Message Reduction in the Local Model is a Free Lunch

TL;DR

This paper introduces a new spanner construction algorithm in the LOCAL model that significantly reduces message complexity, enabling efficient transformation of round complexity into message complexity with minimal overhead.

Contribution

A novel spanner construction algorithm in the LOCAL model achieves low message complexity and enables transforming round complexity into message complexity with minimal increase.

Findings

Constructs a spanner with constant stretch and $O(n^{1+ ext{ε}})$ edges in constant rounds

Transforms any $t$-round LOCAL algorithm into an $O(t)$-round algorithm with $O(t imes n^{1+ ext{ε}})$ messages

Improves upon previous message-reduction schemes with logarithmic blow-up in round complexity

Abstract

A new \emph{spanner} construction algorithm is presented, working under the \emph{LOCAL} model with unique edge IDs. Given an $n$-node communication graph, a spanner with a constant stretch and $O (n^{1 + \varepsilon})$ edges (for an arbitrarily small constant $\varepsilon > 0$) is constructed in a constant number of rounds sending $O (n^{1 + \varepsilon})$ messages whp. Consequently, we conclude that every $t$-round LOCAL algorithm can be transformed into an $O (t)$-round LOCAL algorithm that sends $O (t \cdot n^{1 + \varepsilon})$ messages whp. This improves upon all previous message-reduction schemes for LOCAL algorithms that incur a $\log^{\Omega (1)} n$ blow-up of the round complexity.