Hardness of Exact Distance Queries in Sparse Graphs Through Hub Labeling

Abstract

A distance labeling scheme is an assignment of bit-labels to the vertices of an undirected, unweighted graph such that the distance between any pair of vertices can be decoded solely from their labels. An important class of distance labeling schemes is that of hub labelings, where a node ν ∈ G stores its distance to the so-called hubs Sν ⊆ V, chosen so that for any u,ν ∈ V there is w ∈ Su ∩ Sv belonging to some shortest uv path. Notice that for most existing graph classes, the best distance labelling constructions existing use at some point a hub labeling scheme at least as a key building block.Our interest lies in hub labelings of sparse graphs, i.e., those with |E(G)| = O (n), for which we show a lowerbound of n 2O (√log n) for the average size of the hubsets. Additionally, we show a hub-labeling construction for sparse graphs of average size O(√n RS (n)c) for some 0