Exact size counting in uniform population protocols in nearly logarithmic time

We study population protocols: networks of anonymous agents that interact under a scheduler that picks pairs of agents uniformly at random. The _size counting problem_ is that of calculating the exact number $n$ of agents in the population, assuming no leader (each agent starts in the same state). We give the first protocol that solves this problem in sublinear time. The protocol converges in $O(\log n \log \log n)$ time and uses $O(n^{60})$ states ($O(1) + 60 \log n$ bits of memory per agent) with probability $1-O(\frac{\log \log n}{n})$. The time complexity is also $O(\log n \log \log n)$ in expectation. The time to converge is also $O(\log n \log \log n)$ in expectation. Crucially, unlike most published protocols with $\omega(1)$ states, our protocol is _uniform_: it uses the same transition algorithm for any population size, so does not need an estimate of the population size to be embedded into the algorithm. A sub-protocol is the first uniform sublinear-time leader election population protocol, taking $O(\log n \log \log n)$ time and $O(n^{18})$ states. The state complexity of both the counting and leader election protocols can be reduced to $O(n^{30})$ and $O(n^{9})$ respectively, while increasing the time to $O(\log^2 n)$

Brief announcement: Exact size counting in uniform population protocols in nearly logarithmic time

We study population protocols: networks of anonymous agents whose pairwise interactions are chosen uniformly at random. The size counting problem is that of calculating the exact number n of agents in the population, assuming no leader (each agent starts in the same state). We give the first protocol that solves this problem in sublinear time. The protocol converges in O(log n log log n) time and uses O(n 60 ) states (O(1)+60 log n bits of memory per agent) with probability 1−O( lognlog n ). The time to converge is also O(log n log log n) in expectation. Crucially, unlike most published protocols with ω(1) states, our protocol is uniform: it uses the same transition algorithm for any population size, so does not need an estimate of the population size to be embedded into the algorithm

A time and space optimal stable population protocol solving exact majority

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 well-studied *majority* problem is that of determining in an initial population of $n$ agents, each with one of two opinions $A$ 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 $\mathsf{A}$, $\mathsf{B}$, or $\mathsf{T}$, from which the consensus cannot change. We describe a protocol that solves this problem using $O(\log n)$ states ($\log \log n + O(1)$ bits of memory) and optimal expected time $O(\log n)$. The number of states $O(\log n)$ is known to be optimal for the class of polylogarithmic time stable protocols that are "output dominant" and "monotone". 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 $\left \lceil {\log n} \right \rceil$ encoded in it. We show that the protocol can be modified to be uniform, while increasing the state complexity to $\Theta(\log n \log \log n)$