### Sublinear-Space Distance Labeling using Hubs [article]

Paweł Gawrychowski, Adrian Kosowski, Przemysław Uznański
2016 arXiv   pre-print
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
more » ... length of a shortest u-v-path passing through a hub node from S(u)∩ S(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 exact 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 _Δ T / _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 improvement 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 almost the same label size-time tradeoff L = O(n ^2 T / T), for any T ≤ n. To our knowledge, this is the first additive scheme with constant absolute error to use labels of sublinear size. The corresponding decoding time is then small (any T=ω(1) is sufficient).