arXiv subjects

Lucas Boczkowski

Publications and source records attributed to Lucas Boczkowski.

7 recordsLinked to original sources

arXiv2022

On Mappings on the Hypercube with Small Average Stretch

Let $A \subseteq \{0,1\}^n$ be a set of size $2^{n-1}$, and let $ϕ\colon \{0,1\}^{n-1} \to A$ be a bijection. We define the average stretch of $ϕ$ as ${\sf avgStretch}(ϕ) = {\mathbb E}[{\sf dist}(ϕ(x),ϕ(x'))]$, where the expectation is taken over uniformly random $x,x' \in \{0,1\}^{n-1}$ that differ in exactly one coordinate. In this paper we continue the line of research studying mappings on the discrete hypercube with small average stretch. We prove the following results. (1) For any set $A \subseteq \{0,1\}^n$ of density $1/2$ there exists a bijection $ϕ_A \colon \{0,1\}^{n-1} \to A$ such that ${\sf avgstretch}(ϕ_A) = O(\sqrt{n})$. (2) For $n = 3^k$ let $A_{{\sf rec\text{-}maj}} = \{x \in \{0,1\}^n : {\sf rec\text{-}maj}(x) = 1\}$, where ${\sf rec\text{-}maj} : \{0,1\}^n \to \{0,1\}$ is the function recursive majority of 3's. There exists a bijection $ϕ_{{\sf rec\text{-}maj}} \colon \{0,1\}^{n-1} \to A_{\sf rec\text{-}maj}$ such that ${\sf avgstretch}(ϕ_{\sf rec\text{-}maj}) = O(1)$. (3) Let $A_{\sf tribes} = \{x \in \{0,1\}^n : {\sf tribes}(x) = 1\}$. There exists a bijection $ϕ_{\sf tribes} \colon \{0,1\}^{n-1} \to A_{\sf tribes}$ such that ${\sf avgstretch}(ϕ_{\sf tribes}) = O(\log(n))$. These results answer the questions raised by Benjamini et al.\ (FOCS 2014).

math.CO↗

arXiv2020

Searching Trees with Permanently Noisy Advice: Walking and Query Algorithms

We consider a search problem on trees in which the goal is to find an adversarially placed treasure, while relying on local, partial information. Specifically, each node in the tree holds a pointer to one of its neighbors, termed \emph{advice}. A node is faulty with probability $q$. 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 is {\em permanent}, in the sense that querying the same node again would yield the same answer. Let $Δ$ denote the maximal degree. Roughly speaking, when considering the expected number of {\em moves}, i.e., edge traversals, we show that a phase transition occurs when the {\em noise parameter} $q$ is about $1/\sqrtΔ$. Below the threshold, there exists an algorithm with expected move complexity $O(D\sqrtΔ)$, where $D$ is the depth of the treasure, whereas above the threshold, every search algorithm has expected number of moves which is both exponential in $D$ and 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 $\varepsilon > 0$, if $q<1/Δ^{\varepsilon}$ then there exists a search strategy that with probability $1-δ$ finds the treasure using $(δ^{-1}D)^{O(\frac 1 \varepsilon)}$ moves. Moreover, we show that $(δ^{-1}D)^{Ω(\frac 1 \varepsilon)}$ moves are necessary. Besides the number of moves, we also study the number of advice {\em queries} required to find the treasure. Roughly speaking, for this complexity, we show similar threshold results to those previously stated, where the parameter $D$ is replaced by $\log n$.

cs.DS↗

arXiv2017

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.

cs.MA↗

arXiv2016

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.

cs.DC↗

arXiv2016

Streaming Communication Protocols

We define the Streaming Communication model that combines the main aspects of communication complexity and streaming. We consider two agents that want to compute some function that depends on inputs that are distributed to each agent. The inputs arrive as data streams and each agent has a bounded memory. Agents are allowed to communicate with each other and also update their memory based on the input bit they read and the previous message they received. We provide tight tradeoffs between the necessary resources, i.e. communication 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.

cs.CC↗

arXiv2016

Sensitivity of mixing times in Eulerian digraphs

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.

math.PR↗

arXiv2012

Extended Extremes of Information Combining

Extremes of information combining inequalities play an important role in the analysis of sparse-graph codes under message-passing decoding. We introduce new tools for the derivation of such inequalities, and show by means of a concrete examples how they can be applied to solve some optimization problems in the analysis of low-density parity-check codes.

cs.IT↗