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 loglog n) time and uses O(n^60) states (O(1) + 60 log n bits of memory per agent) with probability 1-O(loglog n/n). The time complexity is also O(log n loglog n) in expectation. The time to converge is also O(log n loglog 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 sub-protocol is the first uniform sublinear-time leader election population protocol, taking O(log n loglog 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).