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Parallel transitive closure and point location in planar structures
 SIAM J. COMPUT
, 1991
"... Parallel algorithms for several graph and geometric problems are presented, including transitive closure and topological sorting in planar stgraphs, preprocessing planar subdivisions for point location queries, and construction of visibility representations and drawings of planar graphs. Most of th ..."
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Cited by 23 (11 self)
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Parallel algorithms for several graph and geometric problems are presented, including transitive closure and topological sorting in planar stgraphs, 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.
HammockonEars Decomposition: A Technique for the Efficient Parallel Solution of Shortest Paths and Other Problems
 Theoretical Computer Science
, 1996
"... 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 decom ..."
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Cited by 8 (5 self)
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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 hammockonears decomposition. We mention that hammockonears 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 nvertex, medge graph or digraph. The hammockonears 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...
A Contraction Procedure for Planar Directed Graphs
 Proc. 4th Annual ACM Symposium on Parallel Algorithms and Architectures
, 1992
"... 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 prop ..."
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Cited by 4 (0 self)
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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 stronglyconnected components algorithm of Kao [Kao91] we can compute multiplesource 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...
An Efficient Parallel Algorithm for the General Planar Monotone Circuit Value Problem
, 1996
"... . 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 ..."
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Cited by 4 (0 self)
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. 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...
Planar Strong Connectivity Helps in Parallel DepthFirst Search
 SIAM Journal on Computing
, 1992
"... . This paper shows that for a strongly connected planar directed graph of size n, a depthfirst 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 depthfirst searc ..."
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Cited by 3 (0 self)
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. This paper shows that for a strongly connected planar directed graph of size n, a depthfirst 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 depthfirst 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. linearprocessor NC algorithms, graph separators, depthfirst search, planar directed graphs, strong connectivity, bubble graphs, st graphs AMS(MOS) subject classification. 68Q10, 05C99 1. Introduction. Depthfirst search is one of the most useful tools in graph theory [32], [4]. The depthfirst search problem is the following: given a graph and a distinguished vertex, construct a tree that corresponds to performing de...
Finding Strongly Connected Components in Parallel using O(log 2 n) Reachability Queries
, 2007
"... We give a randomized (LasVegas) 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 giv ..."
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Cited by 2 (0 self)
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We give a randomized (LasVegas) 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 Breadthfirst and depthfirst search have many applications in the analysis of directed graphs. Breadthfirst search can be used to compute the vertices that are reachable from a given vertex and directed spanning trees. Depthfirst search can: solve these problems, determine if a graph is acyclic, topologically sort an acyclic graph and compute strongly connected components (SCCs) [20]. Efforts
Efficient Parallel Algorithms for Two Graph Layout Problems
, 1991
"... 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 t ..."
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Cited by 1 (1 self)
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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 nonintersecting 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 ...
Efficient Parallel Algorithms for Planar stGraphs 1
, 2002
"... Abstract. Planar stgraphs find applications in a number of areas. In this paper we present efficient parallel algorithms for solving several fundamental problems on planar stgraphs. The problems we consider include allpairs shortest paths in weighted planar stgraphs, singlesource shortest paths ..."
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Cited by 1 (0 self)
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Abstract. Planar stgraphs find applications in a number of areas. In this paper we present efficient parallel algorithms for solving several fundamental problems on planar stgraphs. The problems we consider include allpairs shortest paths in weighted planar stgraphs, singlesource shortest paths in weighted planar layered digraphs (which can be reduced to singlesource shortest paths in certain special planar stgraphs), and depthfirst search in planar stgraphs. Our parallel shortest path techniques exploit the specific geometric and graphic structures of planar stgraphs, and involve schemes for partitioning planar stgraphs into subgraphs in a way that ensures that the resulting path length matrices have a monotonicity property [1], [2]. The parallel algorithms we obtain are a considerable improvement over the previously best known solutions (when they are applied to these stgraph problems), and are in fact relatively simple. The parallel computational models we use are the CREW PRAM and EREW PRAM. Key Words.
Nonnumerical Algorithms and Problems—Computations
"... We give a randomized (LasVegas) 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) multisource parallel reachability queries; i.e. O(log 2 n) calls to a subroutine that computes the union of the descendants ..."
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We give a randomized (LasVegas) 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) multisource 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.