Minimizing Message Size in Stochastic Communication Patterns: Fast Self-Stabilizing Protocols with 3 bits

This paper considers the basic $\mathcal{PULL}$ model of communication, in which in each round, each agent extracts information from few randomly chosen agents. We seek to identify the smallest amount of information revealed in each interaction (message size) that nevertheless allows for efficient and robust computations of fundamental information dissemination tasks. We focus on the Majority Bit Dissemination problem that considers a population of $n$ agents, with a designated subset of source agents. Each source agent holds an input bit and each agent holds an output bit. The goal is to let all agents converge their output bits on the most frequent input bit of the sources (the majority bit). Note that the particular case of a single source agent corresponds to the classical problem of Broadcast. We concentrate on the severe fault-tolerant context of self-stabilization, in which a correct configuration must be reached eventually, despite all agents starting the execution with arbitrary initial states. We first design a general compiler which can essentially transform any self-stabilizing algorithm with a certain property that uses $\ell$-bits messages to one that uses only $\log \ell$-bits messages, while paying only a small penalty in the running time. By applying this compiler recursively we then obtain a self-stabilizing Clock Synchronization protocol, in which agents synchronize their clocks modulo some given integer $T$, within $\tilde O(\log n\log T)$ rounds w.h.p., and using messages that contain $3$ bits only. We then employ the new Clock Synchronization tool to obtain a self-stabilizing Majority Bit Dissemination protocol which converges in $\tilde O(\log n)$ time, w.h.p., on every initial configuration, provided that the ratio of sources supporting the minority opinion is bounded away from half. Moreover, this protocol also uses only 3 bits per interaction.Comment: 28 pages, 4 figure

Streaming Communication Protocols

We define the Streaming Communication model that combines the main aspects of communication complexity and streaming. Input arrives as a stream, spread between several agents across a network. Each agent has a bounded memory, which can be updated upon receiving a new bit, or a message from another agent. We provide tight tradeoffs between the necessary resources, i.e. communication between agents and memory, for some of the canonical problems from communication complexity by proving a strong general lower bound technique. Second, we analyze the Approximate Matching problem and show that the complexity of this problem (i.e. the achievable approximation ratio) in the one-way variant of our model is strictly different both from the streaming complexity and the one-way communication complexity thereof

Searching a Tree with Permanently Noisy Advice

We consider a search problem on trees using unreliable guiding instructions. Specifically, an agent starts a search at the root of a tree aiming to find a treasure hidden at one of the nodes by an adversary. Each visited node holds information, called advice, regarding the most promising neighbor to continue the search. However, the memory holding this information may be unreliable. Modeling this scenario, we focus on a probabilistic setting. That is, the advice at a node is a pointer to one of its neighbors. With probability q each node is faulty, independently of other nodes, in which case its advice points at an arbitrary neighbor, chosen uniformly at random. Otherwise, the node is sound and points at the correct neighbor. Crucially, the advice is permanent, in the sense that querying a node several times would yield the same answer. We evaluate efficiency by two measures: The move complexity denotes the expected number of edge traversals, and the query complexity denotes the expected number of queries. Let Delta denote the maximal degree. Roughly speaking, the main message of this paper is that a phase transition occurs when the noise parameter q is roughly 1/sqrt{Delta}. More precisely, we prove that above the threshold, every search algorithm has query complexity (and move complexity) which is both exponential in the depth d of the treasure and polynomial in the number of nodes n. Conversely, below the threshold, there exists an algorithm with move complexity O(d sqrt{Delta}), and an algorithm with query complexity O(sqrt{Delta}log Delta log^2 n). Moreover, for the case of regular trees, we obtain an algorithm with query complexity O(sqrt{Delta}log n log log n). For q that is below but close to the threshold, the bound for the move complexity is tight, and the bounds for the query complexity are not far from the lower bound of Omega(sqrt{Delta}log_Delta n). In addition, we also consider a semi-adversarial variant, in which an adversary chooses the direction of advice at faulty nodes. For this variant, the threshold for efficient moving algorithms happens when the noise parameter is roughly 1/Delta. Above this threshold a simple protocol that follows each advice with a fixed probability already achieves optimal move complexity

Limits for Rumor Spreading in Stochastic Populations

Biological systems can share and collectively process information to yield emergent effects, despite inherent noise in communication. While man-made systems often employ intricate structural solutions to overcome noise, the structure of many biological systems is more amorphous. It is not well understood how communication noise may affect the computational repertoire of such groups. To approach this question we consider the basic collective task of rumor spreading, in which information from few knowledgeable sources must reliably flow into the rest of the population. In order to study the effect of communication noise on the ability of groups that lack stable structures to efficiently solve this task, we consider a noisy version of the uniform PULL model. We prove a lower bound which implies that, in the presence of even moderate levels of noise that affect all facets of the communication, no scheme can significantly outperform the trivial one in which agents have to wait until directly interacting with the sources. Our results thus show an exponential separation between the uniform PUSH and PULL communication models in the presence of noise. Such separation may be interpreted as suggesting that, in order to achieve efficient rumor spreading, a system must exhibit either some degree of structural stability or, alternatively, some facet of the communication which is immune to noise. We corroborate our theoretical findings with a new analysis of experimental data regarding recruitment in Cataglyphis Niger desert ants

