E. Dahlhaus, J. Gustedt, R.M. McConnell, E#cient and practical modular decomposition, J. Algorithms 41 (2001) 360-387  Home/Search   Document Not in Database   Summary   Related Articles   Check

....with an adjacency matrix. Then, given X a k subset of V , it is possible to determine in constant time if X is a hyperedge of H. All underlying graphs are represented by means of adjacency lists: this is required to obtain the linear time complexity bounds of the algorithms described in [11, 2, 3]. Let V be the vertex set of the hypergraph H = V, E) and let W V : by EM(W ) we denote the set of all maximal modules of H W excluding a vertex v, for some v W ; then to each set X EM(W ) is associated the set Excl(X) of the vertices v in W such that X M(H, v) Moreover we assume ....

....if each 3 subset of vertices in P containing v and not contained in C v is a hyperedge or not. Clearly, the number of edges of (v, P ) is at most #(v, P ) Then, the decomposition tree of GH1 (v, P ) can be computed in time O(n #(v, P ) by using one of the linear time algorithms described in [11, 2, 3]. Note that, once the modular decomposition tree of (v, P ) is computed inside the call to Partition(v, P ) the cost of visiting such a tree to compute #H (v, P ) is O(n) In fact, this step requires to find internal nodes of such a tree which are labeled q complete. The children of such a ....

E. Dahlhaus, J. Gustedt, and R.M. McConnell. E#cient and practical modular decomposition. In Proceedings of the Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA'97, pages 26--35, 1997.

....tree are also known as modular decomposition algorithms. The first published O(n 2 ) algorithm for finding the substitution decomposition is due to Spinrad [72] and the data structure used in this algorithm was later simplified [58] The current best algorithms have execution time O(m n) [9,10,71]. Note that O(m n) is a better bound than O(n 2 ) if m = O(n) although often m = #(n 2 ) Algorithms for building a substitution decomposition tree generally require that the graph be given in transitive closure form. The best known transitive closure algorithms require O(n 2.5 ) time, ....

E. Dahlhaus, J. Gustedt, and R. M. McConnell. E#cient and practical modular decomposition. In Proceedings of the Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 26--35, New Orleans, Louisiana, 5--7 January 1997.

....tree are also known as modular decomposition algorithms. The first published O(n 2 ) algorithm for finding the substitution decomposition is due to Spinrad [72] and the data structure used in this algorithm was later simplified [58] The current best algorithms have execution time O(m n) [9,10,71]. Note that O(m n) is a better bound than O(n 2 ) if m = O(n) although often m = #(n 2 ) Algorithms for building a substitution decomposition tree generally require that the graph be given in transitive closure form. The best known transitive closure algorithms require O(n 2.5 ) time, ....

....section concludes with a discussion of the decomposition tree. 40 CHAPTER 4. GRAPH DECOMPOSITION 41 4.1.1 Modules The concept of a module is central in the decomposition of algebraic structures. The style of exposition here is based on the presentation for undirected graphs by Dahlhaus et al. [9]. Note that the results presented here are not new, being simply special cases of more general results for relational structures [55] The importance of modules is well established in algorithmic graph theory [71] since they form the natural building blocks in graph decomposition. First, a ....

E. Dahlhaus, J. Gustedt, and R. M. McConnell. E#cient and Practical Modular Decomposition. Technical Report 524/

....N(v) or N(v) and a label that is either standard or complemented that tells which of the two cases applies for v. We de ne the size of a mixed representation to be n m 0 , where n is the number of vertices, and m 0 is the sum of cardinalities of their associated lists. 13 De nition 4. 2 (Dahlhaus et al. 1997)) Let the bipartite complement of a directed bipartite graph (V 1 ; V 2 ; A) be the graph V 1 ; V 2 ; V 1 V 2 [ V 2 V 1 ) n A : 1) In a mixed bipartite representation, each vertex in V 1 (resp. V 2 ) has either a list of those members of V 2 (resp. V 1 ) that are ....

....are out neighbors, or else a list of those members of V 2 (resp. V 1 ) that are not out neighbors. This representation has many interesting properties which are beyond the scope of this paper. However, to make ecient algorithmic use of Lemma 5. 3, below, we use the following result: Theorem 2 (Dahlhaus et al. 1997)) Given a mixed representation of a directed graph or directed bipartite graph, nding the strongly connected components takes time linear in the size of the representation. 5 Modular Decomposition in O(n m (m;n) Time We now describe an algorithm that is based on Algorithm 3 and that runs in ....

Dahlhaus, E., Gustedt, J., and McConnell, R. M. (1997). E- cient and practical modular decomposition. Proceedings of the eighth annual ACMSIAM symposium on discrete algorithms, 8:26-35.

....of a sequence of classes, one might get a sublinear time algorithms for other members of the classes. Theorem 1 is essential to the O(n m log n) algorithm for modular decomposition of digraphs we give here and for the linear time modular decomposition algorithm for undirected graphs we give in Dahlhaus et al. 1997, 1999) In the present paper, we also extend this to outward equivalence classes in the following way: Theorem 2 The modular decomposition of an undirected member F of the outward equivalence class of a given digraph G can be found in time O( n(G) m(G) log n(G) Modular decomposition ....

Dahlhaus, E., Gustedt, J., and McConnell, R. M. (1997). Ecient and practical modular decomposition. Proceedings of the eighth annual ACM-SIAM symposium on discrete algorithms, 8:2635.

....algorithms are lengthy and quite challenging to understand. We derive a conceptually simpler linear time sequential algorithm than was previously available. In addition, we give O(n m (m;n) variant that we believe is simple enough to be characterized as practical. We sketched this version in Dahlhaus et al. 1997). 2 Preliminaries This section gives the basic de nitions of the objects and algorithmic features that we are talking about. It is split into two parts. The rst part introduces the basic notation and facts about modular decomposition. The second describes some essential data structures that are ....

....either N(v) or N(v) and a label that is either standard or complemented that tells which of the two cases applies for v. We de ne the size of a mixed representation to be n m 0 , where n is the number of vertices, and m 0 is the sum of cardinalities of their associated lists. De nition 2. 6 (Dahlhaus et al. 1997)) Let the bipartite complement of a directed bipartite graph (V 1 ; V 2 ; A) be the graph V 1 ; V 2 ; V 1 V 2 [ V 2 V 1 ) n A : 1) In a mixed bipartite representation, each vertex in V 1 (resp. V 2 ) has either a list of those members of V 2 (resp. V 1 ) that are ....

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Dahlhaus, E., Gustedt, J., and McConnell, R. M. (1997). Ecient and practical modular decomposition. Proceedings of the eighth annual ACM-SIAM symposium on discrete algorithms, 8:2635.

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E. Dahlhaus, J. Gustedt, R.M. McConnell, E#cient and practical modular decomposition, J. Algorithms 41 (2001) 360-387

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