The Landscape of Distributed Complexities on Trees and Beyond

TL;DR

This paper fully classifies the local complexity of LCL problems on trees and grids, showing that complexities below ( ext{log}^* n) are actually constant, thus refining previous speedup results.

Contribution

It completes the classification of LCL problem complexities on trees and grids, establishing that complexities below ( ext{log}^* n) are constant, improving prior bounds.

Findings

Every LCL problem with complexity o(( ext{log}^* n)) has complexity O(1).

The speedup from o(( ext{log} ext{log}^* n)) to O(1) on trees is proven.

The same speedup applies to bounded degree graphs in LCA and Volume models.

Abstract

We study the local complexity landscape of locally checkable labeling (LCL) problems on constant-degree graphs with a focus on complexities below $\log^* n$. Our contribution is threefold: Our main contribution is that we complete the classification of the complexity landscape of LCL problems on trees in the LOCAL model, by proving that every LCL problem with local complexity $o(\log^* n)$ has actually complexity $O(1)$. This result improves upon the previous speedup result from $o(\log \log^* n)$ to $O(1)$ by [Chang, Pettie, FOCS 2017]. In the related LCA and Volume models [Alon, Rubinfeld, Vardi, Xie, SODA 2012, Rubinfeld, Tamir, Vardi, Xie, 2011, Rosenbaum, Suomela, PODC 2020], we prove the same speedup from $o(\log^* n)$ to $O(1)$ for all bounded degree graphs. Similarly, we complete the classification of the LOCAL complexity landscape of oriented $d$-dimensional grids by proving that any LCL problem with local complexity $o(\log^* n)$ has actually complexity $O(1)$. This improves upon the previous speed-up from $o(\sqrt[d]{\log^* n})$ by Suomela in [Chang, Pettie, FOCS 2017].