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26
Efficient parallel graph algorithms for coarse grained multicomputers and BSP (Extended Abstract)
 in Proc. 24th International Colloquium on Automata, Languages and Programming (ICALP'97
, 1997
"... In this paper, we present deterministic parallel algorithms for the coarse grained multicomputer (CGM) and bulksynchronous parallel computer (BSP) models which solve the following well known graph problems: (1) list ranking, (2) Euler tour construction, (3) computing the connected components and s ..."
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Cited by 59 (23 self)
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In this paper, we present deterministic parallel algorithms for the coarse grained multicomputer (CGM) and bulksynchronous parallel computer (BSP) models which solve the following well known graph problems: (1) list ranking, (2) Euler tour construction, (3) computing the connected components and spanning forest, (4) lowest common ancestor preprocessing, (5) tree contraction and expression tree evaluation, (6) computing an ear decomposition or open ear decomposition, (7) 2edge connectivity and biconnectivity (testing and component computation), and (8) cordal graph recognition (finding a perfect elimination ordering). The algorithms for Problems 17 require O(log p) communication rounds and linear sequential work per round. Our results for Problems 1 and 2, i.e.they are fully scalable, and for Problems hold for arbitrary ratios n p 38 it is assumed that n p,>0, which is true for all commercially
Fully dynamic algorithms for chordal graphs
, 2001
"... We present the first dynamic algorithm that maintains a clique tree representation of a chordal graph and supports the following operations: (1) query whether deleting or inserting an arbitrary edge preserves chordality, (2) delete or insert an arbitrary edge, provided it preserves chordality. We gi ..."
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Cited by 29 (1 self)
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We present the first dynamic algorithm that maintains a clique tree representation of a chordal graph and supports the following operations: (1) query whether deleting or inserting an arbitrary edge preserves chordality, (2) delete or insert an arbitrary edge, provided it preserves chordality. We give two implementations. In the first, each operation runs in O(n) time, where n is the number of vertices. In the second, an insertion query runs in O(log² n) time, an insertion in O(n) time, a deletion query in O(n) time, and a deletion in O(n log n) time. We also present a data structure that allows a deletion query to run in O ( p m) time in either implementation, where m is the current number of edges. Updating this data structure after a deletion or insertion requires O(m) time. We also present a very simple dynamic algorithm that supports each of the following operations in O(1) time on a general graph: (1) query whether the graph is split, (2) delete or insert an arbitrary edge.
Parallel Algorithms for Hierarchical Clustering and Applications to Split Decomposition and Parity Graph Recognition
 JOURNAL OF ALGORITHMS
, 1998
"... We present efficient (parallel) algorithms for two hierarchical clustering heuristics. We point out that these heuristics can also be applied to solve some algorithmic problems in graphs. This includes split decomposition. We show that efficient parallel split decomposition induces an efficient para ..."
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Cited by 23 (1 self)
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We present efficient (parallel) algorithms for two hierarchical clustering heuristics. We point out that these heuristics can also be applied to solve some algorithmic problems in graphs. This includes split decomposition. We show that efficient parallel split decomposition induces an efficient parallel parity graph recognition algorithm. This is a consequence of the result of [7] that parity graphs are exactly those graphs that can be split decomposed into cliques and bipartite graphs.
I/Oefficient algorithms for graphs of bounded treewidth
 In Proceedings of the 12th Annual ACMSIAM Symposium on Discrete Algorithms (SODA’2001
, 2001
"... We present an algorithm that takes O(sort(N)) I/Os 1 to compute a tree decomposition of width at most k, for any graph G of treewidth at most k and size N. Given such a tree decomposition, we use a dynamic programming framework to solve a wide variety of problems on G in O(N/(DB)) I/Os, including th ..."
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Cited by 15 (5 self)
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We present an algorithm that takes O(sort(N)) I/Os 1 to compute a tree decomposition of width at most k, for any graph G of treewidth at most k and size N. Given such a tree decomposition, we use a dynamic programming framework to solve a wide variety of problems on G in O(N/(DB)) I/Os, including the singlesource shortest path problem and a number of problems that are NPhard on general graphs. The tree decomposition can also be used to obtain an optimal separator decomposition of G. We use such a decomposition to perform depthfirst search in G in O(N/(DB)) I/Os. As important tools that are used in the tree decomposition algorithm, we introduce flippable DAGs and present an algorithm that computes a perfect elimination ordering of a ktree in O(sort(N)) I/Os. The second contribution of our paper, which is of independent interest, is a general and simple framework for obtaining I/Oefficient algorithms for a number of graph problems that can be solved using greedy algorithms in internal memory. We apply this framework in order to obtain an improved algorithm for finding a maximal matching and the first deterministic I/Oefficient algorithm for finding a maximal independent set of an arbitrary graph. Both algorithms take O(sort(V +E)) I/Os. The maximal matching algorithm is used in the tree decomposition algorithm.
An Efficient Parallel Algorithm for the Minimal Elimination Ordering (MEO) of an Arbitrary Graph
, 1989
"... . We design the first efficient parallel algorithm for computing the minimal elimination ordering (MEO) of an arbitrary graph. The algorithm works in O(log 3 n) parallel time and O(nm) processors on a CREW PRAM, for an nvertex, medge graph, and is optimal up to a polylogarithmic factor with resp ..."
