Abstract. Parallel algorithms for several graph and geometric problems are presented, including transitive closure and topological sorting in planar st-graphs, preprocessing planar subdivisions for point location queries, and construction of visibility representations and drawings of planar graphs. Most of these algorithms achieve optimal O(log n) running time using n = log n processors in the EREW PRAM model, n being the number of vertices. Key words. parallel algorithms, parallel computation, graph algorithms, planar st-graphs, transitive closure, reachability, planar point location, computational geometry, fractional cascading, graph drawing, visibility AMS(MOS) subject classi cations. 68E05, 68C05, 68C25 1. Introduction. Planar st-graphs

We show how to decompose efficiently in parallel any graph into a number, ~ fl, of outerplanar subgraphs (called hammocks) satisfying certain separator properties. Our work combines and extends the sequential hammock decomposition technique introduced by G. Frederickson and the parallel ear decomposition technique, thus we call it the hammock-on-ears decomposition. We mention that hammock-on-ears decomposition also draws from techniques in computational geometry and that an embedding of the graph does not need to be provided with the input. We achieve this decomposition in O(logn log log n) time using O(n + m) CREW PRAM processors, for an n-vertex, m-edge graph or digraph. The hammock-on-ears decomposition implies a general framework for solving graph problems efficiently. Its value is demonstrated by a variety of applications on a significant class of (di)graphs, namely that of sparse (di)graphs. This class consists of all (di)graphs which have a ~ fl between 1 and \Theta(n...

We show that testing reachability in a planar DAG can be performed in parallel in O(log n log n) time (O(log n) time using randomization) using O(n) processors. In general we give a paradigm for contracting a planar DAG to a point and then expanding it back. This paradigm is developed from a property of planar directed graphs we refer to as the Poincar'e index formula. Using this new paradigm we then "overlay" our application in a fashion similar to parallel tree contraction [MR85, MR89]. We also discuss some of the changes needed to extend the reduction procedure to work for general planar digraphs. Using the strongly-connected components algorithm of Kao [Kao91] we can compute multiple-source reachability for general planar digraphs in O(log 3 n) time using O(n) processors. This improves the results of Kao and Klein [KK90] who showed that this problem could be performed in O(log 5 n) time using O(n) processors. This work represents initial results of an effort to develop effi...

. This paper shows that for a strongly connected planar directed graph of size n, a depth-first search tree rooted a specified vertex can be computed in O(log 5 n) time using n= log n processors. Previously, for planar directed graphs that may not be strongly connected, the best depth-first search algorithm runs in O(log 10 n) time using n processors. Both algorithms run on a parallel random access machine that allows concurrent reads and concurrent writes in its shared memory, and in case of a write conflict, permits an arbitrary processor to succeed. Key words. linear-processor NC algorithms, graph separators, depth-first search, planar directed graphs, strong connectivity, bubble graphs, s-t graphs AMS(MOS) subject classification. 68Q10, 05C99 1. Introduction. Depth-first search is one of the most useful tools in graph theory [32], [4]. The depth-first search problem is the following: given a graph and a distinguished vertex, construct a tree that corresponds to performing de...

. A planar monotone circuit (PMC) is a Boolean circuit that can be embedded in the plane and that contains only AND and OR gates. Goldschlager, Cook & Dymond and others have developed NC 2 algorithms to evaluate a special layered form of a PMC. These algorithms require a large number of processors(\Omega\Gamma n 6 ), where n is the size of the input circuit). Yang, and more recently, Delcher & Kosaraju have given NC algorithms for the general planar monotone circuit value problem. These algorithms use at least as many processors as the algorithms for the layered case. This paper gives an efficient parallel algorithm that evaluates a general PMC of size n in polylog time using only a linear number of processors on an EREW PRAM. This parallel algorithm is the best possible to within a polylog factor, and is a substantial improvement over the earlier algorithms for the problem. The algorithm uses several novel techniques to perform the evaluation, including the use of the dual of the...

We give a randomized (Las-Vegas) parallel algorithm for computing strongly connected components of a graph with n vertices and m edges. The runtime is dominated by O(log 2 n) parallel reachability queries; i.e. O(log 2 n) calls to a subroutine that computes the descendants of a given vertex in a given digraph. Our algorithm also topologically sorts the strongly connected components. Using Ullman and Yannakakis’s [21] techniques for the reachability subroutine gives our algorithm runtime Õ(t) using mn/t2 processors for any (n 2 /m) 1/3 ≤ t ≤ n. On sparse graphs, this improves the number of processors needed to compute strongly connected components and topological sort within time n 1/3 ≤ t ≤ n from the previously best known (n/t) 3 [19] to (n/t) 2. 1 Introduction and main results Breadth-first and depth-first search have many applications in the analysis of directed graphs. Breadth-first search can be used to compute the vertices that are reachable from a given vertex and directed spanning trees. Depth-first search can: solve these problems, determine if a graph is acyclic, topologically sort an acyclic graph and compute strongly connected components (SCCs) [20]. Efforts

We present efficient parallel algorithms for solving two graph layout problems: Find a F'ary Embedding on a grid and construct a rectangular dual for planar graphs. The algorithm for the first problem takes O(log n log n) time with O(n) processors on a PRAM. The algorithm for the second problem takes O(log 2 n) time with O(n) processors. 1. Introduction We present efficient parallel algorithms for solving two graph layout problems: Find a F'ary embedding on a grid and construct a rectangular dual for planar triangular graphs. A F'ary embedding of a planar graph G is a plane embedding in which edges are represented by non-intersecting straight line segments joining their end vertices. It is known every planar graph has a F'ary embedding [5, 17]. Such embeddings can be useful in practice to visualize planar graphs on a graphic screen. Most initial F'ary embedding algorithms require high precision real arithmetic and the vertices tend to bunch together and thus making them useless ...

We give a randomized (Las-Vegas) parallel algorithm for computing strongly connected components of a graph with n vertices and m edges. The runtime is dominated by O(log 2 n) multi-source parallel reachability queries; i.e. O(log 2 n) calls to a subroutine that computes the union of the descendants of a given set of vertices in a given digraph. Our algorithm also topologically sorts the strongly connected components. Using Ullman and Yannakakis’s [23] techniques for the reachability subroutine gives our algorithm runtime Õ(t) using mn/t 2 processors for any (n 2 /m) 1/3 ≤ t ≤ n. On sparse graphs, this improves the number of processors needed to compute strongly connected components and topological sort within time n 1/3 ≤ t ≤ n from the previously best known (n/t) 3 [21] to (n/t) 2.