A number of recent papers – e.g. Brandt et al. (STOC 2016), Chang et al. (FOCS 2016), Ghaffari & Su (SODA 2017), Brandt et al. (PODC 2017), and Chang & Pettie (FOCS 2017) – have advanced our understanding of one of the most fundamental questions in theory of distributed computing: what are the possible time complexity classes of LCL problems in the LOCAL model? In essence, we have a graph problem $\mathrm{\Pi }$ in which a solution can be verified by checking all radius-$O(1)$ neighbourhoods, and the question is what is the smallest $T$ such that a solution can be computed so that each node chooses its own output based on its radius-$T$ neighbourhood. Here $T$ is the distributed time complexity of $\mathrm{\Pi }$.

The time complexity classes for deterministic algorithms in bounded-degree graphs that are known to exist by prior work are $\mathrm{\Theta }(1)$, $\mathrm{\Theta }({\log }^{\ast }n)$, $\mathrm{\Theta }(\log n)$, $\mathrm{\Theta }(n^{1/k})$, and $\mathrm{\Theta }(n)$. It is also known that there are two gaps: one between $\omega (1)$ and $o(\log {\log }^{\ast }n)$, and another between $\omega ({\log }^{\ast }n)$ and $o(\log n)$. It has been conjectured that many more gaps exist, and that the overall time hierarchy is relatively simple – indeed, this is known to be the case in restricted graph families such as cycles and grids.

We show that the picture is much more diverse than previously expected. We present a general technique for engineering LCL problems with numerous different deterministic time complexities, including $\mathrm{\Theta }({\log }^{\alpha }n)$ for any $\alpha \ge 1$, $2^{\mathrm{\Theta }({\log }^{\alpha }n)}$ for any $\alpha \le 1$, and $\mathrm{\Theta }(n^{\alpha })$ for any $\alpha <1/2$ in the high end of the complexity spectrum, and $\mathrm{\Theta }({\log }^{\alpha }{\log }^{\ast }n)$ for any $\alpha \ge 1$, $2^{\mathrm{\Theta }({\log }^{\alpha }{\log }^{\ast }n)}$ for any $\alpha \le 1$, and $\mathrm{\Theta }(({\log }^{\ast }n{)}^{\alpha })$ for any $\alpha \le 1$ in the low end of the complexity spectrum; here $\alpha$ is a positive rational number.