This paper gives drastically faster gossip algorithms to compute exact and
approximate quantiles.
Gossip algorithms, which allow each node to contact a uniformly random other
node in each round, have been intensely studied and been adopted in many
applications due to their fast convergence and their robustness to failures.
Kempe et al. [FOCS'03] gave gossip algorithms to compute important aggregate
statistics if every node is given a value. In particular, they gave a beautiful
O(logn+logϵ1) round algorithm to ϵ-approximate
the sum of all values and an O(log2n) round algorithm to compute the exact
ϕ-quantile, i.e., the the ⌈ϕn⌉ smallest value.
We give an quadratically faster and in fact optimal gossip algorithm for the
exact ϕ-quantile problem which runs in O(logn) rounds. We furthermore
show that one can achieve an exponential speedup if one allows for an
ϵ-approximation. We give an O(loglogn+logϵ1)
round gossip algorithm which computes a value of rank between ϕn and
(ϕ+ϵ)n at every node.% for any 0≤ϕ≤1 and 0<ϵ<1. Our algorithms are extremely simple and very robust - they can
be operated with the same running times even if every transmission fails with
a, potentially different, constant probability. We also give a matching
Ω(loglogn+logϵ1) lower bound which shows that
our algorithm is optimal for all values of ϵ