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Planar Orientations with Low OutDegree and Compaction of Adjacency Matrices
 Theoretical Computer Science
, 1991
"... We consider the problem of orienting the edges of a planar graph in such a way that the outdegree of each vertex is minimized. If, for each vertex v, the outdegree is at most d, then we say that such an orientation is dbounded. We prove the following results: ffl Each planar graph has a 5bounde ..."
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Cited by 34 (3 self)
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We consider the problem of orienting the edges of a planar graph in such a way that the outdegree of each vertex is minimized. If, for each vertex v, the outdegree is at most d, then we say that such an orientation is dbounded. We prove the following results: ffl Each planar graph has a 5bounded acyclic orientation, which can be constructed in linear time. ffl Each planar graph has a 3bounded orientation, which can be constructed in linear time. ffl A 6bounded acyclic orientation, and a 3bounded 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...
RealTime Minimum Vertex Cover For TwoTerminal SeriesParallel Graphs
 Proceedings of the Thirteenth Conference on Parallel and Distributed Computing and Systems
, 2000
"... 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 bottomup fashion, such that the solution to the problem is ..."
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Cited by 8 (8 self)
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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 bottomup 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 twoterminal seriesparallel graph. We then construct a realtime 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 ...
A new algorithm for the recognition of series parallel graphs
 CWI  Centrum voor Wiskunde en Informatica
, 1995
"... In this paper we develop a new lineartime 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 ..."
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Cited by 7 (0 self)
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In this paper we develop a new lineartime 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 breadthfirst 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.
Parallel Algorithms for Series Parallel Graphs
 Algorithmica
, 1996
"... 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(lo ..."
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Cited by 6 (4 self)
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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 treedecomposition 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.
Coarse Grained Parallel Algorithms for Detecting Convex Bipartite Graphs
 In 26th Workshop on GraphTheoretic Concepts in Computer Science (WG 2000), volume 1928 of Lecture Notes in Computer Science
, 1928
"... 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 r ..."
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Cited by 4 (3 self)
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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(...
Solving Problems on Recursively Constructed Graphs
"... 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 gr ..."
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Cited by 2 (0 self)
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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 kterminal graphs, trees, seriesparallel graphs, ktrees, partial ktrees, Halin graphs, bandwidthk graphs, pathwidthk graphs, treewidthk graphs, branchwidthk graphs, cographs, cliquewidthk graphs, kNLC graphs, kHB graphs, and rankwidthk graphs. The definition of each class is provided, after which some typical algorithms are applied to solve problems on instances of each class.
An NC Algorithm for Finding Minimum Weighted Completion Time Schedule on Series Parallel Graphs
, 1992
"... 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 algori ..."
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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...
SeriesParallel Automata and Short Regular Expressions
, 2009
"... 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 propert ..."
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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 seriesparallel 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.