### Abstract

We study how to label the vertices of a tree in such a way that we can decide the distance of two vertices in the tree given only their labels. For trees, Gavoille et al. [7] proved that for any such distance labelling scheme, the maximum label length is at least 1/8 log^{2} n - O(log n) bits. They also gave a separator-based labelling scheme that has the optimal label length Θ(log n · log(H_{n}(T))), where H_{n}(T) is the height of the tree. In this paper, we present two new distance labelling schemes that not only achieve the optimal label length Θ (log n · log(H_{n}(T))), but also have a much smaller expected label length under certain tree distributions. With these new schemes, we also can efficiently find the least common ancestor of any two vertices based on their labels only.

Original language | English (US) |
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Title of host publication | Algorithms and Computation - 16th International Symposium, ISAAC 2005, Proceedings |

Pages | 136-145 |

Number of pages | 10 |

DOIs | |

State | Published - Dec 1 2005 |

Event | 16th International Symposium on Algorithms and Computation, ISAAC 2005 - Hainan, China Duration: Dec 19 2005 → Dec 21 2005 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 3827 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Other

Other | 16th International Symposium on Algorithms and Computation, ISAAC 2005 |
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Country | China |

City | Hainan |

Period | 12/19/05 → 12/21/05 |

### Fingerprint

### ASJC Scopus subject areas

- Theoretical Computer Science
- Computer Science(all)

### Cite this

*Algorithms and Computation - 16th International Symposium, ISAAC 2005, Proceedings*(pp. 136-145). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 3827 LNCS). https://doi.org/10.1007/11602613_15

}

*Algorithms and Computation - 16th International Symposium, ISAAC 2005, Proceedings.*Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 3827 LNCS, pp. 136-145, 16th International Symposium on Algorithms and Computation, ISAAC 2005, Hainan, China, 12/19/05. https://doi.org/10.1007/11602613_15

**Average case analysis for tree labelling schemes.** / Kao, Ming-Yang; Li, Xiang Yang; Wang, Weizhao.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

TY - GEN

T1 - Average case analysis for tree labelling schemes

AU - Kao, Ming-Yang

AU - Li, Xiang Yang

AU - Wang, Weizhao

PY - 2005/12/1

Y1 - 2005/12/1

N2 - We study how to label the vertices of a tree in such a way that we can decide the distance of two vertices in the tree given only their labels. For trees, Gavoille et al. [7] proved that for any such distance labelling scheme, the maximum label length is at least 1/8 log2 n - O(log n) bits. They also gave a separator-based labelling scheme that has the optimal label length Θ(log n · log(Hn(T))), where Hn(T) is the height of the tree. In this paper, we present two new distance labelling schemes that not only achieve the optimal label length Θ (log n · log(Hn(T))), but also have a much smaller expected label length under certain tree distributions. With these new schemes, we also can efficiently find the least common ancestor of any two vertices based on their labels only.

AB - We study how to label the vertices of a tree in such a way that we can decide the distance of two vertices in the tree given only their labels. For trees, Gavoille et al. [7] proved that for any such distance labelling scheme, the maximum label length is at least 1/8 log2 n - O(log n) bits. They also gave a separator-based labelling scheme that has the optimal label length Θ(log n · log(Hn(T))), where Hn(T) is the height of the tree. In this paper, we present two new distance labelling schemes that not only achieve the optimal label length Θ (log n · log(Hn(T))), but also have a much smaller expected label length under certain tree distributions. With these new schemes, we also can efficiently find the least common ancestor of any two vertices based on their labels only.

UR - http://www.scopus.com/inward/record.url?scp=33744960547&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33744960547&partnerID=8YFLogxK

U2 - 10.1007/11602613_15

DO - 10.1007/11602613_15

M3 - Conference contribution

AN - SCOPUS:33744960547

SN - 3540309357

SN - 9783540309352

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 136

EP - 145

BT - Algorithms and Computation - 16th International Symposium, ISAAC 2005, Proceedings

ER -