## 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

Funding Information:OF, PG and OW were supported in part by Israel Science Foundation grant 794/13. Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from permissions@acm.org. PODC ’17, July 25-27, 2017, Washington, DC, USA © 2017 Association for Computing Machinery. ACM ISBN 978-1-4503-4992-5/17/07...$15.00 http://dx.doi.org/10.1145/3087801.3087804

Publisher Copyright:

© 2017 Association for Computing Machinery.

## Keywords

- Labeling scheme
- Universal tree

## ASJC Scopus subject areas

- Software
- Hardware and Architecture
- Computer Networks and Communications