Hardness of exact distance queries in sparse graphs through hub labeling
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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 ∈ G stores its distance to the so-called hubs S_v ⊆ V, chosen so that for any u,v ∈ V there is w ∈ S_u ∩ 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 |E(G)| = O(n), for which we show a lowerbound of n/2^O(√( 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-Szemerédi 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^( n)^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^O(√( 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 Θ(n/2^( n)^c) for some 0<c < 1.
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