Õptimal Fault-Tolerant Labeling for Reachability and Approximate Distances in Directed Planar Graphs

We present a labeling scheme that assigns labels of size Õ(1) to the vertices of a directed weighted planar graph G, such that for any fixed ε>0 from the labels of any three vertices s, t and f one can determine in Õ(1) time a (1+ε)-approximation of the s-to-t distance in the graph G∖f. For approximate distance queries, prior to our work, no efficient solution existed, not even in the centralized oracle setting. Even for the easier case of reachability, Õ(1) queries were known only with a centralized oracle of size Õ(n) [SODA 21].