A randomized distributed algorithm called RAES was introduced in [Becchetti et al., 2020] to extract a bounded-degree expander from a dense n-vertex expander graph G = (V, E). The algorithm relies on a simple threshold-based procedure. A key assumption in [Becchetti et al., 2020] is that the input graph G is static - i.e., both its vertex set V and edge set E remain unchanged throughout the process - while the analysis of raes in dynamic models is left as a major open question. In this work, we investigate the behavior of RAES under a dynamic graph model induced by a streaming node-churn process (also known as the sliding window model), where, at each discrete round, a new node joins the graph and the oldest node departs. This process yields a bounded-degree dynamic graph 𝒢 = {G_t = (V_t, E_t) : t ∈ ℕ} that captures essential characteristics of peer-to-peer networks - specifically, node churn and threshold on the number of connections each node can manage. We prove that every snapshot G_t in the dynamic graph sequence has good expansion properties with high probability. Furthermore, we leverage this property to establish a logarithmic upper bound on the completion time of the well-known PUSH and PULL rumor spreading protocols over the dynamic graph 𝒢.

We propose a protocol to solve Leader Election within weak communication models such as the beeping model or the stone-age model. Unlike most previous work, our algorithm operates on only six states, does not require unique identifiers, and assumes no prior knowledge of the network's size or topology, i.e., it is uniform. We show that under our protocol, the system converges almost surely to a configuration in which a single node is in a leader state. With high probability, this occurs in fewer than O(D2 log n) rounds, where D is the network diameter. We also show that this can be decreased to O(D log n) when an approximation of D is known. The main drawbacks of our approach are a [EQUATION] overhead in the running time compared to algorithms with stronger requirements, and the fact that nodes are unaware of when a single-leader configuration is reached. Nevertheless, the minimal assumptions and natural appeal of our solution make it particularly well-suited for implementation in the simplest distributed systems, especially biological ones.

We reconsider two well-known distributed randomized algorithms computing a maximal independent set, proposed in the seminal work of Luby (1986). We enhance these algorithms such that they become self-stabilizing without sacrificing their run-time, i.e., both stabilize in O(log⁡n) synchronous rounds with high probability on any n-node graph. The first algorithm gets along with three states, but needs to know an upper bound on the maximum degree. The second does not need any information about the graph, but uses a number of states that is linear in the node degree. Both algorithms use messages of logarithmic size.

International audience

We consider self-stabilizing algorithms to compute a Maximal Independent Set (MIS) in the extremely weak beeping communication model. The model consists of an anonymous network with synchronous rounds. In each round, each vertex can optionally transmit a signal to all its neighbors (beep). After the transmission of a signal, each vertex can only differentiate between no signal received, or at least one signal received. We assume that vertices have some knowledge about the topology of the network.

We study the minority-opinion dynamics over a fully-connected network of n nodes with binary opinions. Upon activation, a node receives a sample of opinions from a limited number of neighbors chosen uniformly at random. Each activated node then adopts the opinion that is least common within the received sample.

We consider self-stabilizing algorithms to compute a Maximal Independent Set (MIS) in the extremely weak beeping communication model. The model consists of an anonymous network with synchronous rounds. In each round, each vertex can optionally transmit a signal to all its neighbors (beep). After the transmission of a signal, each vertex can only differentiate between no signal received, or at least one signal received. We also consider an extension of this model where vertices can transmit signals through two distinguishable beeping channels. We assume that vertices have some knowledge about the topology of the network. We revisit the not self-stabilizing algorithm proposed by Jeavons, Scott, and Xu (2013), which computes an MIS in the beeping model. We enhance this algorithm to be self-stabilizing, and explore three different variants, which differ in the knowledge about the topology available to the vertices and the number of beeping channels. In the first variant, every vertex knows an upper bound on the maximum degree $Δ$ of the graph. For this case, we prove that the proposed self-stabilizing version maintains the same run-time as the original algorithm, i.e., it stabilizes after $O(\log n)$ rounds w.h.p. on any $n$-vertex graph. In the second variant, each vertex only knows an upper bound on its own degree. For this case, we prove that the algorithm stabilizes after $O(\log n\cdot \log \log n)$ rounds on any $n$-vertex graph, w.h.p. In the third variant, we consider the model with two beeping channels, where every vertex knows an upper bound of the maximum degree of the nodes in the $1$-hop neighborhood. We prove that this variant stabilizes w.h.p. after $O(\log n)$ rounds.

