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 $O(\log n + \log \frac{1}ε)$ round algorithm to $ε$-approximate the sum of all values and an $O(\log^2 n)$ round algorithm to compute the exact $φ$-quantile, i.e., the the $\lceil φn \rceil$ smallest value. We give an quadratically faster and in fact optimal gossip algorithm for the exact $φ$-quantile problem which runs in $O(\log n)$ rounds. We furthermore show that one can achieve an exponential speedup if one allows for an $ε$-approximation. We give an $O(\log \log n + \log \frac{1}ε)$ round gossip algorithm which computes a value of rank between $φn$ and $(φ+ε)n$ at every node.% for any $0 \leq φ\leq 1$ 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 $Ω(\log \log n + \log \frac{1}ε)$ lower bound which shows that our algorithm is optimal for all values of $ε$.