Grooming

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

Le problème des ouvroirs (Hypergraph gossip problem)

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Hypergraph-designs

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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

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

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

Traffic Grooming in Unidirectional WDM Ring Networks using Design Theory

International audienceWe address the problem of traffic grooming in WDM rings with all-to-all uniform unitary traffic. We want to minimize the total number of SONET add-drop multiplexers (ADMs) required. We show that this problem corresponds to a partition of the edges of the complete graph into subgraphs, where each subgraph has at most C edges (where C is the grooming ratio) and where the total number of vertices has to be minimized. Using tools of graph and design theory, we optimally solve the problem for practical values and infinite congruence classes of values for a given C , and thus improve and unify all the preceding results. We disprove a conjecture of [7] saying that the minimum number of ADMs cannot be achieved with the minimum number of wavelengths, and also another conjecture of [17]

Efficient Gathering in Radio Grids with Interference

International audienceWe study the problem of gathering information from the nodes of a radio network into a central destination node. A transmission can be received by a node if it is sent from a distance of at most d T and there is no interference from other transmissions. One transmission interferes with the reception of another transmission if the sender of the first transmission is at distance d I or less from the receiver of the second transmission. In this paper we study the case d T = 1 and d I > 1 for two-dimensional grid networks with unit time transmissions. We prove lower bounds on the number of rounds required for any two-dimensional grid and we describe protocols for n × n grids with n odd that are optimal for odd d I and near-optimal for even d I