## Abstract

A finite automaton, simply referred to as a robot, has to explore a graph whose nodes are unlabeled and whose edge ports are locally labeled at each node. The robot has no a priori knowledge of the topology of the graph or of its size. Its task is to traverse all the edges of the graph. We first show that, for any K-state robot and any d≥3, there exists a planar graph of maximum degree d with at most K+1 nodes that the robot cannot explore. This bound improves all previous bounds in the literature. More interestingly, we show that, in order to explore all graphs of diameter D and maximum degree d, a robot needs Ω(Dlogd) memory bits, even if we restrict the exploration to planar graphs. This latter bound is tight. Indeed, a simple DFS up to depth D+1 enables a robot to explore any graph of diameter D and maximum degree d using a memory of size O(Dlogd) bits. We thus prove that the worst case space complexity of graph exploration is Θ(Dlogd) bits.

Original language | English |
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Pages (from-to) | 331-344 |

Number of pages | 14 |

Journal | Theoretical Computer Science |

Volume | 345 |

Issue number | 2-3 |

DOIs | |

State | Published - 22 Nov 2005 |

Externally published | Yes |

### Bibliographical note

Funding Information:A preliminary version of this paper appeared in the Proceedings of the 29th International Symposium on Mathematical Foundations of Computer Science (MFCS) [29]. E-mail addresses: (P. Fraigniaud), [email protected] (D. Ilcinkas), [email protected] (G. Peer), [email protected] (A. Pelc), [email protected] (D. Peleg). 1 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. 2Supported in part by NSERC Grant OGP 0008136 and by the Research Chair in Distributed Computing of the Université du Québec en Outaouais.

### Funding

A preliminary version of this paper appeared in the Proceedings of the 29th International Symposium on Mathematical Foundations of Computer Science (MFCS) [29]. E-mail addresses: (P. Fraigniaud), [email protected] (D. Ilcinkas), [email protected] (G. Peer), [email protected] (A. Pelc), [email protected] (D. Peleg). 1 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. 2Supported in part by NSERC Grant OGP 0008136 and by the Research Chair in Distributed Computing of the Université du Québec en Outaouais.

Funders | Funder number |
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ACI Masses de Données | |

ACI Sécurité et Informatique | |

Université du Québec en Outaouais | |

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

Natural Sciences and Engineering Research Council of Canada | OGP 0008136 |

## Keywords

- Finite automaton
- Graph exploration
- Labyrinth
- Mobile agent
- Robot