  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 $O(\log^2 n)$-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 $\log^2 n$ because of
$\Theta(\log n)$ synchronization time per phase. We show how this framework can
be tightened with {\em weak synchronization} to break the $O(\log^2 n)$ upper
bound of previous protocols.
