Eccentricity queries and beyond using hub labels

Abstract

Hub labeling schemes are popular methods for computing distances on road networks and other large complex networks, often answering to a query within a few microseconds for graphs with millions of edges. In this work, we study their algorithmic applications beyond distance queries. Indeed, several implementations of hub labels were reported to have good practical performances, both in terms of pre-processing time and maximum label size. There are also a few relevant graph classes for applications for which we know how to compute hub labelings with sublinear labels in quasi linear time. These positive results raise the question of what can be computed efficiently given a graph and a hub labeling of small maximum label size as input. We focus on eccentricity queries and distance-sum queries, for several versions of these problems on directed weighted graphs, that is in part motivated by their importance in facility location problems. On the negative side, we show conditional lower bounds for these above problems on unweighted undirected sparse graphs, via standard constructions from “Fine-grained” complexity. Specifically, under the Strong Exponential-Time Hypothesis (SETH), answering to these above queries requires Ω(|V|2−o(1)) pre-processing time or Ω(|V|1−o(1)) query time, even if we are given as input a hub labeling of maximum label size in ω(log⁡|V|). However, things take a different turn when the hub labels have a sublogarithmic size. Indeed, for every ε>0 there exists a δε>0 such that the following hold, being given a hub labeling of maximum label size ≤k: after pre-processing the labels in total O(2δε⋅k⋅|V|1+ε) time, we can compute both the eccentricity and the distance-sum of any vertex in O(2δε⋅k⋅|V|ε) time. Our data structure is a novel application of the orthogonal range query framework of Cabello and Knauer (2009) [17]. It can also be applied to the fast global computation of some topological indices. Finally, as a by-product of our approach, on any fixed class of unweighted graphs with bounded expansion, we can decide whether the diameter of an n-vertex graph in the class is at most k in fε(k)⋅n1+ε time for any ε>0, for some “explicit” function fε that depends on ε and the graph class considered. This result is further motivated by the empirical evidence that many classes of complex networks have bounded expansion (Demaine et al., 2019 [24]), and that some of these networks, such as the online social networks, have a relatively small diameter.