Shortest, Fastest, and Foremost Broadcast in Dynamic Networks

Highly dynamic networks rarely offer end-to-end connectivity at a given time. Yet, connectivity in these networks can be established over time and space, based on temporal analogues of multi-hop paths (also called {\em journeys}). Attempting to optimize the selection of the journeys in these networks naturally leads to the study of three cases: shortest (minimum hop), fastest (minimum duration), and foremost (earliest arrival) journeys. Efficient centralized algorithms exists to compute all cases, when the full knowledge of the network evolution is given. In this paper, we study the {\em distributed} counterparts of these problems, i.e. shortest, fastest, and foremost broadcast with termination detection (TDB), with minimal knowledge on the topology. We show that the feasibility of each of these problems requires distinct features on the evolution, through identifying three classes of dynamic graphs wherein the problems become gradually feasible: graphs in which the re-appearance of edges is {\em recurrent} (class R), {\em bounded-recurrent} (B), or {\em periodic} (P), together with specific knowledge that are respectively $n$ (the number of nodes), $\Delta$ (a bound on the recurrence time), and $p$ (the period). In these classes it is not required that all pairs of nodes get in contact -- only that the overall {\em footprint} of the graph is connected over time. Our results, together with the strict inclusion between $P$, $B$, and $R$, implies a feasibility order among the three variants of the problem, i.e. TDB[foremost] requires weaker assumptions on the topology dynamics than TDB[shortest], which itself requires less than TDB[fastest]. Reversely, these differences in feasibility imply that the computational powers of $R_n$, $B_\Delta$, and $P_p$ also form a strict hierarchy

Time-Varying Graphs and Dynamic Networks

The past few years have seen intensive research efforts carried out in some apparently unrelated areas of dynamic systems -- delay-tolerant networks, opportunistic-mobility networks, social networks -- obtaining closely related insights. Indeed, the concepts discovered in these investigations can be viewed as parts of the same conceptual universe; and the formal models proposed so far to express some specific concepts are components of a larger formal description of this universe. The main contribution of this paper is to integrate the vast collection of concepts, formalisms, and results found in the literature into a unified framework, which we call TVG (for time-varying graphs). Using this framework, it is possible to express directly in the same formalism not only the concepts common to all those different areas, but also those specific to each. Based on this definitional work, employing both existing results and original observations, we present a hierarchical classification of TVGs; each class corresponds to a significant property examined in the distributed computing literature. We then examine how TVGs can be used to study the evolution of network properties, and propose different techniques, depending on whether the indicators for these properties are a-temporal (as in the majority of existing studies) or temporal. Finally, we briefly discuss the introduction of randomness in TVGs.Comment: A short version appeared in ADHOC-NOW'11. This version is to be published in Internation Journal of Parallel, Emergent and Distributed System

Shortest, Fastest, and Foremost Broadcast in Dynamic Networks *

Highly dynamic networks rarely offer end-to-end connectivity at a given time. Connectivity in these networks can be established over time and space, based on temporal analogues of multi-hop paths (also called journeys). In a seminal work, Our results, together with the strict inclusion between P, B, and R, implies a feasibility order among the three variants of the problem, i.e. TDB[f oremost] requires weaker assumptions on the topology dynamics than TDB [shortest], which itself requires less than TDB[f astest]. Reversely, these differences in feasibility imply that the computational powers of Rn, B ∆ , and Pp also form a strict hierarchy

When Should You Wait Before Updating? - Toward a Robustness Refinement

Consider a dynamic network and a given distributed problem. At any point in time, there might exist several solutions that are equally good with respect to the problem specification, but that are different from an algorithmic perspective, because some could be easier to update than others when the network changes. In other words, one would prefer to have a solution that is more robust to topological changes in the network; and in this direction the best scenario would be that the solution remains correct despite the dynamic of the network. In [Arnaud Casteigts et al., 2020], the authors introduced a very strong robustness criterion: they required that for any removal of edges that maintain the network connected, the solution remains valid. They focus on the maximal independent set problem, and their approach consists in characterizing the graphs in which there exists a robust solution (the existential problem), or even stronger, where any solution is robust (the universal problem). As the robustness criteria is very demanding, few graphs have a robust solution, and even fewer are such that all of their solutions are robust. In this paper, we ask the following question: Can we have robustness for a larger class of networks, if we bound the number k of edge removals allowed? To answer this question, we consider three classic problems: maximal independent set, minimal dominating set and maximal matching. For the universal problem, the answers for the three cases are surprisingly different. For minimal dominating set, the class does not depend on the number of edges removed. For maximal matching, removing only one edge defines a robust class related to perfect matchings, but for all other bounds k, the class is the same as for an arbitrary number of edge removals. Finally, for maximal independent set, there is a strict hierarchy of classes: the class for the bound k is strictly larger than the class for bound k+1. For the robustness notion of [Arnaud Casteigts et al., 2020], no characterization of the class for the existential problem is known, only a polynomial-time recognition algorithm. We show that the situation is even worse for bounded k: even for k = 1, it is NP-hard to decide whether a graph has a robust maximal independent set

Design of a Scalable Path Service for the Internet

Despite the world-changing success of the Internet, shortcomings in its routing and forwarding system have become increasingly apparent. One symptom is an escalating tension between users and providers over the control of routing and forwarding of packets: providers understandably want to control use of their infrastructure, and users understandably want paths with sufficient quality-of-service (QoS) to improve the performance of their applications. As a result, users resort to various “hacks” such as sending traffic through intermediate end-systems, and the providers fight back with mechanisms to inspect and block such traffic. To enable users and providers to jointly control routing and forwarding policies, recent research has considered various architectural approaches in which provider- level route determination occurs separately from forwarding. With this separation, provider-level path computation and selection can be provided as a centralized service: users (or their applications) send path queries to a path service to obtain provider- level paths that meet their application-specific QoS requirements. At the same time, providers can control the use of their infrastructure by dictating how packets are forwarded across their network. The separation of routing and forwarding offers many advantages, but also brings a number of challenges such as scalability. In particular, the path service must respond to path queries in a timely manner and periodically collect topology information containing load-dependent (i.e., performance) routing information. We present a new design for a path service that makes use of expensive pre- computations, parallel on-demand computations on performance information, and caching of recently computed paths to achieve scalability. We demonstrate that, us- ing commodity hardware with a modest amount of resources, the path service can respond to path queries with acceptable latency under a realistic workload. The ser- vice can scale to arbitrarily large topologies through parallelism. Finally, we describe how to utilize the path service in the current Internet with existing Internet applica- tions

The Next 700 Impossibility Results in Time-Varying Graphs

International audienceWe consider highly dynamic distributed systems modelled by time-varying graphs (TVGs). We first address proof of impossibility results that often use informal arguments about convergence. We provide a general framework that formally proves the convergence of the sequence of executions of any deterministic algorithm over TVGs of any convergent sequence of TVGs. Next, we focus of the weakest class of long-lived TVGs, i.e., the class of TVGs where any node can communicate any other node infinitely often. We illustrate the relevance of our result by showing that no deterministic algorithm is able to compute various distributed covering structure on any TVG of this class. Namely, our impossibility results focus on the eventual footprint, the minimal dominating set and the maximal matching problems