Optimal Gossip Algorithms for Exact and Approximate Quantile Computations

Abstract

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