Sublinear-Space Distance Labeling Using Hubs

We propose a labeling scheme for unweighted undirected graphs, that is an assignment of binary labels to vertices of a graph such that the distance between any pair of vertices can be decoded solely from their labels. Recently, Alstrup et al. [3] have shown a labeling scheme for sparse graphs with labels of size O( n D log D) where D = logn log m+n n . We present a simpler approach achieving size O( n D logD) with O(n) decoding time. Based on similar techniques we also present a scheme for general graphs with labels of size log 3 2 n+ o(n) with small decoding time (any ω(1) decoding time is possible), or labels of size ( log 3 2 + ε)n and decoding time O(1). This almost matches the currently best scheme of Alstrup et al. [4] which uses labels of size log 3 2 n+ o(n) with decoding time O(1). We believe our simple techniques are of independent value and provide a desirable simplification of the previous approaches.