Complexity landscape for local certification

Abstract: An impressive recent line of work has charted the complexity landscape of distributed graph algorithms. For many settings, it has been determined which time complexities exist, and which do not (in the sense that no local problem could have an optimal algorithm with that complexity). In this paper, we initiate the study of the landscape for space complexity of distributed graph algorithms. More precisely, we focus on the local certification setting, where a prover assigns certificates to nodes to certify a property, and where the space complexity is measured by the size of the certificates. Already for anonymous paths and cycles, we unveil a surprising landscape: - There is a gap between complexity $O(1)$ and $\Theta(\log \log n)$ in paths. This is the first gap established in local certification. - There exists a property that has complexity $\Theta(\log \log n)$ in paths, a regime that was not known to exist for a natural property. - There is a gap between complexity $O(1)$ and $\Theta(\log n)$ in cycles, hence a gap that is exponentially larger than for paths. We then generalize our result for paths to the class of trees. Namely, we show that there is a gap between complexity $O(1)$ and $\Theta(\log \log d)$ in trees, where $d$ is the diameter. We finally describe some settings where there are no gaps at all. To prove our results we develop a new toolkit, based on various results of automata theory and arithmetic, which is of independent interest.