Results 1  10
of
24
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 ..."
Abstract

Cited by 88 (2 self)
 Add to MetaCart
(Show Context)
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 ..."
Abstract

Cited by 37 (4 self)
 Add to MetaCart
(Show Context)
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
Planarity Testing in Parallel
, 1994
"... We present a parallel algorithm based on open ear decomposition to construct an embedding of a graph onto the plane or report that the graph is nonplanar. Our parallel algorithm runs on a CRCW PRAM in logarithmic time with a number of processors bounded by that needed for finding connected component ..."
Abstract

Cited by 33 (6 self)
 Add to MetaCart
We present a parallel algorithm based on open ear decomposition to construct an embedding of a graph onto the plane or report that the graph is nonplanar. Our parallel algorithm runs on a CRCW PRAM in logarithmic time with a number of processors bounded by that needed for finding connected components in a graph and for performing bucket sort.
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 ..."
Abstract

Cited by 28 (0 self)
 Add to MetaCart
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 ..."
Abstract

Cited by 20 (3 self)
 Add to MetaCart
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 ..."
Abstract

Cited by 10 (2 self)
 Add to MetaCart
(Show Context)
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.
A linear work, O(n^1/6) time, parallel algorithm for solving planar Laplacians
"... We present a linear work parallel iterative algorithm for solving linear systems involving Laplacians of planar graphs. In particular, if Ax = b, where A is the Laplacian of any planar graph with n nodes, the algorithm produces a vector ¯x such that x − ¯xA ≤ ɛ, in O(n 1/6+c log(1/ɛ)) parallel t ..."
Abstract

Cited by 10 (3 self)
 Add to MetaCart
(Show Context)
We present a linear work parallel iterative algorithm for solving linear systems involving Laplacians of planar graphs. In particular, if Ax = b, where A is the Laplacian of any planar graph with n nodes, the algorithm produces a vector ¯x such that x − ¯xA ≤ ɛ, in O(n 1/6+c log(1/ɛ)) parallel time, doing O(n log(1/ɛ)) work, where c is any positive constant. One of the key ingredients of the solver, is an O(nk log 2 k) work, O(k log n) time, parallel algorithm for decomposing any embedded planar graph into components of size O(k) that are delimited by O(n / √ k) boundary edges. The result also applies to symmetric diagonally dominant matrices of planar structure.
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 ..."
Abstract

Cited by 7 (4 self)
 Add to MetaCart
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.