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Robust Comparison in Population Protocols

13 March 2020
Dan Alistarh
Martin Töpfer
P. Uznański
ArXiv (abs)PDFHTML
Abstract

There has recently been a surge of interest in the computational and complexity properties of the population model, which assumes nnn anonymous, computationally-bounded nodes, interacting at random, and attempting to jointly compute global predicates. In particular, a significant amount of work, has gone towards investigating majority and consensus dynamics in this model: assuming that each node is initially in one of two states XXX or YYY, determine which state had higher initial count. In this paper, we consider a natural generalization of majority/consensus, which we call comparison. We are given two baseline states, X0X_0X0​ and Y0Y_0Y0​, present in any initial configuration in fixed, possibly small counts. Importantly, one of these states has higher count than the other: we will assume ∣X0∣≥C∣Y0∣|X_0| \ge C |Y_0|∣X0​∣≥C∣Y0​∣ for some constant CCC. The challenge is to design a protocol which can quickly and reliably decide on which of the baseline states X0X_0X0​ and Y0Y_0Y0​ has higher initial count. We propose a simple algorithm solving comparison: the baseline algorithm uses O(log⁡n)O(\log n)O(logn) states per node, and converges in O(log⁡n)O(\log n)O(logn) (parallel) time, with high probability, to a state where whole population votes on opinions XXX or YYY at rates proportional to initial ∣X0∣|X_0|∣X0​∣ vs. ∣Y0∣|Y_0|∣Y0​∣ concentrations. We then describe how such output can be then used to solve comparison. The algorithm is self-stabilizing, in the sense that it converges to the correct decision even if the relative counts of baseline states X0X_0X0​ and Y0Y_0Y0​ change dynamically during the execution, and leak-robust, in the sense that it can withstand spurious faulty reactions. Our analysis relies on a new martingale concentration result which relates the evolution of a population protocol to its expected (steady-state) analysis, which should be broadly applicable in the context of population protocols and opinion dynamics.

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