Efficient size estimation and impossibility of termination in uniform dense population protocols

We study uniform population protocols: networks of anonymous agents whose pairwise interactions are chosen at random, where each agent uses an identical transition algorithm that does not depend on the population size $n$. Many existing polylog$(n)$ time protocols for leader election and majority computation are nonuniform: to operate correctly, they require all agents to be initialized with an approximate estimate of $n$ (specifically, the exact value $\lfloor \log n \rfloor$). Our first main result is a uniform protocol for calculating $\log(n) \pm O(1)$ with high probability in $O(\log^2 n)$ time and $O(\log^4 n)$ states ($O(\log \log n)$ bits of memory). The protocol is converging but not terminating: it does not signal when the estimate is close to the true value of $\log n$. If it could be made terminating, this would allow composition with protocols, such as those for leader election or majority, that require a size estimate initially, to make them uniform (though with a small probability of failure). We do show how our main protocol can be indirectly composed with others in a simple and elegant way, based on the leaderless phase clock, demonstrating that those protocols can in fact be made uniform. However, our second main result implies that the protocol cannot be made terminating, a consequence of a much stronger result: a uniform protocol for any task requiring more than constant time cannot be terminating even with probability bounded above 0, if infinitely many initial configurations are dense: any state present initially occupies $\Omega(n)$ agents. (In particular, no leader is allowed.) Crucially, the result holds no matter the memory or time permitted. Finally, we show that with an initial leader, our size-estimation protocol can be made terminating with high probability, with the same asymptotic time and space bounds.Comment: Using leaderless phase cloc

Statistical structures for internet-scale data management

Efficient query processing in traditional database management systems relies on statistics on base data. For centralized systems, there is a rich body of research results on such statistics, from simple aggregates to more elaborate synopses such as sketches and histograms. For Internet-scale distributed systems, on the other hand, statistics management still poses major challenges. With the work in this paper we aim to endow peer-to-peer data management over structured overlays with the power associated with such statistical information, with emphasis on meeting the scalability challenge. To this end, we first contribute efficient, accurate, and decentralized algorithms that can compute key aggregates such as Count, CountDistinct, Sum, and Average. We show how to construct several types of histograms, such as simple Equi-Width, Average-Shifted Equi-Width, and Equi-Depth histograms. We present a full-fledged open-source implementation of these tools for distributed statistical synopses, and report on a comprehensive experimental performance evaluation, evaluating our contributions in terms of efficiency, accuracy, and scalability

Stable Leader Election in Population Protocols Requires Linear Time

A population protocol *stably elects a leader* if, for all $n$, starting from an initial configuration with $n$ agents each in an identical state, with probability 1 it reaches a configuration $\mathbf{y}$ that is correct (exactly one agent is in a special leader state $\ell$) and stable (every configuration reachable from $\mathbf{y}$ also has a single agent in state $\ell$). We show that any population protocol that stably elects a leader requires $\Omega(n)$ expected "parallel time" --- $\Omega(n^2)$ expected total pairwise interactions --- to reach such a stable configuration. Our result also informs the understanding of the time complexity of chemical self-organization by showing an essential difficulty in generating exact quantities of molecular species quickly.Comment: accepted to Distributed Computing special issue of invited papers from DISC 2015; significantly revised proof structure and intuitive explanation

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