M. Habib, M. Morvan and J.-X. Rampon. On the calculation of transitive reductionclosure of orders, Discrete Mathematics 111 (1993) 289--303.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....equal to the product of the number of edges and vertices of D that are stored in p. 1 Introduction The problem of computing the transitive closure of a digraph was first considered in 1959 by B. Roy [15] and a variety of sequential algorithms to solve this problem have been proposed ever since [1, 2, 7, 8, 9, 14, 16, 19]. These sequential solutions, usually are based on the use of the adjacency matrix of the digraph, considered as a Boolean matrix or use the adjacency matrix in more directed terms as a problem representation [13] Parallel algorithms for this problem where presented by [10, 12] PRAM) 11] ....

....2 In the sequel, we evaluate the complexity of the method. Basically, the algorithm consists of at most 1 dlog pe parallel computations of a sequential transitive closure algorithm. We employ a sequential algorithm whose complexity is the product of the number of vertices and edges of the closure [9, 16]. Consider a worst case example, where D consists of a single path. Then D is a complete acyclic digraph. In this case, each processor j may compute the transitive closure D (S j ) of a digraph D(S j ) where jV (D(S j ) j = p and jE(D (S j ) j = n=p) n Gamma n=p) O(n =p) Since at ....

M. Habib, M. Morvan and J.-X. Rampon. On the calculation of transitive reductionclosure of orders, Discrete Mathematics 111 (1993) 289--303.

....in p. KEY WORDS Parallel algorithm, transitive closure, graph algorithm, CGM, BSP 1 Introduction The problem of computing the transitive closure of a digraph was first considered in 1959 by B. Roy [15] and a variety of sequential algorithms to solve this problem have been proposed ever since [1, 2, 7, 8, 9, 14, 16, 19]. These sequential solutions, usually are based on the use of the adjacency matrix of the digraph, considered as a Boolean matrix or use the adjacency matrix in more directed terms as a problem representation [13] Parallel algorithms for this problem where presented by [10, 12] PRAM) 11] ....

....In the sequel, we evaluate the complexity of the method. Basically, the algorithm consists of at most 1 dlog pe parallel computations of a sequential transitive closure algorithm. We employ a sequential algorithm whose complexity is the product of the number of vertices and edges of the closure [9, 16]. Consider a worst case example, where D consists of a single path. Then D is a complete acyclic digraph. In this case, each processor j may compute the transitive closure D (S j ) of a digraph D(S j ) where jV (D(S j ) j = p and jE(D (S j ) j = n=p) n n=p) O(n =p) Since at most ....

M. Habib, M. Morvan and J.-X. Rampon. On the calculation of transitive reduction-closure of orders, Discrete Mathematics 111 (1993) 289--303.

....y is reachable from x in Eg) R(L) corresponds to the graph of the re exive, transitive closure of the partial order of L. Transitive closure computation can be done via boolean matrix multiplication; the best known algorithm for computing the product of n n bit matrices works in O(n 2:376 ) [AHU74, HMR93]) Theorem 12. Let L p , L t be nite lattices. L t embeds L p i R(L t ) is (subgraph) homeomorphic to L p . Proof. Let be a homeomorphism from L p to R(L t ) If (x; y) 2 E p then a simple path p in R(L t ) exists, such that, p = p 1 p 2 : pn , p 1 = x) and pn = y) Thus, y) is ....

M. Habib, M. Morvan, and J.X. Rampon. On the calculation of transitive reduction-closure of orders. Discrete Mathematics, 111:289-303, 1993.

....this is not any union; indeed, it insures that each comparability of P is contained in one P 0 i . The transitive closure of the graph G representing the order P can be realized using the version of the algorithm of Goralcikova Koubek [79] presented by M. Habib, M. Morvan and J. X. Rampon in [HMR93, Mor91], whose time complexity, in the worst case, is O(j X j P y2X P x2TrSucc(y) j TcSucc(x) j) An adjacency matrix representation of the transitive closure costs O(n 2 ) in space and allows a constant time pairwise comparison. Even if in the worst case, our method seems no better than a ....

Michel Habib, Michel Morvan, and Jean-Xavier Rampon. On the calculation of transitive reduction-closure of orders. Discrete Mathematics, (111):289--303, 1993.

.... main result emphasizes the algorithmic applications of a well known representation theorem for distributive lattices [3, 4, 8] We also use part of the knowledge on algorithmic order theory slowly gathered on basic problems such as transitive (closure and reduction) calculations on acyclic graphs [5, 12, 10]. These results show in that example of distributive lattices that ecient algorithms are closely related to structural properties. This approach can be generalized to other classes of orders. An order P is said to be graded if all maximal chains between any two elements of P have the same length. ....

....An order P is said to be graded if all maximal chains between any two elements of P have the same length. It should be noticed that the covering graph of graded orders with a unique minimal element and unique maximal element, can be obtained in linear time using the rank function (see for example [10]) We denote by x y (resp. x y) the join (resp. meet) of x and y. Since we are dealing with algorithms on lattice, the lattices we consider are supposed to be nite. 2 Let L be a lattice. We denote the set of join irreducible elements of L by J(L) and the set of meet irreducible elements by ....

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M. Habib, M. Morvan, and J.-X. Rampon. On the calculation of transitive reduction-closure of orders. Discrete Mathematics, To appear 1993.

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