Optimal Gossip Algorithms for Exact and Approximate Quantile Computations

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. [24] gave gossip algorithms to compute important aggregate statistics if every node is given a value. In particular, they gave a beautiful O(logn + log 1 ε ) round algorithm to ε-approximate the sum of all values and an O(log2 n) round algorithm to compute the exact Φ-quantile, i.e., 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. In particular, we give an O(log logn + log 1 ε ) round gossip algorithm which computes a value of rank between Φn and (Φ + ε)n at every node. 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(log logn + log 1 ε ) lower bound which shows that our algorithm is optimal for all values of ε.