Locally checkable problems in rooted trees

Abstract Consider any locally checkable labeling problem $$\Pi $$ Π in rooted regular trees : there is a finite set of labels $$\Sigma Σ , and for each label $$x \in \Sigma x ∈ we specify what are permitted combinations the children an internal node x (the leaf nodes unconstrained). This formalism expressive enough to capture many classic problems studied distributed computing, including vertex coloring, edge maximal independent set. We show that computational complexity such falls one following classes: it O (1), $$\Theta (\log ^* n)$$ Θ ( log ∗ n ) or $$n^{\Theta (1)}$$ 1 rounds with n (and all these classes nonempty). given same four standard models graph algorithms: deterministic $$\mathsf {LOCAL}$$ LOCAL randomized {CONGEST}$$ CONGEST model. In particular, randomness does not help this setting, class \log exist (while broader setting general trees). also how systematically determine i.e., whether takes rounds. While algorithm may take exponential time size description nevertheless practical: provide freely available implementation classifier algorithm, fast classify interest.