  We consider distributed elections, where there is a center and $k$ sites. In
such distributed elections, each voter has preferences over some set of
candidates, and each voter is assigned to exactly one site such that each site
is aware only of the voters assigned to it. The center is able to directly
communicate with all sites. We are interested in designing
communication-efficient protocols, allowing the center to maintain a candidate
which, with arbitrarily high probability, is guaranteed to be a winner, or at
least close to being a winner. We consider various single-winner voting rules,
such as variants of Approval voting and scoring rules, tournament-based voting
rules, and several round-based voting rules. For the voting rules we consider,
we show that, using communication which is logarithmic in the number of voters,
it is possible for the center to maintain such approximate winners; that is,
upon a query at any time the center can immediately return a candidate which is
guaranteed to be an approximate winner with high probability. We complement our
protocols with lower bounds. Our results are theoretical in nature and relate
to various scenarios, such as aggregating customer preferences in online
shopping websites or supermarket chains and collecting votes from different
polling stations of political elections.
