## 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 ⊖(log2 n) bits1. Gavoille et al. [Gavoille et al., ESA, 2001] show that for very small approximate stretch, labels use ⊖(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 log2 n bits are needed and that 1 2 log2 n bits are sufficient. We also give (1+ϵ)-stretch labeling schemes using ⊖(log n) bits for constant ϵ > 0. (1 + ϵ)-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⊖(log log n). For simple paths with k nodes and edge weights in [1, n], we show that labels must have size (k1)/k log n + ⊖(log k).

Original language | English |
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Title of host publication | 43rd International Colloquium on Automata, Languages, and Programming, ICALP 2016 |

Editors | Yuval Rabani, Ioannis Chatzigiannakis, Davide Sangiorgi, Michael Mitzenmacher |

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

ISBN (Electronic) | 9783959770132 |

DOIs | |

State | Published - 1 Aug 2016 |

Event | 43rd International Colloquium on Automata, Languages, and Programming, ICALP 2016 - Rome, Italy Duration: 12 Jul 2016 → 15 Jul 2016 |

### Publication series

Name | Leibniz International Proceedings in Informatics, LIPIcs |
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Volume | 55 |

ISSN (Print) | 1868-8969 |

### Conference

Conference | 43rd International Colloquium on Automata, Languages, and Programming, ICALP 2016 |
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Country/Territory | Italy |

City | Rome |

Period | 12/07/16 → 15/07/16 |

### Bibliographical note

Publisher Copyright:© Stephen Alstrup, Esben Bistrup Halvorsen, Inge Li Grtz, and Ely Porat.

## Keywords

- Distance labeling
- Distributed computing
- Graph theory
- Routing
- Trees