An Exponential Separation between Randomized and Deterministic Complexity in the LOCAL Model

Over the past 30 years numerous algorithms have been designed for symmetry breaking problems in the LOCAL model, such as maximal matching, MIS, vertex coloring, and edge coloring. For most problems the best randomized algorithm is at least exponentially faster than the best deterministic algorithm. In this paper we prove that these exponential gaps are necessary and establish numerous connections between the deterministic and randomized complexities in the LOCAL model. Each of our results has a very compelling take-away message: Fast $\Delta$-coloring of trees requires random bits. Building on a recent randomized lower bound of Brandt et al. [A lower bound for the distributed Lovász local lemma, in Proceedings of the 48th ACM Symposium on Theory of Computing (STOC), ACM, New York, 2016, pp. 479--488], we prove that the randomized complexity of $\Delta$-coloring a tree with maximum degree $\Delta$ is $O(\log_\Delta \log n + \log^\ast n)$ for any $\Delta \ge 55$, whereas its deterministic complexity is $\Omega(\log_\Delta n)$ for any $\Delta\ge 3$. This also establishes a large separation between the deterministic complexity of $\Delta$-coloring and $(\Delta+1)$-coloring trees. There is a gap in the deterministic complexity hierarchy. We show that any deterministic algorithm for a natural class of problems that runs in $O(1) + o(\log_\Delta n)$ rounds can be transformed to run in $O(\log^* n - \log^*\Delta + 1)$ rounds. If the transformed algorithm violates a lower bound (even allowing randomization), then one can conclude that the problem requires $\Omega(\log_\Delta n)$ time deterministically. This gives an alternate proof that deterministically $\Delta$-coloring a tree with small $\Delta$ takes $\Omega(\log_\Delta n)$ rounds. Graph shattering is necessary. We prove that the randomized complexity of any natural problem on instances of size $n$ is at least its deterministic complexity on instances of size $\sqrt{\log n}$. This shows that any randomized $O(1) + o(\log_\Delta \log n)$-round algorithm can be derandomized to run in deterministically $O(1) + o(\log_\Delta n)$ rounds and hence can be transformed to run in $O(\log^* n - \log^*\Delta + 1)$ rounds. This also shows that a deterministic $\Omega(\log_\Delta n)$ lower bound for any problem ($\Delta$-coloring a tree, for example) implies a randomized $\Omega(\log_\Delta \log n)$ lower bound. It illustrates that the graph shattering technique employed in recent randomized symmetry breaking algorithms is absolutely essential to the LOCAL model. For example, it is provably impossible to improve the $2^{O(\sqrt{\log\log n})}$ terms in the complexities of the best MIS and $(\Delta+1)$-coloring algorithms without also improving the $2^{O(\sqrt{\log n})}$-round Panconesi--Srinivasan algorithms.