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Cited by 11 (5 self)
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. We design the first efficient parallel algorithm for computing the minimal elimination ordering (MEO) of an arbitrary graph. The algorithm works in O(log 3 n) parallel time and O(nm) processors on a CREW PRAM, for an nvertex, medge graph, and is optimal up to a polylogarithmic factor with respect to the best sequential algorithm of Rose, Tarjan and Lueker ([RTL 76]). The MEO problem for arbitrary graphs arises in a number of combinatorial optimization problems, as well as in database applications, scheduling problems, and the An Extended Abstract has appeared in [DK 89]. y present address: Basser Department of Computer Science, University of Sydney, NSW 2006, Australia z Research partially supported by the Leibniz Center for Research in Computer Science, by the DFG Grant KA 673/41, and by the SERC Grant GRE 68297. sparse Gaussian elimination on symmetric matrices. It was believed before to be inherently sequential, and strongly resisting sublinear parallel time (subli...
Fast Parallel Algorithms for the Clique Separator Decomposition
, 1990
"... We give an efficient NC algorithm for finding a clique separator decomposition of an arbitrary graph, that is, a series of cliques whose removal disconnects the graph. This algorithm allows one to extend a large body of results which were originally formulated for chordal graphs to other classes of ..."
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Cited by 5 (1 self)
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We give an efficient NC algorithm for finding a clique separator decomposition of an arbitrary graph, that is, a series of cliques whose removal disconnects the graph. This algorithm allows one to extend a large body of results which were originally formulated for chordal graphs to other classes of graphs. Our algorithm is optimal to within a polylogarithmic factor of Tarjan's O(mn) time sequential algorithm. The decomposition can also be used to find NC algorithms for some optimization problems on special families of graphs, assuming these problems can be solved in NC for the prime graphs of the decomposition. These optimization problems include: finding a maximumweight clique, a minimum coloring, a maximumweight independent set, and a minimum fillin elimination order. We also give the first parallel algorithms for solving these problems by using the clique separator decomposition. Our maximumweight independent set algorithm applied to chordal graphs yields the most efficient know...
LinearTime Counting Algorithms for Independent Sets in Chordal Graphs
"... We study some counting and enumeration problems for chordal graphs, especially concerning independent sets. We first provide the following efficient algorithms for a chordal graph: (1) a lineartime algorithm for counting the number of independent sets; (2) a lineartime algorithm for counting the n ..."
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Cited by 4 (0 self)
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We study some counting and enumeration problems for chordal graphs, especially concerning independent sets. We first provide the following efficient algorithms for a chordal graph: (1) a lineartime algorithm for counting the number of independent sets; (2) a lineartime algorithm for counting the number of maximum independent sets; (3) a polynomialtime algorithm for counting the number of independent sets of a fixed size. With similar ideas, we show that enumeration (namely, listing) of the independent sets, the maximum independent sets, and the independent sets of a fixed size in a chordal graph can be done in constant amortized time per output. On the other hand, we prove that the following problems for a chordal graph are #Pcomplete: (1) counting the number of maximal independent sets; (2) counting the number of minimum maximal independent sets. With similar ideas, we also show that finding a minimum weighted maximal independent set in a chordal graph is NPhard, and even hard to approximate.
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
Recognizing and representing proper interval graphs in parallel using merging and sorting
, 2006
"... We present a parallel algorithm for recognizing and representing a proper interval graph in O(log 2 n) time with O(m + n) processors on the CREW PRAM, where m and n are the number of edges and vertices in the graph. The algorithm uses sorting to compute a weak linear ordering of the vertices, from w ..."
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Cited by 2 (0 self)
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We present a parallel algorithm for recognizing and representing a proper interval graph in O(log 2 n) time with O(m + n) processors on the CREW PRAM, where m and n are the number of edges and vertices in the graph. The algorithm uses sorting to compute a weak linear ordering of the vertices, from which an interval representation is easily obtained. It is simple, uses no complex data structures, and extends ideas from an optimal sequential algorithm for recognizing and representing a proper interval graph [DHH96]. 1
Laminar Structure of Ptolemaic Graphs and Its Applications
 In 16th Annual International Symposium on Algorithms and Computation (ISAAC 2005
, 2005
"... Ptolemaic graphs are graphs that satisfy the Ptolemaic inequality for any four vertices. The graph class coincides with the intersection of chordal graphs and distance hereditary graphs. The graph class can also be seen as a natural generalization of block graphs (and hence trees). In this paper, a ..."
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Cited by 2 (1 self)
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Ptolemaic graphs are graphs that satisfy the Ptolemaic inequality for any four vertices. The graph class coincides with the intersection of chordal graphs and distance hereditary graphs. The graph class can also be seen as a natural generalization of block graphs (and hence trees). In this paper, a new characterization of ptolemaic graphs is presented. It is a laminar structure of cliques, and leads us to a canonical tree representation. The tree representation gives a simple intersection model for ptolemaic graphs. The tree representation is constructed in linear time from a perfect elimination ordering obtained by the lexicographic breadth first search. Hence the recognition and the graph isomorphism for ptolemaic graphs can be solved in linear time. Using the tree representation, we also give an O(n) time algorithm for the Hamiltonian cycle problem. The Hamiltonian cycle problem is NPhard for chordal graphs, and an O(n + m) time algorithm is known for distance hereditary graphs.