Hardness Of Exact Distance Queries In Sparse Graphs Through Hub Labeling

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 v is an element of G stores its distance to the so-called hubs S-v subset of V, chosen so that for any u, v is an element of V there is w is an element of S-u boolean AND S-v 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 vertical bar E(G)vertical bar = O(n), for which we show a lowerbound of n/2(O(root 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 < c < 1, where RS(n) is the so-called Ruzsa-Szemeredi function, linked to structure of induced matchings in dense graphs. This implies that further improving the lower bound on hub labeling size to n/2((logn)o(1)) would require a breakthrough in the study of lower bounds on RS(n), which have resisted substantial improvement in the last 70 years.For general distance labeling of sparse graphs, we show a lowerbound of 1/2(Theta(root log n))SUMINDEX(n), where SUMINDEX(n) is the communication complexity of the Sum-Index problem over Z(n). Our results suggest that the best achievable hub-label size and distance-label size in sparse graphs may be Theta(n/2((log n)c)) for some 0 < c < 1.