Minimizing message size in stochastic communication patterns: fast self-stabilizing protocols with 3 bits

Abstract

This paper considers the basic P U LL 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 -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 , withinÕ(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 inÕ(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.

Universal protocols for information dissemination using emergent signals

“…A protocol solving their problem was presented, giving a stabilized solution inÕ(log n) time, using a number of states of agents which depends on population size, but exchanging messages of bit size O(1) (this assumption can be modeled in the population protocol framework as a restriction on the permitted rule set). In this sense, our result can be seen as providing improved results with respect to their approach, since it is applicable in an asynchronous setting and reduces the number of states to constant (the latter question was open [11]). We remark that their protocol has a more general application to the problem of deciding which of the sources X 1 or X 2 is represented by a larger number of agents, provided these two numbers are separated by a multiplicative constants.…”

“…Other protocols with ω(1)-states: Clock-Sync (in synchronized round model) [11] Time-to-Live [5] detection problem has the intuitive interpretation of detecting if a given type of chemical or biological agent (e.g., a contaminant, cancer cell, or hormonal signal) is present in the population, and spreading this information among all agents.…”

“…Other work on the problems. The BITBROADCAST problem has been previously considered by Boczkowski, Korman, and Natale in [11], in a self-stabilizing but synchronous (round-based) setting. A protocol solving their problem was presented, giving a stabilized solution inÕ(log n) time, using a number of states of agents which depends on population size, but exchanging messages of bit size O(1) (this assumption can be modeled in the population protocol framework as a restriction on the permitted rule set).…”

“…(Our approach could also be used in such a setting, but the required separation of agent numbers to ensure a correct output would have to be much larger: we can compare values if their logarithms are separated by a multiplicative constant.) The protocol of [11] involves a routine which allows the population to create a synchronized modulo clock, working in a synchronous setting. The period of this clock is independent of the input states of the protocol, which should be contrasted with the oscillators we work on in this paper, which encode the input into a signal (with a period depending on the number of agents in a given input state).…”

“…By Lemma 25, after O(n log 2 n) steps, agents in at least one of the states {M +1 , M −1 } are permanently eliminated from the system. Thus, either rule (11) or rule (12) will never again be executed in the future. An agent which is in state L on will spontaneously move to another state following rule (13) within O( 1 q(ε) n log n) steps, with probability 1 − O(1/n 2 ), and will never reenter such a state, since this would require the activation of both rule (11) and rule (12).…”