The Landscape of Distributed Complexities on Trees and Beyond

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].