Slobodan Mitrovic

On the Resiliency of Randomized Routing Against Multiple Edge Failures

We study the Static-Routing-Resiliency problem, motivated by routing on the Internet: Given a graph G = (V,E), a unique destination vertex d, and an integer constant c > 0, does there exist a static and destination-based routing scheme such that the correct delivery of packets from any source s to the destination d is guaranteed so long as (1) no more than c edges fail and (2) there exists a physical path from s to d? We embark upon a study of this problem by relating the edge-connectivity of a graph, i.e., the minimum number of edges whose deletion partitions G, to its resiliency. Following the success of randomized routing algorithms in dealing with a variety of problems (e.g., Valiant load balancing in the network design problem), we embark upon a study of randomized routing algorithms for the Static-Routing-Resiliency problem. For any k-connected graph, we show a surprisingly simple randomized algorithm that has expected number of hops O(|V|k) if at most k-1 edges fail, which reduces to O(|V|) if only a fraction t of the links fail (where t < 1 is a constant). Furthermore, our algorithm is deterministic if the routing does not encounter any failed link.

The quest for resilient (static) forwarding tables

Fast Reroute (FRR) and other forms of immediate failover have long been used to recover from certain classes of failures without invoking the network control plane. While the set of such techniques is growing, the level of resiliency to failures that this approach can provide is not adequately understood. We embark upon a systematic algorithmic study of the resiliency of immediate failover in a variety of models (with/without packet marking/duplication, etc.). We leverage our findings to devise new schemes for immediate failover and show, both theoretically and experimentally, that these outperform existing approaches.

Round compression for parallel matching algorithms

For over a decade now we have been witnessing the success of massive parallel computation (MPC) frameworks, such as MapReduce, Hadoop, Dryad, or Spark. One of the reasons for their success is the fact that these frameworks are able to accurately capture the nature of large-scale computation. In particular, compared to the classic distributed algorithms or PRAM models, these frameworks allow for much more local computation. The fundamental question that arises in this context is though: can we leverage this additional power to obtain even faster parallel algorithms? A prominent example here is the maximum matching problem—one of the most classic graph problems. It is well known that in the PRAM model one can compute a 2-approximate maximum matching in O(logn) rounds. However, the exact complexity of this problem in the MPC framework is still far from understood. Lattanzi et al. (SPAA 2011) showed that if each machine has n1+Ω(1) memory, this problem can also be solved 2-approximately in a constant number of rounds. These techniques, as well as the approaches developed in the follow up work, seem though to get stuck in a fundamental way at roughly O(logn) rounds once we enter the (at most) near-linear memory regime. It is thus entirely possible that in this regime, which captures in particular the case of sparse graph computations, the best MPC round complexity matches what one can already get in the PRAM model, without the need to take advantage of the extra local computation power. In this paper, we finally refute that possibility. That is, we break the above O(logn) round complexity bound even in the case of slightly sublinear memory per machine. In fact, our improvement here is almost exponential: we are able to deliver a (2+є)-approximate maximum matching, for any fixed constant є>0, in O((loglogn)2) rounds. To establish our result we need to deviate from the previous work in two important ways that are crucial for exploiting the power of the MPC model, as compared to the PRAM model. Firstly, we use vertex–based graph partitioning, instead of the edge–based approaches that were utilized so far. Secondly, we develop a technique of round compression. This technique enables one to take a (distributed) algorithm that computes an O(1)-approximation of maximum matching in O(logn) independent PRAM phases and implement a super-constant number of these phases in only a constant number of MPC rounds.