The Landscape of Distributed Complexities on Trees and Beyond
Abstract
We study the local complexity landscape of locally checkable labeling (LCL) problems on constantdegree 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 speedup from $o(\sqrt[d]{\log^* n})$ by Suomela in [Chang, Pettie, FOCS 2017].
 Publication:

arXiv eprints
 Pub Date:
 February 2022
 DOI:
 10.48550/arXiv.2202.04724
 arXiv:
 arXiv:2202.04724
 Bibcode:
 2022arXiv220204724G
 Keywords:

 Computer Science  Data Structures and Algorithms;
 Computer Science  Distributed;
 Parallel;
 and Cluster Computing