A time and space optimal stable population protocol solving exact majority

Abstract

We study population protocols, a model of distributed computing appropriate for modeling well-mixed chemical reaction networks and other physical systems where agents exchange information in pairwise interactions, but have no control over their schedule of interaction partners. The majority problem is that of determining in an initial population of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$n$</tex> agents, each with one of two opinions <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$A$</tex> or B, whether there are more A, more B, or a tie. A stable protocol solves this problem with probability 1 by eventually entering a configuration in which all agents agree on a correct consensus decision of A, B, or T, from which the consensus cannot change. We describe a protocol solving this problem using O(log n) states (log log <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$n$</tex> + O(1) bits of memory) and optimal expected time <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$O$</tex> (log <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$n$</tex> ). The number of states <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$O$</tex> (log <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$n$</tex> ) is known to be optimal for polylogarithmic time stable protocols that are “output dominant” and “monotone” [1]. These are two natural constraints satisfied by our protocol, making it simultaneously time- and state-optimal for that class. We introduce a key technique called a “fixed resolution clock” to achieve partial synchronization. Our protocol is nonuniform: the transition function has the value [log <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$n$</tex> ] encoded in it. We show that the protocol can be modified to be uniform, while increasing the state complexity to Θ (log <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$n$</tex> log log n).