Distance Labelings on Random Power Law Graphs
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A Distance Labeling scheme is a data structure that can answer shortest path queries on a graph. Experiment results from several recent studies (Akiba et al.'13, Delling et al.'14) found very efficient and very accurate labeling schemes, which scale to social and information networks with tens of millions of vertices and edges. Such a finding is not expected in the worst case, since even for graphs with maximum degree 3, it is known that any distance labeling requires Ω(n^3/2) space (Gavoille et al.'03). On the other hand, social and information networks have a heavy-tailed degree distribution and small average distance, which are not captured in the worst case. In this paper, we fill in the gap between empirical and worst case results. We consider distance labeling schemes on random graph models with a power law degree distribution. For such graphs, we show that simple breadth-first-search based algorithm can find near optimal labeling schemes. The intuition behind our proof reveals that the distances between different pairs of vertices are almost independent, even for polynomially many pairs.
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