A distance labeling scheme labels the n nodes of a graph with binary
strings such that, given the labels of any two nodes, one can determine the
distance in the graph between the two nodes by looking only at the labels. A
D-preserving distance labeling scheme only returns precise distances between
pairs of nodes that are at distance at least D from each other. In this paper
we consider distance labeling schemes for the classical case of unweighted
graphs with both directed and undirected edges.
We present a O(Dnlog2D) bit D-preserving distance labeling
scheme, improving the previous bound by Bollob\'as et. al. [SIAM J. Discrete
Math. 2005]. We also give an almost matching lower bound of
Ω(Dn). With our D-preserving distance labeling scheme as a
building block, we additionally achieve the following results:
1. We present the first distance labeling scheme of size o(n) for sparse
graphs (and hence bounded degree graphs). This addresses an open problem by
Gavoille et. al. [J. Algo. 2004], hereby separating the complexity from
distance labeling in general graphs which require Ω(n) bits, Moon [Proc.
of Glasgow Math. Association 1965].
2. For approximate r-additive labeling schemes, that return distances
within an additive error of r we show a scheme of size O(rn⋅lognpolylog(rlogn)) for r≥2.
This improves on the current best bound of O(rn) by
Alstrup et. al. [SODA 2016] for sub-polynomial r, and is a generalization of
a result by Gawrychowski et al. [arXiv preprint 2015] who showed this for
r=2.Comment: A preliminary version of this paper appeared at ESA'1