Consider a computer network that consists of a path with $n$ nodes. The nodes are labeled with inputs from a constant-sized set, and the task is to find output labels from a constant-sized set subject to some local constraints—more formally, we have an LCL (locally checkable labeling) problem. How many communication rounds are needed (in the standard LOCAL model of computing) to solve this problem?
It is well known that the answer is always either $O(1)$ rounds, or $\Theta(\log^* n)$ rounds, or $\Theta(n)$ rounds. In this work we show that this question is decidable (albeit PSPACE-hard): we present an algorithm that, given any LCL problem defined on a path, outputs the distributed computational complexity of this problem and the corresponding asymptotically optimal algorithm.
Peter Robinson and Faith Ellen (Eds.): Proceedings of the 2019 ACM Symposium on Principles of Distributed Computing, pages 262–271, ACM Press, New York, 2019