We consider a search problem on trees aiming to find a treasure that an adversary places at one of the nodes. The algorithm can query nodes and extract directional information from them. That is, each node holds a pointer, termed advice , to one of its neighbors. Ideally, this advice points to the neighbor that is closer to the treasure, however, with probability \(q\) this advice points to a uniformly random neighbor. Crucially, the advice is permanent , hence querying the same node again yields the same answer. Let \(\Delta\) denote the maximal degree. Roughly speaking, we show that the expected number of queries incurs a phase transition when \(q\) is about \(1/\sqrt{\Delta}\) . In a recent paper, at TALG’21, we showed that if \(q\) is above the threshold then the expected number of queries is polynomial in \(n\) . Here, we prove that below the threshold, the expected number of queries is \(\mathcal{O}(\sqrt{\Delta}\log\Delta\cdot\log^{2}n)\) , which is tight up to an \(\mathcal{O}(\log n)\) factor when \(\Delta\) is small. We further show that this factor can be reduced to \(\mathcal{O}(\log\log n)\) in the case of regular trees and assuming that \(q for sufficiently small \(c>0\) . In addition, we study the case that the treasure must be found with some given probability. We show that for every fixed \(\varepsilon,\delta>0\) , if \(q<1/\Delta^{\varepsilon}\) then there exists a search strategy that with probability \(1-\delta\) finds the treasure using \((\delta^{-1}\log n)^{O(\frac{1}{\varepsilon})}\) queries, whereas \((\delta^{-1}\log n)^{\Omega(\frac{1}{\varepsilon})}\) queries are necessary.

We consider a search problem on trees in which an agent starts at the root of a tree and aims to locate an adversarially placed treasure, by moving along the edges, while relying on local, partial information. Specifically, each node in the tree holds a pointer to one of its neighbors, termedadvice. A node is faulty with probabilityq. The advice at a non-faulty node points to the neighbor that is closer to the treasure, and the advice at a faulty node points to a uniformly random neighbor. Crucially, the advice ispermanent, in the sense that querying the same node again would yield the same answer. Let Δ denote the maximum degree. For the expected number of moves (edge traversals) until finding the treasure, we show that a phase transition occurs when thenoise parameterqis roughly 1 √Δ. Below the threshold, there exists an algorithm with expected number of movesO(D√Δ), whereDis the depth of the treasure, whereas above the threshold, every search algorithm has an expected number of moves, which is both exponential inDand polynomial in the number of nodes n. In contrast, if we require to find the treasure with probability at least 1 − δ, then for every fixed ɛ > 0, ifq< 1/Δɛ, then there exists a search strategy that with probability 1 − δ finds the treasure using (Δ−1D)O(1/ε)moves. Moreover, we show that (Δ−1D)Ω(1/ε)moves are necessary.

We consider a problem of searching for an unknown target vertex t in a (possibly edge-weighted) graph. Each vertex-query points to a vertex v and the response either admits that v is the target or provides any neighbor s of v that lies on a shortest path from v to t. This model has been introduced for trees by Onak and Parys [FOCS 2006] and for general graphs by Emamjomeh-Zadeh et al. [STOC 2016]. In the latter, the authors provide algorithms for the error-less case and for the independent noise model (where each query independently receives an erroneous answer with known probability p<1/2 and a correct one with probability 1-p). We study this problem both with adversarial errors and independent noise models. First, we show an algorithm that needs at most (log_2 n)/(1 - H(r)) queries in case of adversarial errors, where the adversary is bounded with its rate of errors by a known constant r<1/2. Our algorithm is in fact a simplification of previous work, and our refinement lies in invoking an amortization argument. We then show that our algorithm coupled with a Chernoff bound argument leads to a simpler algorithm for the independent noise model and has a query complexity that is both simpler and asymptotically better than the one of Emamjomeh-Zadeh et al. [STOC 2016]. Our approach has a wide range of applications. First, it improves and simplifies the Robust Interactive Learning framework proposed by Emamjomeh-Zadeh and Kempe [NIPS 2017]. Secondly, performing analogous analysis for edge-queries (where a query to an edge e returns its endpoint that is closer to the target) we actually recover (as a special case) a noisy binary search algorithm that is asymptotically optimal, matching the complexity of Feige et al. [SIAM J. Comput. 1994]. Thirdly, we improve and simplify upon an algorithm for searching of unbounded domains due to Aslam and Dhagat [STOC 1991].

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.

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. We study the effect of communication noise on the ability of groups that lack stable structures to efficiently solve this task. We present an impossibility result which strongly restricts reliable rumor spreading in such groups. Namely, we prove 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-a process which requires linear time in the population size. Our results imply that in order to achieve efficient rumor spread a system must exhibit either some degree of structural stability or, alternatively, some facet of the communication which is immune to noise. We then corroborate this claim by providing new analyses of experimental data regarding recruitment in Cataglyphis niger desert ants. Finally, in light of our theoretical results, we discuss strategies to overcome noise in other biological systems.

Let $X$ be a lazy random walk on a graph $G$. If $G$ is undirected, then the mixing time is upper bounded by the maximum hitting time of the graph. This fails for directed chains, as the biased random walk on the cycle $\mathbb{Z}_n$ shows. However, we establish that for Eulerian digraphs, the mixing time is $O(mn)$, where $m$ is the number of edges and $n$ is the number of vertices. In the reversible case, the mixing time is robust to the change of the laziness parameter. Surprisingly, in the directed setting the mixing time can be sensitive to such changes. We also study exploration and cover times for random walks on Eulerian digraphs and prove universal upper bounds in analogy to the undirected case.

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This paper considers the basic 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 (also termed Rumor Spreading). 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. In particular, the specification of who is a source and what is its initial input bit may be set by an adversary. We first design a general compiler which can essentially transform any self-stabilizing algorithm with a certain property (called “the bitwise-independence property”) that uses ℓ-bits messages to one that uses only log ¿-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 Õ(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 Õ(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.

This paper is motivated by the aspiration to identify the weakest computational models that allow for efficient, robust distributed computation. We focus on one of the most fundamental building-blocks in distributed computing, namely, Broadcast. In this problem, a unique source agent $s$ needs to disseminate a bit $b$ to the rest of the population. To account for unpredictability issues that may result from uncoordinated executions, we consider a self-stabilizing setting, in which a correct configuration must be reached eventually, despite processors starting the execution with arbitrary initial states (that do not violate the requirement for the existence of a unique source). Similarly to many works on broadcast, we consider a synchronous communication model on a complete anonymous network, in which in each round, each agent can extract information from two other agents, chosen uniformly at random. Our focus is on identifying the smallest message size that is required in order to achieve fast self-stabilizing broadcast. We first observe that with an extra bit added to the message-size and a small additive penalty to the running time, the self-stabilizing broadcast problem can be reduced to a self-stabilizing clock-synchronization problem, where agents aim to synchronize their clocks modulo some integer T. Our main technical contribution lies in solving the latter problem in poly-logarithmic time using only 3 bits per interaction. This allows for a self-stabilizing broadcast protocol that uses only 4 bits per interaction and converges in O log n time.