Graphs with optimal forwarding indices: What is the best throughput you can get with a given number of edges?

The (edge) forwarding index of a graph is the minimum, over all possible routings of all the demands, of the maximum load of an edge. This metric is of a great interest since it captures the notion of global congestion in a precise way: the lesser the forwarding-index, the lesser the congestion. In this paper, we study the following design question: Given a number e of edges and a number n of vertices, what is the least congested graph that we can construct? and what forwarding-index can we achieve? Our problem has some distant similarities with the well-known (∆,D) problem, and we sometimes build upon results obtained on it. The goal of this paper is to study how to build graphs with low forwarding indices and to understand how the number of edges impacts the forwarding index. We answer here these questions for different families of graphs: general graphs, graphs with bounded degree, sparse graphs with a small number of edges by providing constructions, most of them asymptotically optimal. Hence, our results allow to understand how the forwarding-index drops when edges are added to a graph and also to determine what is the best (i.e least congested) structure with e edges. Doing so, we partially answer the practical problem that initially motivated our work: If an operator wants to power only e links of its network, in order to reduce the energy consumption (or wiring cost) of its networks, what should be those links and what performance can be expected

Sur la complexité du routage OSPF

International audienceCe travail montre que dans un réseau (général) où le protocole de routage est OSPF avec la stratégie d'équilibrage de charge ECMP, le problème qui consiste à maximiser un flot simple d'une source vers un puits ne peut être approché à une constante près

Grid spanners with low forwarding index for energy efficient networks

International audienceA routing R of a connected graph G is a collection that contains simple paths connecting every ordered pair of vertices in G. The edge-forwarding index with respect to R (or simply the forwarding index with respect to $R) π(G, R)$ of G is the maximum number of paths in R passing through any edge of G. The forwarding index $π(G)$ of G is the minimum $π(G, R)$ over all routings R's of G. This parameter has been studied for different graph classes (1), (2), (3), (4). Motivated by energy efficiency, we look, for different numbers of edges, at the best spanning graphs of a square grid, namely those with a low forwarding index

A propos de la difficulté du routage égal par plus courts chemins

Les réseaux de communications sont devenus très complexes de nos jours. Cependant, depuis leur tout début avec ARPANET, les chercheurs et les ingénieurs ont conçu plusieurs protocoles aux divers équipements de ces réseaux afin de réduire cette complexité et d'atteindre la meilleure performance possible du réseau. OSPF ("Open Shortest Path First") est l'un des plus anciens protocoles qui ont tenu bon avec cette évolution. Il appartient à la sous-famille des protocoles de routage. Le but d'un protocol de routage est de rendre chaque routeur capable de décider, à chaque paquet recu, le prochain routeur voisin à qui ce paquet doit etre transmis. Quand le protocole de routage activé sur le réseau est OSPF, tout les paquets suivent des plus courts chemins pondérés, ou les poids (à voir comme des longueurs) sont fixés par l'administrateur réseau. Quand ces poids sont définis de manière à avoir plusieurs plus courts chemins pour une paire de noeuds, le routage dépendra de la règle qui est implémentée avec OSPF. Il y a plusieurs règles permettant de balancer le trafic entre les différents plus courts chemins, et l'une des plus connues est ECMP ("Equal Cost Multiple Path"): un noeud qui a plusieurs liens sortant sur le plus court chemin envers une destination d divisera le trafic qui lui arrive et qui est destiné à d de manière égale entre tous les chemins. Si le trafic nécessite de ne pas être partagé entre différents chemins, les poids doivent etre affectés de manière à ce que pour chaque paire de noeuds il y ait un unique plus court chemin. Afin de comprendre la difficulté du routage OSPF, nous avons regardé un problème assez simple, mais qui reste fondamental, où le but est de maximiser le débit quand une seule et unique paire de noeuds com- munique des données sur le réseau. A notre meilleure connaissance, ce problème a été prouvé NP-complet, utilisant une réduction au problème de la couverture d'un ensemble, et aucune approximation, avec garantie non trivial, n'a été proposée jusqu'à présent. Nous montrons, en utilisant une réduction différente que ce problème ne peut être approximée à un facteur constant et nous donnons une approximation qui dépend de la plus longue distance du graphe

