# 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. [FOCS'03] gave gossip algorithms to compute important aggregate statistics if every node is given a value. In particular, they gave a beautiful O( n + 1/ϵ) round algorithm to ϵ-approximate the sum of all values and an O(^2 n) round algorithm to compute the exact ϕ-quantile, i.e., the the ϕ n smallest value. We give an quadratically faster and in fact optimal gossip algorithm for the exact ϕ-quantile problem which runs in O( n) rounds. We furthermore show that one can achieve an exponential speedup if one allows for an ϵ-approximation. We give an O( n + 1/ϵ) round gossip algorithm which computes a value of rank between ϕ n and (ϕ+ϵ)n at every node.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( n + 1/ϵ) lower bound which shows that our algorithm is optimal for all values of ϵ.

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