Random Walks with Multiple Step Lengths

International audienceIn nature, search processes that use randomly oriented steps of different lengths have been observed at both the microscopic and the macroscopic scales. Physicists have analyzed in depth two such processes on grid topologies: Intermittent Search, which uses two step lengths, and Lévy Walk, which uses many. Taking a computational perspective, this paper considers the number of distinct step lengths k as a complexity measure of the considered process. Our goal is to understand what is the optimal achievable time needed to cover the whole terrain, for any given value of k. Attention is restricted to dimension one, since on higher dimensions, the simple random walk already displays a quasi linear cover time. We say X is a k-intermittent search on the one dimensional n-node cycle if there exists a probability distribution $p = (pi) k i=1$ , and integers L1, L2,. .. , L k , such that on each step X makes a jump $±Li$ with probability pi, where the direction of the jump (+ or −) is chosen independently with probability 1/2. When performing a jump of length Li, the process consumes time Li, and is only considered to visit the last point reached by the jump (and not any other intermediate nodes). This assumption is consistent with biological evidence, in which entities do not search while moving ballistically. We provide upper and lower bounds for the cover time achievable by k-intermittent searches for any integer k. In particular, we prove that in order to reduce the cover time $Θ(n 2)$ of a simple random walk to linear in n up to logarithmic factors, roughly log n log log n step lengths are both necessary and sufficient, and we provide an example where the lengths form an exponential sequence. In addition, inspired by the notion of intermittent search, we introduce the Walk or Probe problem, which can be defined with respect to arbitrary graphs. Here, it is assumed that querying (probing) a node takes significantly more time than moving to a random neighbor. Hence, to efficiently probe all nodes, the goal is to balance the time spent walking randomly and the time spent probing. We provide preliminary results for connected graphs and regular graphs

Recherche et diffusion d'informations dans un environnement bruité, une perspective biologique.

This thesis is built around two series of works, each motivated by experiments on ants. We derive and analyse new models, that use computer science concepts and methodology, despite their biological roots and motivation.The first model studied in this thesis takes its inspiration in collaborative transport of food in the P. Longicornis species. We find that some key aspects of the process are well described by a graph search problem with noisy advice. The advice corresponds to characteristic short scent marks laid in front of the load in order to facilitate its navigation. In this thesis, we provide detailed analysis of the model on trees, which are relevant graph structures from a computer science standpoint. In particular our model may be viewed as a noisy extension of binary search to trees. Tight results in expectation and high probability are derived with matching upper and lower bounds. Interestingly, there is a sharp phase transition phenomenon for the expected runtime, but not when the algorithms are only required to succeed with high probability.The second model we work with was initially designed to capture information broadcast amongst desert ants. The model uses a stochastic meeting pattern and noise in the interactions, in a way that matches experimental data. Within this theoretical model, we present in this document a strong lower bound on the number of interactions required before information can be spread reliably. Experimentally, we see that the time required for the recruitment process of even few ants increases sharply with the group size, in accordance with our result. A theoretical consequence of the lower bound is a separation between the uniform noisy PUSH and PULL models of interaction. We also study a close variant of broadcast, without noise this time but under more strict convergence requirements and show that in this case, the problem can be solved efficiently, even with very limited exchange of information on each interaction.Cette thèse s’articule autour de deux séries de travaux motivés par des expériences sur des fourmis. Bien qu’inspirés par la biologie, les modèles que nous développons utilisent une terminologie et une approche typique de l’informatique théorique. Le premier modèle s’inspire du transport collaboratif de nourriture au sein de l’espèce P. Longicornis. Certains aspects fondamentaux du processus peuvent être décrits par un problème de recherche sur un graphe augmenté d'un certain type d’indications bruitées à chaque noeud. Ces indications représentent de courtes traces de phéromones déposées devant l’objet transporté afin de faciliter la navigation. Dans cette thèse, nous donnons une analyse complète du problème lorsque le graphe sous-jacent est un arbre, une hypothèse pertinente dans un cadre informatique. En particulier, notre modèle peut être vu comme une généralisation de la recherche binaire aux arbres en présence de bruit. De manière surprenante, les comportements des algorithmes optimaux dans ce cadre diffèrent suivant le type de garantie que l’on étudie : convergence en moyenne ou avec grande probabilité.Le deuxième modèle présenté dans cette thèse décrit la diffusion d’informations au sein de fourmis du désert. Dans notre modèle, les échanges ont lieu uniformément au hasard, et sont sujets à du bruit. Nous prouvons une borne inférieure sur le nombre d’interactions requis en fonction de la taille du groupe. La borne, de même que les hypothèses du modèle, semblent compatible avec les données expérimentales. Une conséquence théorique de ce résultat est une séparation dans ce cadre des variantes PUSH et PULL pour le problème du broadcast avec bruit. Nous étudions aussi une version du problème avec des garanties de convergence plus fortes. Dans ce cas, le problème peut être résolu efficacement, même si les messages échangés au cours de chaque interaction sont très limités