  Consider a system of coalescing random walks where each individual performs
random walk over a finite graph G, or (more generally) evolves according to
some reversible Markov chain generator Q. Let C be the first time at which all
walkers have coalesced into a single cluster. C is closely related to the
consensus time of the voter model for this G or Q.
  We prove that the expected value of C is at most a constant multiple of the
largest hitting time of an element in the state space. This solves a problem
posed by Aldous and Fill and gives sharp bounds in many examples, including all
vertex-transitive graphs. We also obtain results on the expected time until
only k>1 clusters remain. Our proof tools include a new exponential inequality
for the meeting time of a reversible Markov chain and a deterministic
trajectory, which we believe to be of independent interest.
