Showing papers by "Laurent Viennot published in 2011"

Treewidth and Hyperbolicity of the Internet

Abstract: We study the measurement of the Internet according to two graph parameters: tree width and hyper bolicity. Both tell how far from a tree a graph is. They are computed from snapshots of the Internet released by CAIDA, DIMES, AQUALAB, UCLA, Rocket fuel and Strasbourg University, at the AS or at the router level. On the one hand, the tree width of the Internet appears to be quite large and being far from a tree with that respect, reflecting some high degree of connectivity. This proves the existence of a well linked core in the Internet. On the other hand, the hyper bolicity (as a graph parameter) appears to be very low, reflecting a tree-like structure with respect to distances. Additionally, we compute the tree width and hyper bolicity obtained for classical Internet models and compare with the snapshots.

Toward more localized local algorithms: removing assumptions concerning global knowledge

Abstract: Numerous sophisticated local algorithm were suggested in the literature for various fundamental problems. Notable examples are the MIS and (Δ+1)-coloring algorithms by Barenboim and Elkin [6], by Kuhn [22], and by Panconesi and Srinivasan [33], as well as the OΔ2-coloring algorithm by Linial [27]. Unfortunately, most known local algorithms (including, in particular, the aforementioned algorithms) are non-uniform, that is, they assume that all nodes know good estimations of one or more global parameters of the network, e.g., the maximum degree Δ or the number of nodes n.This paper provides a rather general method for transforming a non-uniform local algorithm into a uniform one. Furthermore, the resulting algorithm enjoys the same asymptotic running time as the original non-uniform algorithm. Our method applies to a wide family of both deterministic and randomized algorithms. Specifically, it applies to almost all of the state of the art non-uniform algorithms regarding MIS and Maximal Matching, as well as to many results concerning the coloring problem. (In particular, it applies to all aforementioned algorithms.)To obtain our transformations we introduce a new distributed tool called pruning algorithms, which we believe may be of independent interest.

Asymptotic modularity of some graph classes

Abstract: Modularity has been introduced as a quality measure for graph partitioning. It has received considerable attention in several disciplines, especially complex systems. In order to better understand this measure from a graph theoretical point of view, we study the modularity of a variety of graph classes. We first consider simple graph classes such as tori and hypercubes. We show that these regular graph families have asymptotic modularity 1 (that is the maximum possible). We extend this result to the general class of unit ball graphs of bounded growth metrics. Our most striking result concerns trees with bounded degree which also appear to have asymptotic modularity 1. This last result can be extended to graphs with constant average degree and to some power-law graphs.

Node-Disjoint multipath spanners and their relationship with fault-tolerant spanners

Abstract: Motivated by multipath routing, we introduce a multi-connected variant of spanners. For that purpose we introduce the p-multipath cost between two nodes u and v as the minimum weight of a collection of p internally vertex-disjoint paths between u and v. Given a weighted graph G, a subgraph H is a p-multipath s-spanner if for all u,v, the p-multipath cost between u and v in H is at most s times the p-multipath cost in G. The s factor is called the stretch. Building upon recent results on fault-tolerant spanners, we show how to build p-multipath spanners of constant stretch and of ${\tilde{O}}(n^{1+1/k})$ edges, for fixed parameters p and k, n being the number of nodes of the graph. Such spanners can be constructed by a distributed algorithm running in O(k) rounds. Additionally, we give an improved construction for the case p=k=2. Our spanner H has O(n3/2) edges and the p-multipath cost in H between any two node is at most twice the corresponding one in G plus O(W), W being the maximum edge weight.

Clustering de métrique et clustering de graphe

Abstract: Ce papier s'interesse aux liens entre deux concepts : d'une part le clustering de graphes, mesure par la modularite de Newman, et d'autre part le clustering d'un espace metrique, mesure par la somme des carres des distances au barycentre des clusters. Nous montrons qu'en passant de l'espace a un graphe de facon ''naturelle'' (par boules unitaires) et en appliquant un algorithme ''naturel'' de clustering (par r-net), on obtient un clustering de modularite bornee inferieurement (en fonction de la dimension de grille de l'espace et pour certains rayons de r-net). Quelques simulations en espace euclidien viennent illustrer le propos en comparant deux algorithmes pour les deux mesures de qualite.

Node-Disjoint Multipath Spanners and their Relationship with Fault-Tolerant Spanners

Abstract: Motivated by multipath routing, we introduce a multi-connected variant of spanners. For that purpose we introduce the $p$-multipath cost between two nodes $u$ and $v$ as the minimum weight of a collection of $p$ internally vertex-disjoint paths between $u$ and $v$. Given a weighted graph $G$, a subgraph $H$ is a $p$-multipath $s$-spanner if for all $u,v$, the $p$-multipath cost between $u$ and $v$ in $H$ is at most $s$ times the $p$-multipath cost in $G$. The $s$ factor is called the stretch. Building upon recent results on fault-tolerant spanners, we show how to build $p$-multipath spanners of constant stretch and of $\tO(n^{1+1/k})$ edges, for fixed parameters $p$ and $k$, $n$ being the number of nodes of the graph. Such spanners can be constructed by a distributed algorithm running in $O(k)$ rounds. Additionally, we give an improved construction for the case $p=k=2$. Our spanner $H$ has $O(n^{3/2})$ edges and the $p$-multipath cost in $H$ between any two node is at most twice the corresponding one in $G$ plus $O(W)$, $W$ being the maximum edge weight.

Toward More Localized Local Algorithms: Removing Assumptions Concerning Global Knowledge (Extended Abstract)

Abstract: Numerous sophisticated local algorithm were suggested in the literature for various fundamental problems. Notable examples are the MIS and ( + 1)-coloring algorithms by