Abstract
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 Dpreserving 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 and undirected graphs.
We present a O(n/D * log^2(D)) bit Dpreserving distance labeling scheme, improving the previous bound by Bollobás et al. [SIAM J. Discrete Math. 2005]. We also give an almost matching lower bound of Omega(n/D). With our Dpreserving 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 Omega(n) bits, Moon [Proc. of Glasgow Math. Association 1965].
2. For approximate radditive labeling schemes, that return distances within an additive error of r we show a scheme of size
O(n/r * polylog(r*log(n))/log(n)) for r >= 2. This improves on the current best bound of O(n/r) by Alstrup et al. [SODA 2016] for subpolynomial r, and is a generalization of a result by Gawrychowski et al. [arXiv preprint 2015] who showed this for r=2.
BibTeX  Entry
@InProceedings{alstrup_et_al:LIPIcs:2016:6347,
author = {Stephen Alstrup and S\oren Dahlgaard and Mathias Bæk Tejs Knudsen and Ely Porat},
title = {{Sublinear Distance Labeling}},
booktitle = {24th Annual European Symposium on Algorithms (ESA 2016)},
pages = {5:15:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959770156},
ISSN = {18688969},
year = {2016},
volume = {57},
editor = {Piotr Sankowski and Christos Zaroliagis},
publisher = {Schloss DagstuhlLeibnizZentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2016/6347},
URN = {urn:nbn:de:0030drops63479},
doi = {10.4230/LIPIcs.ESA.2016.5},
annote = {Keywords: Graph labeling schemes, Distance labeling, Graph theory, Sparse graphs}
}
Keywords: 

Graph labeling schemes, Distance labeling, Graph theory, Sparse graphs 
Collection: 

24th Annual European Symposium on Algorithms (ESA 2016) 
Issue Date: 

2016 
Date of publication: 

18.08.2016 