Sur la complexité du routage OSPF

International audienceCe travail montre que dans un réseau (général) où le protocole de routage est OSPF avec la stratégie d'équilibrage de charge ECMP, le problème qui consiste à maximiser un flot simple d'une source vers un puits ne peut être approché à une constante près

On the Hardness of Equal Shortest Path Routing

International audienceIn telecommunication networks packets are carried from a source s to a destination t on a path that is determined by the underlying routing protocol. Most routing protocols belong to the class of shortest path routing protocols. In such protocols, the network operator assigns a length to each link. A packet going from s to t follows a shortest path according to these lengths. For better protection and efficiency, one wishes to use multiple (shortest) paths between two nodes. Therefore the routing protocol must determine how the traffic from s to t is distributed among the shortest paths. In the protocol called ospf-ecmp (for Open Shortest Path First -Equal Cost Multiple Path) the traffic incoming at every node is uniformly balanced on all outgoing links that are on shortest paths. In that context, the operator task is to determine the "best" link lengths, toward a goal such as maximizing the network throughput for given link capacities. In this work, we show that the problem of maximizing even a single commodity flow for the ospf- ecmp protocol cannot be approximated within any constant factor ratio. Besides this main theorem, we derive some positive results which include polynomial-time approximations and an exponential-time exact algorithm. We also prove that despite their weakness, our approximation and exact algorithms are, in a sense, the best possible

Optimisation de la consommation énergétique dans les réseaux sans fil fixes

International audienceNous étudions le problème d'optimisation énergétique dans les réseaux sans fil fixes dans le cas d'une faible demande de trafic par rapport à la capacité du réseau. Nous proposons un programme linéaire pour résoudre le problème, puis nous présentons une heuristique permettant de trouver rapidement une bonne solution

Graphs with optimal forwarding indices: What is the best throughput you can get with a given number of edges?

The (edge) forwarding index of a graph is the minimum, over all possible routings of all the demands, of the maximum load of an edge. This metric is of a great interest since it captures the notion of global congestion in a precise way: the lesser the forwarding-index, the lesser the congestion. In this paper, we study the following design question: Given a number e of edges and a number n of vertices, what is the least congested graph that we can construct? and what forwarding-index can we achieve? Our problem has some distant similarities with the well-known (∆,D) problem, and we sometimes build upon results obtained on it. The goal of this paper is to study how to build graphs with low forwarding indices and to understand how the number of edges impacts the forwarding index. We answer here these questions for different families of graphs: general graphs, graphs with bounded degree, sparse graphs with a small number of edges by providing constructions, most of them asymptotically optimal. Hence, our results allow to understand how the forwarding-index drops when edges are added to a graph and also to determine what is the best (i.e least congested) structure with e edges. Doing so, we partially answer the practical problem that initially motivated our work: If an operator wants to power only e links of its network, in order to reduce the energy consumption (or wiring cost) of its networks, what should be those links and what performance can be expected

Grid spanners with low forwarding index for energy efficient networks

International audienceA routing R of a connected graph G is a collection that contains simple paths connecting every ordered pair of vertices in G. The edge-forwarding index with respect to R (or simply the forwarding index with respect to R) π(G, R) of G is the maximum number of paths in R passing through any edge of G. The forwarding index π(G) of G is the minimum π(G, R) over all routings R's of G. This parameter has been studied for different graph classes [12], [1], [5], [4]. Motivated by energy efficiency, we look, for different numbers of edges, at the best spanning graphs of a square grid, namely those with a low forwarding index

simulating routing schemes on large-scale topologies

The expansion of the Internet routing system results in a number of research challenges, in particular, the Border Gateway Protocol (BGP) starts to show its limits a.o. in terms of the number of routing table entries it can dynamically process and control. Dynamic routing protocols showing better scaling properties are thus under investigation. However, because deploying under-development routing protocols on the Internet is not practicable at a large-scale (due to the size of the Internet topology), simulation is an unavoidable step to validate the properties of a newly proposed routing scheme. Unfortunately, the simulation of inter-domain routing protocols over large networks (order of tens of thousands of nodes) poses real challenges due to the limited memory and computational power that computers impose. This paper presents the Dynamic Routing Model simulator \drmsim which addresses the specific problem of large-scale simulations of (inter-domain) routing models on large networks. The motivation for developing a new simulator lies in the limitation of existing simulation tools in terms of the number of nodes they can handle and in the models they propose