We study a simple random process that computes a maximal independent set (MIS) on a general n-vertex graph. Each vertex has a binary state, black or white, where black indicates inclusion into the MIS. The vertex states are arbitrary initially, and are updated in parallel: In each round, every vertex whose state is "inconsistent" with its neighbors, i.e., it is black and has a black neighbor, or it is white and all neighbors are white, changes its state with probability 1/2. The process stabilizes with probability 1 on any graph, and the resulting set of black vertices is an MIS. We show that the expected stabilization time is O(log n) on certain graph families, such as cliques and graphs of bounded arboricity.

Abstract We study the problem of designing a resilient data structure maintaining a tree under the Faulty-RAM model [Finocchi and Italiano, STOC’04] in which up to $$\delta $$ δ memory words can be corrupted by an adversary. Our data structure stores a rooted dynamic tree that can be updated via the addition of new leaves, requires linear size, and supports resilient (weighted) level ancestor queries, lowest common ancestor queries, and bottleneck vertex queries in $$O(\delta )$$ O ( δ ) worst-case time per operation.

International audience

We study expansion and information diffusion properties of dynamic networks, i.e., networks whose topologies evolve over time as nodes enter or leave the system and edges are continuously created or destroyed. In this scenario, we investigate flooding as a basic information diffusion mechanism. We are interested in models that are likely to result in sparse networks, i.e., in networks containing$O(n)$edges, with$n$the number of nodes that are present at any given time of interest, with a focus on models in which edges are created randomly according to simple probabilistic mechanisms, rather than according to carefully designed distributed algorithms. In this perspective, in all models we consider, upon joining the network, a node connects to$d=O(1)$random nodes currently in the system. On the other hand, an edge remains alive as long as both its endpoints are. For the case in which edges that fail (because one endpoint left the network) are not replaced, we show that, although the network is likely to contain$\Omega_{d}(n)$isolated nodes, flooding still informs a fraction$1-\exp(-\Omega(d))$of the nodes in time$\mathrm{O}(\log n)$with large, constant probability. Moreover, we are able to show, that at any given time, the graph exhibits a “large-set expansion” property. We further investigate models that exhibit edge regeneration, meaning that, whenever an edge$(v, w)$established by$v$fails because$w$leaves the network, it is replaced by a new random edge$(v, z)$. We show that models with edge regeneration result in evolving networks that, at any given time, are vertex expanders with high probability, so that flooding takes$\mathrm{O}(\log n)$time. The above results hold both for a simplfied streaming model of node churn and in a more realistic, continuous-time setting, in which the interval between two consecutive node arrivals follows a Poisson distribution, while nodes' lifetimes follow an exponential distribution. Previous work considered models in which either the vertex set is fixed or edges are established according to more or less sophisticated algorithms. Our motivation for studying models with simple and random edge creation mechanisms is to move one step further towards models that may eventually capture key aspects of the formation of social or peer-to-peer networks.

We study the problem of designing a \emph{resilient} data structure maintaining a tree under the Faulty-RAM model [Finocchi and Italiano, STOC'04] in which up to $\delta$ memory words can be corrupted by an adversary. Our data structure stores a rooted dynamic tree that can be updated via the addition of new leaves, requires linear size, and supports \emph{resilient} (weighted) level ancestor queries, lowest common ancestor queries, and bottleneck vertex queries in $O(\delta)$ worst-case time per operation.

We study parallel Load Balancing protocols for a client-server distributed model defined as follows. There is a set C of n clients and a set S of n servers where each client has (at most) a constant number d ≥ 1 of requests that must be assigned to some server. The client set and the server one are connected to each other via a fixed bipartite graph: the requests of client v can only be sent to the servers in its neighborhood N(v). The goal is to assign every client request so as to minimize the maximum load of the servers.