A lower bound for distributed averaging algorithms

TL;DR

This paper establishes fundamental lower bounds on the convergence speed of certain distributed averaging algorithms, highlighting the importance of increased memory or state space for faster convergence.

Contribution

It proves that algorithms with single real number states and smooth update functions cannot converge faster than quadratic time in the number of nodes, setting a theoretical performance limit.

Findings

Any such algorithm has a worst-case running time of at least order n^2.

Increasing memory or expanding the state space is necessary for faster convergence.

The results apply to a broad class of nonlinear distributed averaging algorithms.

Abstract

We derive lower bounds on the convergence speed of a widely used class of distributed averaging algorithms. In particular, we prove that any distributed averaging algorithm whose state consists of a single real number and whose (possibly nonlinear) update function satisfies a natural smoothness condition has a worst case running time of at least on the order of $n^2$ on a network of $n$ nodes. Our results suggest that increased memory or expansion of the state space is crucial for improving the running times of distributed averaging algorithms.