Abstract
We consider distance labeling schemes for trees: given a tree with n nodes, label the nodes with binary strings such that, given the labels of any two nodes, one can determine, by looking only at the labels, the distance in the tree between the two nodes.
A lower bound by Gavoille et al. [Gavoille et al., J. Alg., 2004] and an upper bound by Peleg [Peleg, J. Graph Theory, 2000] establish that labels must use Theta(log^2(n)) bits. Gavoille et al. [Gavoille et al., ESA, 2001] show that for very small approximate stretch, labels use Theta(log(n) log(log(n))) bits. Several other papers investigate various variants such as, for example, small distances in trees [Alstrup et al., SODA, 2003].
We improve the known upper and lower bounds of exact distance labeling by showing that 1/4*log^2(n) bits are needed and that 1/2*log^2(n) bits are sufficient. We also give (1 + epsilon)stretch labeling schemes using Theta(log(n)) bits for constant epsilon > 0. (1 + epsilon)stretch labeling schemes with polylogarithmic label size have previously been established for doubling dimension graphs by Talwar [Talwar, STOC, 2004].
In addition, we present matching upper and lower bounds for distance labeling for caterpillars, showing that labels must have size 2*log(n)  Theta(log(log(n))). For simple paths with k nodes and edge weights in [1,n], we show that labels must have size (k  1)/k*log(n) + Theta(log(k)).
BibTeX  Entry
@InProceedings{alstrup_et_al:LIPIcs:2016:6266,
author = {Stephen Alstrup and Inge Li G\ortz and Esben Bistrup Halvorsen and Ely Porat},
title = {{Distance Labeling Schemes for Trees}},
booktitle = {43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)},
pages = {132:1132:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959770132},
ISSN = {18688969},
year = {2016},
volume = {55},
editor = {Ioannis Chatzigiannakis and Michael Mitzenmacher and Yuval Rabani and Davide Sangiorgi},
publisher = {Schloss DagstuhlLeibnizZentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2016/6266},
URN = {urn:nbn:de:0030drops62661},
doi = {10.4230/LIPIcs.ICALP.2016.132},
annote = {Keywords: Distributed computing, Distance labeling, Graph theory, Routing, Trees}
}
Keywords: 

Distributed computing, Distance labeling, Graph theory, Routing, Trees 
Collection: 

43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016) 
Issue Date: 

2016 
Date of publication: 

23.08.2016 