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Modular Decomposition and Transitive Orientation
, 1999
"... A module of an undirected graph is a set X of nodes such for each node x not in X, either every member of X is adjacent to x, or no member of X is adjacent to x. There is a canonical linearspace representation for the modules of a graph, called the modular decomposition. Closely related to modular ..."
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Cited by 88 (13 self)
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A module of an undirected graph is a set X of nodes such for each node x not in X, either every member of X is adjacent to x, or no member of X is adjacent to x. There is a canonical linearspace representation for the modules of a graph, called the modular decomposition. Closely related to modular decomposition is the transitive orientation problem, which is the problem of assigning a direction to each edge of a graph so that the resulting digraph is transitive. A graph is a comparability graph if such an assignment is possible. We give O(n +m) algorithms for modular decomposition and transitive orientation, where n and m are the number of vertices and edges of the graph. This gives linear time bounds for recognizing permutation graphs, maximum clique and minimum vertex coloring on comparability graphs, and other combinatorial problems on comparability graphs and their complements.
A Compact Data Structure and Parallel Algorithms for Permutation Graphs
 WG '95 21ST WORKSHOP ON GRAPHTHEORETIC CONCEPTS IN COMPUTER SCIENCE
, 1995
"... . Starting from a permutation of f0; : : : ; n \Gamma 1g we compute in parallel with a workload of O(n log n) a compact data structure of size O(n log n). This data structure allows to obtain the associated permutation graph and the transitive closure and reduction of the associated order of dimens ..."
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Cited by 2 (2 self)
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. Starting from a permutation of f0; : : : ; n \Gamma 1g we compute in parallel with a workload of O(n log n) a compact data structure of size O(n log n). This data structure allows to obtain the associated permutation graph and the transitive closure and reduction of the associated order of dimension 2 efficiently. The parallel algorithms obtained have a workload of O(m + n log n) where m is the number of edges of the permutation graph. They run in time O(log 2 n) on a CREW PRAM. 1 Introduction Permutation graphs are combinatorial objects that found a lot of attention in recent years. This interest led to many results under a structural point of view as well as algorithmically, see [1, 2, 6, 7, 8]. By definition permutation graphs have a compact encoding of size n, where n is the number of vertices. In sequential computing model, it is possible to pass from the graph to the permutation and vice versa with a workload of O(n 2 ). For parallel computing this has been open up to no...
Some Results on Ongoing Research on Parallel Implementation of Graph Algorithms
, 1997
"... In high performance computing, three recognized important points are usability, scalability and portability. No models seemed to satisfy these three steps till recently: a few proposed models try to fulfill the previous goals. Among them, the BSPlike CGM model seemed adapted to us to facilitate the ..."
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Cited by 2 (2 self)
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In high performance computing, three recognized important points are usability, scalability and portability. No models seemed to satisfy these three steps till recently: a few proposed models try to fulfill the previous goals. Among them, the BSPlike CGM model seemed adapted to us to facilitate the way between algorithms design and real implementations. Many algorithms have been designed but few implementations have been carried out to demonstrate the practical relevance of this model. In this article, we propose to test this model actually on an irregular problem. We present the results of implementations of permutation graph algorithms written in two different models: the PRAM and the BSPlike CGM model. These implementation have been made on a CM5 and a PC cluster. We compare the results of these implementations with the performances of sequential code for this problem. With a classical problem in gaph theory, we validate BSPlike CGM model: it is possible to write portable code o...
Feasibility, Portability, . . . Grained Graph Algorithms
, 2000
"... We study the relationship between the design and analysis of graph algorithms in the coarsed grained parallel models and the behavior of the resulting code on todays parallel machines and clusters. We conclude that the coarse grained multicomputer model (CGM) is well suited to design competitive al ..."
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We study the relationship between the design and analysis of graph algorithms in the coarsed grained parallel models and the behavior of the resulting code on todays parallel machines and clusters. We conclude that the coarse grained multicomputer model (CGM) is well suited to design competitive algorithms, and that it is thereby now possible to aim to develop portable, predictable and efficient parallel algorithms code for graph problems.
The Handling of Graphs on PC Clusters: A Coarse Grained Approach
, 2000
"... We study the relationship between the design and analysis of graph algorithms in the coarsed grained parallel models and the behavior of the resulting code on clusters. We conclude that the coarse grained multicomputer model (CGM) is well suited to design competitive algorithms, and that it is there ..."
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We study the relationship between the design and analysis of graph algorithms in the coarsed grained parallel models and the behavior of the resulting code on clusters. We conclude that the coarse grained multicomputer model (CGM) is well suited to design competitive algorithms, and that it is thereby now possible to aim to develop portable, predictable and efficient parallel code for graph problems on clusters.
O(m log n) Split Decomposition of Strongly Connected Graphs
"... Abstract. In the early 1980’s, Cunningham described a unique decomposition of a stronglyconnected graph. A linear time bound for finding it in the special case of an undirected graph has been given previously, but up until now, the best bound known for the general case has been O(n 3). We give an O ..."
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Abstract. In the early 1980’s, Cunningham described a unique decomposition of a stronglyconnected graph. A linear time bound for finding it in the special case of an undirected graph has been given previously, but up until now, the best bound known for the general case has been O(n 3). We give an O(m log n) bound.