Space-Efficient Routing Tables for Almost All Networks and the Incompressibility Method

We use the incompressibility method based on Kolmogorov complexity to determine the total number of bits of routing information for almost all network topologies. In most models for routing, for almost all labeled graphs $\Theta (n^2)$ bits are necessary and sufficient for shortest path routing. By `almost all graphs' we mean the Kolmogorov random graphs which constitute a fraction of $1-1/n^c$ of all graphs on $n$ nodes, where $c > 0$ is an arbitrary fixed constant. There is a model for which the average case lower bound rises to $\Omega(n^2 \log n)$ and another model where the average case upper bound drops to $O(n \log^2 n)$. This clearly exposes the sensitivity of such bounds to the model under consideration. If paths have to be short, but need not be shortest (if the stretch factor may be larger than 1), then much less space is needed on average, even in the more demanding models. Full-information routing requires $\Theta (n^3)$ bits on average. For worst-case static networks we prove a $\Omega(n^2 \log n)$ lower bound for shortest path routing and all stretch factors $<2$ in some networks where free relabeling is not allowed.Comment: 19 pages, Latex, 1 table, 1 figure; SIAM J. Comput., To appea

Compact routing on the Internet AS-graph

Compact routing algorithms have been presented as candidates for scalable routing in the future Internet, achieving near-shortest path routing with considerably less forwarding state than the Border Gateway Protocol. Prior analyses have shown strong performance on power-law random graphs, but to better understand the applicability of compact routing algorithms in the context of the Internet, they must be evaluated against real- world data. To this end, we present the first systematic analysis of the behaviour of the Thorup-Zwick (TZ) and Brady-Cowen (BC) compact routing algorithms on snapshots of the Internet Autonomous System graph spanning a 14 year period. Both algorithms are shown to offer consistently strong performance on the AS graph, producing small forwarding tables with low stretch for all snapshots tested. We find that the average stretch for the TZ algorithm increases slightly as the AS graph has grown, while previous results on synthetic data suggested the opposite would be true. We also present new results to show which features of the algorithms contribute to their strong performance on these graphs

Compact Routing on Internet-Like Graphs

The Thorup-Zwick (TZ) routing scheme is the first generic stretch-3 routing scheme delivering a nearly optimal local memory upper bound. Using both direct analysis and simulation, we calculate the stretch distribution of this routing scheme on random graphs with power-law node degree distributions, $P_k \sim k^{-\gamma}$. We find that the average stretch is very low and virtually independent of $\gamma$. In particular, for the Internet interdomain graph, $\gamma \sim 2.1$, the average stretch is around 1.1, with up to 70% of paths being shortest. As the network grows, the average stretch slowly decreases. The routing table is very small, too. It is well below its upper bounds, and its size is around 50 records for $10^4$-node networks. Furthermore, we find that both the average shortest path length (i.e. distance) $\bar{d}$ and width of the distance distribution $\sigma$ observed in the real Internet inter-AS graph have values that are very close to the minimums of the average stretch in the $\bar{d}$- and $\sigma$-directions. This leads us to the discovery of a unique critical quasi-stationary point of the average TZ stretch as a function of $\bar{d}$ and $\sigma$. The Internet distance distribution is located in a close neighborhood of this point. This observation suggests the analytical structure of the average stretch function may be an indirect indicator of some hidden optimization criteria influencing the Internet's interdomain topology evolution.Comment: 29 pages, 16 figure

Scalable Routing Easy as PIE: a Practical Isometric Embedding Protocol (Technical Report)

We present PIE, a scalable routing scheme that achieves 100% packet delivery and low path stretch. It is easy to implement in a distributed fashion and works well when costs are associated to links. Scalability is achieved by using virtual coordinates in a space of concise dimensionality, which enables greedy routing based only on local knowledge. PIE is a general routing scheme, meaning that it works on any graph. We focus however on the Internet, where routing scalability is an urgent concern. We show analytically and by using simulation that the scheme scales extremely well on Internet-like graphs. In addition, its geometric nature allows it to react efficiently to topological changes or failures by finding new paths in the network at no cost, yielding better delivery ratios than standard algorithms. The proposed routing scheme needs an amount of memory polylogarithmic in the size of the network and requires only local communication between the nodes. Although each node constructs its coordinates and routes packets locally, the path stretch remains extremely low, even lower than for centralized or less scalable state-of-the-art algorithms: PIE always finds short paths and often enough finds the shortest paths.Comment: This work has been previously published in IEEE ICNP'11. The present document contains an additional optional mechanism, presented in Section III-D, to further improve performance by using route asymmetry. It also contains new simulation result

Crux: Locality-Preserving Distributed Services

Distributed systems achieve scalability by distributing load across many machines, but wide-area deployments can introduce worst-case response latencies proportional to the network's diameter. Crux is a general framework to build locality-preserving distributed systems, by transforming an existing scalable distributed algorithm A into a new locality-preserving algorithm ALP, which guarantees for any two clients u and v interacting via ALP that their interactions exhibit worst-case response latencies proportional to the network latency between u and v. Crux builds on compact-routing theory, but generalizes these techniques beyond routing applications. Crux provides weak and strong consistency flavors, and shows latency improvements for localized interactions in both cases, specifically up to several orders of magnitude for weakly-consistent Crux (from roughly 900ms to 1ms). We deployed on PlanetLab locality-preserving versions of a Memcached distributed cache, a Bamboo distributed hash table, and a Redis publish/subscribe. Our results indicate that Crux is effective and applicable to a variety of existing distributed algorithms.Comment: 11 figure