertices.