arXiv subjects

Felix Biermeier

Publications and source records attributed to Felix Biermeier.

7 recordsLinked to original sources

arXiv2024

Undecided State Dynamics with Stubborn Agents

In the classical Approximate Majority problem with two opinions there are agents with Opinion 1 and with Opinion 2. The goal is to reach consensus and to agree on the majority opinion if the bias is sufficiently large. It is well known that the problem can be solved efficiently using the Undecided State Dynamics (USD) where an agent interacting with an agent of the opposite opinion becomes undecided. In this paper, we consider a variant of the USD with a preferred Opinion 1. That is, agents with Opinion 1 behave stubbornly -- they preserve their opinion with probability $p$ whenever they interact with an agent having Opinion 2. Our main result shows a phase transition around the stubbornness parameter $p \approx 1-x_1/x_2$. If $x_1 = \Theta(n)$ and $p \geq 1-x_1/x_2 + o(1)$, then all agents agree on Opinion 1 after $O(n\cdot \log n)$ interactions. On the other hand, for $p \leq 1-x_1/x_2 - o(1)$, all agents agree on Opinion 2, again after $O(n\cdot \log n)$ interactions. Finally, if $p \approx 1-x_1/x_2$, then all agents do agree on one opinion after $O(n\cdot \log^2 n)$ interactions, but either of the two opinions can survive. All our results hold with high probability.

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arXiv2024

Population Protocols for Exact Plurality Consensus -- How a small chance of failure helps to eliminate insignificant opinions

We consider the \emph{exact plurality consensus} problem for \emph{population protocols}. Here, $n$ anonymous agents start each with one of $k$ opinions. Their goal is to agree on the initially most frequent opinion (the \emph{plurality opinion}) via random, pairwise interactions. The case of $k = 2$ opinions is known as the \emph{majority problem}. Recent breakthroughs led to an always correct, exact majority population protocol that is both time- and space-optimal, needing $O(\log n)$ states per agent and, with high probability, $O(\log n)$ time~[Doty, Eftekhari, Gasieniec, Severson, Stachowiak, and Uznanski; 2021]. We know that any always correct protocol requires $\Omega(k^2)$ states, while the currently best protocol needs $O(k^{11})$ states~[Natale and Ramezani; 2019]. For ordered opinions, this can be improved to $O(k^6)$~[Gasieniec, Hamilton, Martin, Spirakis, and Stachowiak; 2016]. We design protocols for plurality consensus that beat the quadratic lower bound by allowing a negligible failure probability. While our protocols might fail, they identify the plurality opinion with high probability even if the bias is $1$. Our first protocol achieves this via $k-1$ tournaments in time $O(k \cdot \log n)$ using $O(k + \log n)$ states. While it assumes an ordering on the opinions, we remove this restriction in our second protocol, at the cost of a slightly increased time $O(k \cdot \log n + \log^2 n)$. By efficiently pruning insignificant opinions, our final protocol reduces the number of tournaments at the cost of a slightly increased state complexity $O(k \cdot \log\log n + \log n)$. This improves the time to $O(n / x_{\max} \cdot \log n + \log^2 n)$, where $x_{\max}$ is the initial size of the plurality. Note that $n/x_{\max}$ is at most $k$ and can be much smaller (e.g., in case of a large bias or if there are many small opinions).

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arXiv2023

Fast Convergence of $k$-Opinion Undecided State Dynamics in the Population Protocol Model

We analyze the convergence of the $k$-opinion Undecided State Dynamics (USD) in the population protocol model. For $k$=2 opinions it is well known that the USD reaches consensus with high probability within $O(n \log n)$ interactions. Proving that the process also quickly solves the consensus problem for $k>2$ opinions has remained open, despite analogous results for larger $k$ in the related parallel gossip model. In this paper we prove such convergence: under mild assumptions on $k$ and on the initial number of undecided agents we prove that the USD achieves plurality consensus within $O(k n \log n)$ interactions with high probability, regardless of the initial bias. Moreover, if there is an initial additive bias of at least $\Omega(\sqrt{n} \log n)$ we prove that the initial plurality opinion wins with high probability, and if there is a multiplicative bias the convergence time is further improved. Note that this is the first result for $k > 2$ for the USD in the population protocol model. Furthermore, it is the first result for the unsynchronized variant of the USD with $k>2$ which does not need any initial bias.

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arXiv2021

Loosely-Stabilizing Phase Clocks and the Adaptive Majority Problem

We present a loosely-stabilizing phase clock for population protocols. In the population model we are given a system of $n$ identical agents which interact in a sequence of randomly chosen pairs. Our phase clock is leaderless and it requires $O(\log n)$ states. It runs forever and is, at any point of time, in a synchronous state w.h.p. When started in an arbitrary configuration, it recovers rapidly and enters a synchronous configuration within $O(n\log n)$ interactions w.h.p. Once the clock is synchronized, it stays in a synchronous configuration for at least poly $n$ parallel time w.h.p. We use our clock to design a loosely-stabilizing protocol that solves the comparison problem introduced by Alistarh et al., 2021. In this problem, a subset of agents has at any time either $A$ or $B$ as input. The goal is to keep track which of the two opinions is (momentarily) the majority. We show that if the majority has a support of at least $\Omega(\log n)$ agents and a sufficiently large bias is present, then the protocol converges to a correct output within $O(n\log n)$ interactions and stays in a correct configuration for poly $n$ interactions, w.h.p.

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arXiv2021

Fast Consensus via the Unconstrained Undecided State Dynamics

We consider the plurality consensus problem among $n$ agents. Initially, each agent has one of $k$ different opinions. Agents choose random interaction partners and revise their state according to a fixed transition function, depending on their own state and the state of the interaction partners. The goal is to reach a consensus configuration in which all agents agree on the same opinion, and if there is initially a sufficiently large bias towards one opinion, that opinion should prevail. We analyze a synchronized variant of the undecided state dynamics defined as follows. The agents act in phases, consisting of a decision and a boosting part. In the decision part, any agent that encounters an agent with a different opinion becomes undecided. In the boosting part, undecided agents adopt the first opinion they encounter. We consider this dynamics in the population model and the gossip model. For the population model, our protocol reaches consensus (w.h.p.) in $O(\log^2 n)$ parallel time, providing the first polylogarithmic result for $k > 2$ (w.h.p.) in this model. Without any assumption on the bias, fast consensus has only been shown for $k = 2$ for the unsynchronized version of the undecided state dynamics [Clementi et al., MFCS'18]. We show that the synchronized variant of the undecided state dynamics reaches consensus (w.h.p.) in time $O(\log^2 n)$, independently of the initial number, bias, or distribution of opinions. In both models, we guarantee that if there is an initial bias of $\Omega(\sqrt{n \log n})$, then (w.h.p.) that opinion wins. A simple extension of our protocol in the gossip model yields a dynamics that does not depend on $n$ or $k$, is anonymous, and has (w.h.p.) runtime $O(\log^2 n)$. This solves an open problem formulated by Becchetti et al.~[Distributed Computing,~2017].

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arXiv2017

A Communication-Efficient Distributed Data Structure for Top-k and k-Select Queries

We consider the scenario of $n$ sensor nodes observing streams of data. The nodes are connected to a central server whose task it is to compute some function over all data items observed by the nodes. In our case, there exists a total order on the data items observed by the nodes. Our goal is to compute the $k$ currently lowest observed values or a value with rank in $[(1-\varepsilon)k,(1+\varepsilon)k]$ with probability $(1-δ)$. We propose solutions for these problems in an extension of the distributed monitoring model where the server can send broadcast messages to all nodes for unit cost. We want to minimize communication over multiple time steps where there are $m$ updates to a node's value in between queries. The result is composed of two main parts, which each may be of independent interest: (1) Protocols which answer Top-k and k-Select queries. These protocols are memoryless in the sense that they gather all information at the time of the request. (2) A dynamic data structure which tracks for every $k$ an element close to $k$. We describe how to combine the two parts to receive a protocol answering the stated queries over multiple time steps. Overall, for Top-$k$ queries we use $O(k + \log m + \log \log n)$ and for $k$-Select queries $O(\frac{1}{\varepsilon^2} \log \frac{1}δ + \log m + \log^2 \log n)$ messages in expectation. These results are shown to be asymptotically tight if $m$ is not too small.

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arXiv2017

Monitoring of Domain-Related Problems in Distributed Data Streams

Consider a network in which $n$ distributed nodes are connected to a single server. Each node continuously observes a data stream consisting of one value per discrete time step. The server has to continuously monitor a given parameter defined over all information available at the distributed nodes. That is, in any time step $t$, it has to compute an output based on all values currently observed across all streams. To do so, nodes can send messages to the server and the server can broadcast messages to the nodes. The objective is the minimisation of communication while allowing the server to compute the desired output. We consider monitoring problems related to the domain $D_t$ defined to be the set of values observed by at least one node at time $t$. We provide randomised algorithms for monitoring $D_t$, (approximations of) the size $|D_t|$ and the frequencies of all members of $D_t$. Besides worst-case bounds, we also obtain improved results when inputs are parameterised according to the similarity of observations between consecutive time steps. This parameterisation allows to exclude inputs with rapid and heavy changes, which usually lead to the worst-case bounds but might be rather artificial in certain scenarios.

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