Brief announcement: Sublinear-space distance labeling using hubs

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. We propose a series of new labeling schemes within the framework of so-called hub labeling (HL, also known as landmark labeling or 2-hop-cover labeling), in which each node u stores its distance to all nodes from an appropriately chosen set of hubs S(u) ⊆ V . For a queried pair of nodes (u; v), the length of a shortest u..v-path passing through a hub node from S(u)∩(v) is then used as an upper bound on the distance between u and v. We present a hub labeling which allows us to decode ex- act distances in sparse graphs using labels of size sublinear in the number of nodes. For graphs with at most n nodes and average degree δ, the tradeoff between label bit size L and query decoding time T for our approach is given by L = O(n log log_δ T= log_δ T), for any T ≤ n. Our simple approach is thus the first sublinear-space distance labeling for sparse graphs that simultaneously admits small decoding time (for constant δ, we can achieve any T = ω(1) while maintaining L = o(n)), and it also provides an improve- ment in terms of label size with respect to previous slower approaches. By using similar techniques, we then present a 2-additive labeling scheme for general graphs, i.e., one in which the decoder provides a 2-additive-approximation of the distance between any pair of nodes. We achieve the same label size- time tradeoff L = O(n log² log T= log T), for any T ≤ n. To our knowledge, this is the first additive scheme with con- stant absolute error to use labels of sublinear size. The corresponding decoding time is then small (any T = ω(1) is sufficient).