Section: New Results

Dynamic graph algorithms

Routing: design, modeling and analysis

Multi-path routing

Participants : Luca Muscariello [ Orange Labs, Issy Les Moulineaux, France ] , Bruno Nardelli [ Rice University, Texas, USA ] , Diego Perino [ Orange Labs, Issy Les Moulineaux, France ] , Dario Rossi [ Telecom ParisTech, France ] .

Network traffic is increasing in size and is becoming more and more dynamic leading to unpredictable and highly variable traffic patters. Multi-path routing is considered a powerful approach to deal with the aforementioned traffic patterns, mostly for intra-domain TE, inter-AS path selection under the same ISP, routing in wireless networks and metropolitan access networks. Through an optimization framework in [22] we quantify the benefit in using multi-path routing and we analyze the role network topology and traffic matrix play on multi-path routing performance. Starting from the insights inferred from this analysis in [20] , [19] , we introduce MIRTO, which is a distributed multi-path routing protocol that jointly uses best path selection and flow control for optimality and stability. In [20] , [19] we introduce an analytical model in order to compare different routing schemes using fluid ordinary differential equations (ODEs). We model the sending rate of MIRTO and of other two recently proposed algorithms, TEXCP and TRUMP, and we compare their performance on the Abilene network topology with FIFO and Fair-Queuing scheduling. We show MIRTO consumes less network resource but suffers of longer convergence times because it relies on simpler feedbacks and it is not equation based. We also highlight that Fair-Queuing scheduling leads to sub-optimal global resource allocations because it imposes fairness at link level. We implement MIRTO, TEXCP and TRUMP in a software prototype in order to analyze their performance in real networks under real traffic conditions. Results reported in [21] confirm the trends already highlighted by our analytical analysis, and show that dynamic yet stable traffic engineering is not only feasible but expected with increasing interest by network operators.

Compact Routing Schemes for Dynamic Trees in the Fixed Port Model

Participant : Amos Korman.

This paper considers the routing problem in dynamic trees under the fixed-port model, in which an adversary chooses the port numbers assigned to each node. We present two routing schemes for dynamic trees that maintain labels of asymptotically optimal size using extremely low average message complexity (per node). Specifically, we first present a dynamic routing scheme that supports additions of both leaves and internal nodes, maintains asymptotically optimal labels and incurs only O(log²n/log²logn) average message complexity. This routing scheme is then extended to supports also deletions of nodes of degree at most 2. The extended scheme incurs O(log²n) average message complexity and still maintains asymptotically optimal labels.

We would like to point out that the best known routing scheme for dynamic trees that maintains asymptotically optimal labels in the fixed port model has very high average message complexity, namely, . Moreover, that scheme supports additions and removals of leaf nodes only.

Adversarial Queuing Theory with Setups

Participant : Mauricio Soto.

We look at routing and scheduling problems on Kelly type networks [3] where the injection process is under the control of an adversary. The novelty of the model we consider is that the adversary injects requests of distinct types. Resources are subject to switch-over delays or setups when they begin servicing a new request class. In this new setting, we study the behavior of sensible policies as introduced by Dai and Jennings  [37] .

We first show that the model is robust in the sense that under some mild conditions universal stability of work conserving packet routing protocols is preserved for natural variants of the underlying model. Also, the model's equivalence to so called token networks is established.

We adapt, to the multi-type request and setup setting, standard arguments for proving stability. Nevertheless, we provide counterexamples that show that for several reasonable adaptations of contention resolution protocols to the multi-type case, stability results do not carry over from the single-type scenario. This motivates us to explore fluid model based arguments that could be used for proving stability for a given network. Specifically we show analogues of results obtained by Gamarnik  [44] but in the multi-type request with setups scenario.

Online algorithms

New Bounds for the Controller Problem

Participants : Yuval Emek [ University of Tel Aviv, Israel ] , Amos Korman.

The (M, W) -controller, originally studied by Afek, Awerbuch, Plotkin, and Saks, is a basic distributed tool that provides an abstraction for managing the consumption of a global resource in a distributed dynamic network. The input to the controller arrives online in the form of requests presented at arbitrary nodes. A request presented at node u corresponds to the “desire” of some entity to consume one unit of the global resource at u and the controller should handle this request within finite time by either granting it with a permit or denying it. Initially, M permits (corresponding to M units of the global resource) are stored at a designated root node. Throughout the execution permits can be transported from place to place along the network's links so that they can be granted to requests presented at various nodes; when a permit is granted to some request, it is eliminated from the network. The fundamental rule of an (M, W) -controller is that a request should not be denied unless it is certain that at least M-W permits are eventually granted. The most efficient (M, W) -controller known to date has message complexity , where N is the number of nodes that ever existed in the network (the dynamic network may undergo node insertions and deletions).

In this paper we establish two new lower bounds on the message complexity of the controller problem. We first prove a simple lower bound stating that any (M, W) -controller must send messages. Second, for the important case when W is proportional to M (this is the common case in most applications), we use a surprising reduction from the (centralized) monotonic labeling problem to show that any (M, W) -controller must send (NlogN) messages. In fact, under a long lasting conjecture regarding the complexity of the monotonic labeling problem, this lower bound is improved to a tight (Nlog²N) . The proof of this lower bound requires that N = O(M) which turns out to be somewhat inevitable due to a new construction of an (M, M/2) -controller with message complexity O(Nlog²M) .

Online Computation with Advice

Participants : Yuval Emek [ University of Tel Aviv, Israel ] , Pierre Fraigniaud, Amos Korman, Adi Rosen [ CNRS LRI, University of Paris Sud, France ] .

In [11] , we consider a model for online computation in which the online algorithm receives, together with each request, some information regarding the future, referred to as advice . The advice provided to the online algorithm may allow an improvement in its performance, compared to the classical model of complete lack of information regarding the future. We are interested in the impact of such advice on the competitive ratio, and in particular, in the relation between the size b of the advice, measured in terms of bits of information per request, and the (improved) competitive ratio. Since b = 0 corresponds to the classical online model, and , where is the algorithm's action space, corresponds to the optimal (offline) one, our model spans a spectrum of settings ranging from classical online algorithms to offline ones. We illustrate the applicability of this model by considering two of the most extensively studied online problems, namely, metrical task systems (MTS) and the k -server problem. For MTS we establish tight (up to constant factors) upper and lower bounds on the competitive ratio of deterministic and randomized online algorithms with advice for any choice of 1b(logn) , where n is the number of states in the system: we prove that any randomized online algorithm for MTS has competitive ratio (log(n)/b) and we present a deterministic online algorithm for MTS with competitive ratio O(log(n)/b) . For the k -server problem we construct a deterministic online algorithm for general metric spaces with competitive ratio k^O(1/b) for any choice of (1)blogk .

In [10] , we consider the Work Function Algorithm for the k-server problem. We show that if the Work Function Algorithm is c-competitive, then it is also strictly (2c)-competitive. As a consequence of [Koutsoupias and Papadimitriou, JACM 1995] this also shows that the Work Function Algorithm is strictly (4k-2)-competitive.