  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].
