## 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 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 and undirected graphs. We present a O(n/D log^{2} D) bit D-preserving 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 Ω(n/D). 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].^{1} For approximate r-additive 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 sub-polynomial r, and is a generalization of a result by Gawrychowski et al. [arXiv preprint 2015] who showed this for r = 2.

Original language | English |
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Title of host publication | 24th Annual European Symposium on Algorithms, ESA 2016 |

Editors | Christos Zaroliagis, Piotr Sankowski |

Publisher | Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing |

ISBN (Electronic) | 9783959770156 |

DOIs | |

State | Published - 1 Aug 2016 |

Event | 24th Annual European Symposium on Algorithms, ESA 2016 - Aarhus, Denmark Duration: 22 Aug 2016 → 24 Aug 2016 |

### Publication series

Name | Leibniz International Proceedings in Informatics, LIPIcs |
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Volume | 57 |

ISSN (Print) | 1868-8969 |

### Conference

Conference | 24th Annual European Symposium on Algorithms, ESA 2016 |
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Country/Territory | Denmark |

City | Aarhus |

Period | 22/08/16 → 24/08/16 |

### Bibliographical note

Publisher Copyright:© Stephen Alstrup, Søren Dahlgaard, Mathias Bæk Tejs Knudsen, and Ely Porat.

## Keywords

- Distance labeling
- Graph labeling schemes
- Graph theory
- Sparse graphs