## Abstract

Labeling schemes seek to assign a short label to each node in a network, so that a function on two nodes (such as distance or adjacency) can be computed by examining their labels alone. For the particular case of trees, following a long line of research, optimal bounds (up to loworder terms)were recently obtained for adjacency labeling [FOCS '15], nearest common ancestor labeling [SODA '14], and ancestry labeling [SICOMP '06]. In this paper we obtain optimal bounds for distance labeling. We present labels of size 1/4 log^{2} n + o(log^{2} n), matching (up to low order terms) the recent 1/4 log^{2} n - O(log n) lower bound [ICALP '16]. Prior to our work, all distance labeling schemes for trees could be reinterpreted as universal trees. A tree T is said to be universal if any tree on n nodes can be found as a subtree of T. A universal tree with /T/ nodes implies a distance labeling scheme with label size log /T/. In 1981, Chung et al. proved that any distance labeling scheme based on universal trees requires labels of size 1/2 log^{2} n - log n · log log n + O(log n). Our scheme is the first to break this lower bound, showing a separation between distance labeling and universal trees. The Θ(log^{2} n) barrier for distance labeling in trees has led researchers to consider distances bounded by k. The size of such labels was shown to be log n + O(k √log n) in [WADS '01], and then improved to log n + O(k^{2} log(k log n)) in [SODA '03] and finally to log n + O(k log(k log(n/k))) in [PODC '07]. We show how to construct labels whose size is the minimum between log n + O(k log((log n)/k)) and O(log n·log(k/ log n)). We complement this with almost tight lower bounds of log n + Ω(k log(log n/(k log k))) and Ω(log n·log(k/ log n)). Finally, we consider (1+ϵ)-approximate distances. We show that the recent labeling scheme of [ICALP '16] can be easily modified to obtain an O(log(1/ϵ) · log n) upper bound and we prove a matching Ω(log(1/ϵ) · log n) lower bound.

Original language | English |
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Title of host publication | PODC 2017 - Proceedings of the ACM Symposium on Principles of Distributed Computing |

Publisher | Association for Computing Machinery |

Pages | 185-194 |

Number of pages | 10 |

ISBN (Electronic) | 9781450349925 |

DOIs | |

State | Published - 26 Jul 2017 |

Event | 36th ACM Symposium on Principles of Distributed Computing, PODC 2017 - Washington, United States Duration: 25 Jul 2017 → 27 Jul 2017 |

### Publication series

Name | Proceedings of the Annual ACM Symposium on Principles of Distributed Computing |
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Volume | Part F129314 |

### Conference

Conference | 36th ACM Symposium on Principles of Distributed Computing, PODC 2017 |
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Country/Territory | United States |

City | Washington |

Period | 25/07/17 → 27/07/17 |

### Bibliographical note

Publisher Copyright:© 2017 Association for Computing Machinery.

## Keywords

- Labeling scheme
- Universal tree

## ASJC Scopus subject areas

- Software
- Hardware and Architecture
- Computer Networks and Communications