Greedy routing and virtual coordinates for future networks

At the core of the Internet, routers are continuously struggling with ever-growing routing and forwarding tables. Although hardware advances do accommodate such a growth, we anticipate new requirements e.g. in data-oriented networking where each content piece has to be referenced instead of hosts, such that current approaches relying on global information will not be viable anymore, no matter the hardware progress. In this thesis, we investigate greedy routing methods that can achieve similar routing performance as today but use much less resources and which rely on local information only. To this end, we add specially crafted name spaces to the network in which virtual coordinates represent the addressable entities. Our scheme enables participating routers to make forwarding decisions using only neighbourhood information, as the overarching pseudo-geometric name space structure already organizes and incorporates "vicinity" at a global level. A first challenge to the application of greedy routing on virtual coordinates to future networks is that of "routing dead-ends" that are local minima due to the difficulty of consistent coordinates attribution. In this context, we propose a routing recovery scheme based on a multi-resolution embedding of the network in low-dimensional Euclidean spaces. The recovery is performed by routing greedily on a blurrier view of the network. The different network detail-levels are obtained though the embedding of clustering-levels of the graph. When compared with higher-dimensional embeddings of a given network, our method shows a significant diminution of routing failures for similar header and control-state sizes. A second challenge to the application of virtual coordinates and greedy routing to future networks is the support of "customer-provider" as well as "peering" relationships between participants, resulting in a differentiated services environment. Although an application of greedy routing within such a setting would combine two very common fields of today's networking literature, such a scenario has, surprisingly, not been studied so far. In this context we propose two approaches to address this scenario. In a first approach we implement a path-vector protocol similar to that of BGP on top of a greedy embedding of the network. This allows each node to build a spatial map associated with each of its neighbours indicating the accessible regions. Routing is then performed through the use of a decision-tree classifier taking the destination coordinates as input. When applied on a real-world dataset (the CAIDA 2004 AS graph) we demonstrate an up to 40% compression ratio of the routing control information at the network's core as well as a computationally efficient decision process comparable to methods such as binary trees and tries. In a second approach, we take inspiration from consensus-finding in social sciences and transform the three-dimensional distance data structure (where the third dimension encodes the service differentiation) into a two-dimensional matrix on which classical embedding tools can be used. This transformation is achieved by agreeing on a set of constraints on the inter-node distances guaranteeing an administratively-correct greedy routing. The computed distances are also enhanced to encode multipath support. We demonstrate a good greedy routing performance as well as an above 90% satisfaction of multipath constraints when relying on the non-embedded obtained distances on synthetic datasets. As various embeddings of the consensus distances do not fully exploit their multipath potential, the use of compression techniques such as transform coding to approximate the obtained distance allows for better routing performances

Improved embeddings of graph metrics into random trees

Over the past decade, numerous algorithms have been developed using the fact that the distances in any n-point metric (V, d) can be approximated to within O(log n) by distributions D over trees on the point set V [3, 10]. However, when the metric (V, d) is the shortest-path metric of an edge weighted graph G = (V, E), a natural requirement is to obtain such a result where the support of the distribution D is only over subtrees of G. For a long time, the best result satisfying this stronger requirement was a exp { √ log n log log n} distortion result of Alon et al. [1]. In a recent breakthrough, Elkin et al. [9] improved the distortion to O(log 2 n log log n). (The best lower bound on the distortion is Ω(log n), say, for the n-vertex grid [1].) In this paper, we give a construction that improves the distortion to O(log 2 n), improving slightly on the EEST construction. The main contribution of this paper is in the analysis: we use an algorithm which is similar to one used by EEST to give a distortion of O(log 3 n), but using a new probabilistic analysis, we eliminate one of the logarithmic factors. The ideas and techniques we use to obtain this logarithmic improvement seem orthogonal to those used earlier in such situations—e.g., Seymour’s decomposition scheme [4, 9] or the cutting procedures of CKR/FRT [5, 10], both which do not seem to give a guarantee of better than O(log 2 n log log n) for this problem. We hope that our ideas (perhaps in conjunction with some of these others) will ultimately lead to an O(log n) distortion embedding of graph metrics into distributions over their spanning trees.