Optimal Gossip-Based Aggregate Computation

TL;DR

This paper introduces nearly optimal gossip algorithms for aggregate computation in networks, achieving minimal time and message complexity, and establishes fundamental lower bounds for address-oblivious methods.

Contribution

The paper presents the first provably almost-optimal gossip algorithms for aggregate computation with time and message efficiency, and introduces a new technique called distributed random ranking.

Findings

Algorithms compute aggregates in O(log n) time using O(n log log n) messages.

Proves a lower bound of Ω(n log n) messages for address-oblivious algorithms.

Shows non-address oblivious algorithms are necessary for message efficiency.

Abstract

We present the first provably almost-optimal gossip-based algorithms for aggregate computation that are both time optimal and message-optimal. Given a $n$-node network, our algorithms guarantee that all the nodes can compute the common aggregates (such as Min, Max, Count, Sum, Average, Rank etc.) of their values in optimal $O(\log n)$ time and using $O(n \log \log n)$ messages. Our result improves on the algorithm of Kempe et al. \cite{kempe} that is time-optimal, but uses $O(n \log n)$ messages as well as on the algorithm of Kashyap et al. \cite{efficient-gossip} that uses $O(n \log \log n)$ messages, but is not time-optimal (takes $O(\log n \log \log n)$ time). Furthermore, we show that our algorithms can be used to improve gossip-based aggregate computation in sparse communication networks, such as in peer-to-peer networks. The main technical ingredient of our algorithm is a technique called {\em distributed random ranking (DRR)} that can be useful in other applications as well. DRR gives an efficient distributed procedure to partition the network into a forest of (disjoint) trees of small size. Our algorithms are non-address oblivious. In contrast, we show a lower bound of $\Omega(n\log n)$ on the message complexity of any address-oblivious algorithm for computing aggregates. This shows that non-address oblivious algorithms are needed to obtain significantly better message complexity. Our lower bound holds regardless of the number of rounds taken or the size of the messages used. Our lower bound is the first non-trivial lower bound for gossip-based aggregate computation and also gives the first formal proof that computing aggregates is strictly harder than rumor spreading in the address-oblivious model.