Grooming

chapter VI.27International audienceState-of-the-art on traffic grooming with a design theory approac

On a combinatorial problem of antennas in radio astronomy

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Bus interconnection networks

AbstractIn bus interconnection networks every bus provides a communication medium between a set of processors. These networks are modeled by hypergraphs where vertices represent the processors and edges represent the buses. We survey the results obtained on the construction methods that connect a large number of processors in a bus network with given maximum processor degree Δ, maximum bus size r, and network diameter D. (In hypergraph terminology this problem is known as the (Δ,D, r)-hypergraph problem.)The problem for point-to-point networks (the case r = 2) has been extensively studied in the literature. As a result, several families of networks have been proposed. Some of these point-to-point networks can be used in the construction of bus networks. One approach is to consider the dual of the network. We survey some families of bus networks obtained in this manner. Another approach is to view the point-to-point networks as a special case of the bus networks and to generalize the known constructions to bus networks. We provide a summary of the tools developed in the theory of hypergraphs and directed hypergraphs to handle this approach

Traffic Grooming in Bidirectional WDM Ring Networks

We study the minimization of ADMs (Add-Drop Multiplexers) in optical WDM bidirectional rings considering symmetric shortest path routing and all-to-all unitary requests. We precisely formulate the problem in terms of graph decompositions, and state a general lower bound for all the values of the grooming factor $C$ and $N$, the size of the ring. We first study exhaustively the cases $C=1$, $C = 2$, and $C=3$, providing improved lower bounds, optimal constructions for several infinite families, as well as asymptotically optimal constructions and approximations. We then study the case $C>3$, focusing specifically on the case $C = k(k+1)/2$ for some $k \geq 1$. We give optimal decompositions for several congruence classes of $N$ using the existence of some combinatorial designs. We conclude with a comparison of the cost functions in unidirectional and bidirectional WDM rings

Le problème des ouvroirs (Hypergraph gossip problem)

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Hypergraph-designs

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Traffic Grooming in Bidirectional WDM Ring Networks

We study the minimization of ADMs (Add-Drop Multiplexers) in optical WDM bidirectional rings considering symmetric shortest path routing and all-to-all unitary requests. We precisely formulate the problem in terms of graph decompositions, and state a general lower bound for all the values of the grooming factor $C$ and $N$, the size of the ring. We first study exhaustively the cases $C=1$, $C = 2$, and $C=3$, providing improved lower bounds, optimal constructions for several infinite families, as well as asymptotically optimal constructions and approximations. We then study the case $C>3$, focusing specifically on the case $C = k(k+1)/2$ for some $k \geq 1$. We give optimal decompositions for several congruence classes of $N$ using the existence of some combinatorial designs. We conclude with a comparison of the cost functions in unidirectional and bidirectional WDM rings

Minimum number of wavelengths equals load in a DAG without internal cycle

International audienceLet P be a family of dipaths. The load of an arc is the number of dipaths containing this arc. Let (G,P) be the maximum of the load of all the arcs and let w(G,P) be the minimum number of wavelengths (colors) needed to color the family of dipathsP in such a way that two dipaths with the same wavelength are arc-disjoint. Let G be a DAG (Directed Acyclic Graph). An internal cycle is an oriented cycle such that all the vertices have at least one predecessor and one successor in G (said otherwise every cycle contain neither a source nor a sink of G). Here we prove that if G is a DAG without internal cycle, then for any family of dipaths P, w(G,P) = (G,P). On the opposite we give examples of DAGs with internal cycles such that the ratio between w(G,P) and (G,P) cannot be bounded. We also consider an apparently new class of DAGs, which is of interest in itself, those for which there is at most one dipath from a vertex to another. We call these digraphs UPP-DAGs. For these UPP-DAGs we show that the load is equal to the maximum size of a clique of the conflict graph. We show that if an UPP-DAG has only one internal cycle, then for any family of dipaths w(G,P) = 4 3(G,P) and we exhibit an UPP-DAG and a family of dipaths reaching the bound. We conjecture that the ratio between w(G,P) and (G,P) cannot be bounded

Minimizing SONET ADMs in unidirectional WDM rings with grooming ratio 3

We consider traffic grooming in WDM unidirectional rings with all-to-all uniform unitary traffic. We determine the minimum number of SONET/SDH add-drop multiplexers (ADMs) required when the grooming ratio is 3. In fact, using tools of design theory, we solve the equivalent edge partitioning problem: find a partition of the edges of the complete graph on n vertices (K_n) into subgraphs having at most 3 edges and in which the total number of vertices has to be minimized

Internet et la théorie des graphes

National audienceLa théorie des graphes constitue un domaine des mathématiques qui s'est développé au sein de disciplines diverses telles que la chimie (modélisation de structures), la biologie (génome), les sciences sociales (modélisation des relations) et le transport (réseaux routiers, électriques, etc.). Le cycle eulérien et le cycle hamiltonien Réseaux internet et graphes " petit-monde " Comment calculer un plus court chemin