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23
Parallel SymmetryBreaking in Sparse Graphs
 SIAM J. Disc. Math
, 1987
"... We describe efficient deterministic techniques for breaking symmetry in parallel. These techniques work well on rooted trees and graphs of constant degree or genus. Our primary technique allows us to 3color a rooted tree in O(lg n) time on an EREW PRAM using a linear number of processors. We use th ..."
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Cited by 71 (2 self)
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We describe efficient deterministic techniques for breaking symmetry in parallel. These techniques work well on rooted trees and graphs of constant degree or genus. Our primary technique allows us to 3color a rooted tree in O(lg n) time on an EREW PRAM using a linear number of processors. We use these techniques to construct fast linear processor algorithms for several problems, including (\Delta + 1)coloring constantdegree graphs and 5coloring planar graphs. We also prove lower bounds for 2coloring directed lists and for finding maximal independent sets in arbitrary graphs. 1 Introduction Some problems for which trivial sequential algorithms exist appear to be much harder to solve in a parallel framework. When converting a sequential algorithm to a parallel one, at each step of the parallel algorithm we have to choose a set of operations which may be executed in parallel. Often, we have to choose these operations from a large set A preliminary version of this paper appear...
PC trees and circularones arrangements
 Theoretical Computer Science
"... A 01 matrix has the consecutiveones property if its columns can be ordered so that the ones in every row are consecutive. It has the circularones property if its columns can be ordered so that, in every row, either the ones or the zeros are consecutive. PQ trees are used for representing all cons ..."
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Cited by 35 (4 self)
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A 01 matrix has the consecutiveones property if its columns can be ordered so that the ones in every row are consecutive. It has the circularones property if its columns can be ordered so that, in every row, either the ones or the zeros are consecutive. PQ trees are used for representing all consecutiveones orderings of the columns of a matrix that has the consecutiveones property. We give an analogous structure, called a PC tree, for representing all circularones orderings of the columns of a matrix that has the circularones property. No such representation has been given previously. In contrast to PQ trees, PC trees are unrooted. We obtain a much simpler algorithm for computing PQ trees that those that were previously available, by adding a zero column, x, to a matrix, computing the PC tree, and then picking the PC tree up by x to root it. 1
Efficient parallel algorithms for chordal graphs
"... We give the first efficient parallel algorithms for recognizing chordal graphs, finding a maximum clique and a maximum independent set in a chordal graph, finding an optimal coloring of a chordal graph, finding a breadthfirst search tree and a depthfirst search tree of a chordal graph, recognizing ..."
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Cited by 26 (0 self)
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We give the first efficient parallel algorithms for recognizing chordal graphs, finding a maximum clique and a maximum independent set in a chordal graph, finding an optimal coloring of a chordal graph, finding a breadthfirst search tree and a depthfirst search tree of a chordal graph, recognizing interval graphs, and testing interval graphs for isomorphism. The key to our results is an efficient parallel algorithm for finding a perfect elimination ordering.
A Simple Parallel Algorithm for the SingleSource Shortest Path Problem on Planar Digraphs
 OF LNCS
, 1996
"... We present a simple parallel algorithm for the singlesource shortest path problem in planar digraphs with nonnegative real edge weights. The algorithm runs on the EREW PRAM model of parallel computation in O((n 2ffl +n 1\Gammaffl ) log n) time, performing O(n 1+ffl log n) work for any 0 ! f ..."
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Cited by 18 (3 self)
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We present a simple parallel algorithm for the singlesource shortest path problem in planar digraphs with nonnegative real edge weights. The algorithm runs on the EREW PRAM model of parallel computation in O((n 2ffl +n 1\Gammaffl ) log n) time, performing O(n 1+ffl log n) work for any 0 ! ffl ! 1=2. The strength of the algorithm is its simplicity, making it easy to implement, and presumably quite efficient in practice. The algorithm improves upon the work of all previous algorithms. The work can be further reduced to O(n 1+ffl ), by plugging in a less practical, sequential planar shortest path algorithm. Our algorithm is based on a region decomposition of the input graph, and uses a wellknown parallel implementation of Dijkstra's algorithm.
