Almost global problems in the LOCAL model

The landscape of the distributed time complexity is nowadays well-understood for subpolynomial complexities. When we look at deterministic algorithms in the $$\mathsf {LOCAL}$$ LOCAL model and locally checkable problems ( $$\mathsf {LCL}$$ LCL s) in bounded-degree graphs, the following picture emerges: There are lots of problems with time complexities of $$\varTheta (\log ^* n)$$ Θ ( log ∗ n ) or $$\varTheta (\log n)$$ Θ ( log n ) . It is not possible to have a problem with complexity between $$\omega (\log ^* n)$$ ω ( log ∗ n ) and $$o(\log n)$$ o ( log n ) . In general graphs , we can construct $$\mathsf {LCL}$$ LCL problems with infinitely many complexities between $$\omega (\log n)$$ ω ( log n ) and $$n^{o(1)}$$ n o ( 1 ) . In trees , problems with such complexities do not exist. However, the high end of the complexity spectrum was left open by prior work. In general graphs there are $$\mathsf {LCL}$$ LCL problems with complexities of the form $$\varTheta (n^\alpha )$$ Θ ( n α ) for any rational $$0 < \alpha \le 1/2$$ 0 < α ≤ 1 / 2 , while for trees only complexities of the form $$\varTheta (n^{1/k})$$ Θ ( n 1 / k ) are known. No $$\mathsf {LCL}$$ LCL problem with complexity between $$\omega (\sqrt{n})$$ ω ( n ) and o ( n ) is known, and neither are there results that would show that such problems do not exist. We show that: In general graphs , we can construct $$\mathsf {LCL}$$ LCL problems with infinitely many complexities between $$\omega (\sqrt{n})$$ ω ( n ) and o ( n ). In trees , problems with such complexities do not exist. Put otherwise, we show that any $$\mathsf {LCL}$$ LCL with a complexity o ( n ) can be solved in time $$O(\sqrt{n})$$ O ( n ) in trees, while the same is not true in general graphs.