A population protocol for exact majority with $O(\log^{5/3} n)$ stabilization time and asymptotically optimal number of states

A population protocol can be viewed as a sequence of pairwise interactions of $n$ agents (nodes). During one interaction, two agents selected uniformly at random update their states by applying a specified deterministic transition function. In a long run, the whole system should stabilize at the correct output property. The main performance objectives in designing population protocols are small number of states per agent and fast stabilization time. We present a fast population protocol for the exact-majority problem which uses $\Theta(\log n)$ states (per agent) and stabilizes in $O(\log^{5/3} n)$ parallel time (i.e., $O(n\log^{5/3} n)$ interactions) in expectation and with high probability. Alistarh et al. [SODA 2018] showed that any exact-majority protocol which stabilizes in expected $O(n^{{1-\epsilon})$} parallel time, for any constant $\epsilon > 0$, requires $\Omega(\log n)$ states. They also showed an $O(\log² n)$-time protocol with $O(\log n)$ states, the currently fastest exact-majority protocol with polylogarithmic number of states. The standard design framework for majority protocols is based on $O(\log n)$ phases and requires that all nodes are well synchronized within each phase, leading naturally to upper bounds of the order of at least $\log² n$ because of $\Theta(\log n)$ synchronization time per phase. We show how this framework can be tightened with {\em weak synchronization} to break the $O(\log² n)$ upper bound of previous protocols.