PCtrees vs. PQtrees
 Lecture Notes in Computer Science
"... A data structure called PCtree is introduced as a generalization of PQtrees. PCtrees were originally introduced in a planarity test of Shih and Hsu [7] where they represent partial embeddings of planar graphs. PQtrees were invented by Booth and Lueker [1] to test the consecutive ones property in ..."
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Cited by 8 (2 self)
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A data structure called PCtree is introduced as a generalization of PQtrees. PCtrees were originally introduced in a planarity test of Shih and Hsu [7] where they represent partial embeddings of planar graphs. PQtrees were invented by Booth and Lueker [1] to test the consecutive ones property in matrices. The original implementation of the PQtree algorithms by Booth and Lueker using nine templates in each bottomup iteration is rather complicated. Also the complexity analysis is rather intricate. We give a very simple linear time PCtree algorithm with the following advantages: (1) it does not use any template; (2) at each iteration, it does all necessary treemodification operations in one batch and does not involve the nodebynode bottomup matching; (3) it can be used naturally to test the circular ones property in matrices; (4) the induced PQtree algorithm can considerably simplify Booth and Lueker’s modification of Lempel, Even and Cederbaum’s planarity test. 1.
Fully Dynamic Planarity Testing with Applications
"... The fully dynamic planarity testing problem consists of performing an arbitrary sequence of the following three kinds of operations on a planar graph G: (i) insert an edge if the resultant graph remains planar; (ii) delete an edge; and (iii) test whether an edge could be added to the graph without ..."
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Cited by 6 (0 self)
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The fully dynamic planarity testing problem consists of performing an arbitrary sequence of the following three kinds of operations on a planar graph G: (i) insert an edge if the resultant graph remains planar; (ii) delete an edge; and (iii) test whether an edge could be added to the graph without violating planarity. We show how to support each of the above operations in O(n2=3) time, where n is the number of vertices in the graph. The bound for tests and deletions is worstcase, while the bound for insertions is amortized. This is the first algorithm for this problem with sublinear running time, and it affirmatively answers a question posed in [11]. The same data structure has further applications in maintaining the biconnected and triconnected components of a dynamic planar graph. The time bounds are the same: O(n2=3) worstcase time per edge deletion, O(n2=3) amortized time per edge insertion, and O(n2=3) worstcase time to check whether two vertices are either biconnected or triconnected.
EFFICIENT PARALLEL ALGORITHMS FOR SHORTEST PATHS IN PLANAR DIGRAPHS
 BIT 32 (1992),215236
, 1992
"... Efficient parallel algorithms are presented, on the CREW PRAM model, for generating a succinct encoding of all pairs shortest path information in a directed planar graph G with realvalued edge costs but no negative cycles. We assume that a planar embedding of G is given, togetber with a set of q fa ..."
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Cited by 6 (3 self)
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Efficient parallel algorithms are presented, on the CREW PRAM model, for generating a succinct encoding of all pairs shortest path information in a directed planar graph G with realvalued edge costs but no negative cycles. We assume that a planar embedding of G is given, togetber with a set of q faces that cover all the vertices. Then our algorithm runs in O(log 2 n) time and employs O{nq + M(q)) processors (where M(t) is the number of processors required to multiply two t x t matrices in O(log t) time). Let us note here that whenever q < n then our processor bound is better than the best previous one (M(n)). O(log 2 n) time, nprocessor algorithms are presented for various subproblems, including that of generating all pairs shortest path information in a directed outerplanar graph. Our work is based on the fundamental hammockdecomposition technique ofG. Frederickson. We achieve this decomposition in O(log n log * n) parallel time by using O(n) processors. The hammockdecomposition seems to be a fundamental operation that may help in improving efficiency of many parallel (and sequential) graph algorithms.
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
, 2000
"... 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(log p) andO(log 2 p) communication rounds, respectively, and linear sequential work per round on a CGM with p processors and N=p local memory per processor, N=jGj. The algorithms assume that N p p for some xed>0, which is true for all commercially available multiprocessors. Our results imply BSP algorithms with O(log p) andO(log 2 p) supersteps, respectively, O(g log(p) N