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Optimal upward planarity testing of singlesource digraphs
 SIAM Journal on Computing
, 1998
"... Abstract. A digraph is upward planar if it has a planar drawing such that all the edges are monotone with respect to the vertical direction. Testing upward planarity and constructing upward planar drawings is important for displaying hierarchical network structures, which frequently arise in softwar ..."
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Cited by 34 (4 self)
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Abstract. A digraph is upward planar if it has a planar drawing such that all the edges are monotone with respect to the vertical direction. Testing upward planarity and constructing upward planar drawings is important for displaying hierarchical network structures, which frequently arise in software engineering, project management, and visual languages. In this paper we investigate upward planarity testing of singlesource digraphs; we provide a new combinatorial characterization of upward planarity and give an optimal algorithm for upward planarity testing. Our algorithm tests whether a singlesource digraph with n vertices is upward planar in O(n) sequential time, and in O(log n) time on a CRCW PRAM with n log log n / log n processors, using O(n) space. The algorithm also constructs an upward planar drawing if the test is successful. The previously known best result is an O(n2)time algorithm by Hutton and Lubiw [Proc. 2nd ACM–SIAM Symposium on Discrete Algorithms, SIAM, Philadelphia, 1991, pp. 203–211]. No efficient parallel algorithms for upward planarity testing were previously known.
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...
Finding the Closed Partition of a Planar Graph
 ALGORITHMICA
, 1994
"... We consider the problem of finding a closed partition in a directed graph. This problem has applications in concurrent probabilistic program verification. The best sequential algorithm known for this problem runs in O(mn) time where m is the number of directed edges and n is the number of vertices i ..."
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Cited by 1 (1 self)
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We consider the problem of finding a closed partition in a directed graph. This problem has applications in concurrent probabilistic program verification. The best sequential algorithm known for this problem runs in O(mn) time where m is the number of directed edges and n is the number of vertices in the given digraph. In this paper, we present a linear time sequential algorithm to solve the closed partition problem for planar digraphs that are compact. We then build on this algorithm to obtain an O(n 1:5 ) time sequential algorithm to solve the closed partition problem for a general planar digraph.