## Abstract

This paper deals with compact label-based representations for trees. Consider an n-node undirected connected graph G with a predefined numbering on the ports of each node. The all-ports tree labeling L _{all} gives each node v of G a label containing the port numbers of all the tree edges incident to v. The upward tree labeling L _{up} labels each node v by the number of the port leading from v to its parent in the tree. Our measure of interest is the worst case and total length of the labels used by the scheme, denoted M _{up} (T) and S _{up} (T) for L _{up} and M _{all} (T) and S _{all} (T) for L _{all} . The problem studied in this paper is the following: Given a graph G and a predefined port labeling for it, with the ports of each node v numbered by 0,..., deg∈(v)-1, select a rooted spanning tree for G minimizing (one of) these measures. We show that the problem is polynomial for M _{up} (T), S _{up} (T) and S _{all} (T) but NP-hard for M _{all} (T) (even for 3-regular planar graphs). We show that for every graph G and port labeling there exists a spanning tree T for which S _{up} (T)=O(nlog∈log∈n). We give a tight bound of O(n) in the cases of complete graphs with arbitrary labeling and arbitrary graphs with symmetric port labeling. We conclude by discussing some applications for our tree representation schemes.

Original language | English |
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Pages (from-to) | 1-15 |

Number of pages | 15 |

Journal | Algorithmica |

Volume | 53 |

Issue number | 1 |

DOIs | |

State | Published - Jan 2009 |

Externally published | Yes |

### Bibliographical note

Funding Information:A. Korman supported in part by an Aly Kaufman fellowship.

Funding Information:

R. Cohen supported by the Pacific Theaters Foundation.

Funding Information:

P. Fraigniaud and D. Ilcinkas supported by the project “PairAPair” of the ACI Masses de Données, the project “Fragile” of the ACI Sécurité et Informatique, and by the project “Grand Large” of INRIA.

### Funding

A. Korman supported in part by an Aly Kaufman fellowship. R. Cohen supported by the Pacific Theaters Foundation. P. Fraigniaud and D. Ilcinkas supported by the project “PairAPair” of the ACI Masses de Données, the project “Fragile” of the ACI Sécurité et Informatique, and by the project “Grand Large” of INRIA.

Funders | Funder number |
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ACI Masses de Données | |

ACI Sécurité et Informatique | |

Pacific Theaters Foundation | |

Institut national de recherche en informatique et en automatique (INRIA) | |

Israel Science Foundation |

## Keywords

- Labeling scheme
- Spanning tree
- Tree representation