## Abstract

We consider a search problem on trees using unreliable guiding instructions. Specifically, an agent starts a search at the root of a tree aiming to find a treasure hidden at one of the nodes by an adversary. Each visited node holds information, called advice, regarding the most promising neighbor to continue the search. However, the memory holding this information may be unreliable. Modeling this scenario, we focus on a probabilistic setting. That is, the advice at a node is a pointer to one of its neighbors. With probability q each node is faulty, independently of other nodes, in which case its advice points at an arbitrary neighbor, chosen uniformly at random. Otherwise, the node is sound and points at the correct neighbor. Crucially, the advice is permanent, in the sense that querying a node several times would yield the same answer. We evaluate efficiency by two measures: The move complexity denotes the expected number of edge traversals, and the query complexity denotes the expected number of queries. Let Δ denote the maximal degree. Roughly speaking, the main message of this paper is that a phase transition occurs when the noise parameter q is roughly 1/v Δ. More precisely, we prove that above the threshold, every search algorithm has query complexity (and move complexity) which is both exponential in the depth d of the treasure and polynomial in the number of nodes n. Conversely, below the threshold, there exists an algorithm with move complexity O(d√Δ), and an algorithm with query complexity O(√Δ log Δlog^{2} n). Moreover, for the case of regular trees, we obtain an algorithm with query complexity O(√Δ log n log log n). For q that is below but close to the threshold, the bound for the move complexity is tight, and the bounds for the query complexity are not far from the lower bound of Ω(√Δlog_{Δ} n). In addition, we also consider a semi-adversarial variant, in which an adversary chooses the direction of advice at faulty nodes. For this variant, the threshold for efficient moving algorithms happens when the noise parameter is roughly 1/Δ. Above this threshold a simple protocol that follows each advice with a fixed probability already achieves optimal move complexity.

Original language | American English |
---|---|

Title of host publication | 26th European Symposium on Algorithms, ESA 2018 |

Editors | Hannah Bast, Grzegorz Herman, Yossi Azar |

Publisher | Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing |

ISBN (Print) | 9783959770811 |

DOIs | |

State | Published - 1 Aug 2018 |

Externally published | Yes |

Event | 26th European Symposium on Algorithms, ESA 2018 - Helsinki, Finland Duration: 20 Aug 2018 → 22 Aug 2018 |

### Publication series

Name | Leibniz International Proceedings in Informatics, LIPIcs |
---|---|

Volume | 112 |

### Conference

Conference | 26th European Symposium on Algorithms, ESA 2018 |
---|---|

Country/Territory | Finland |

City | Helsinki |

Period | 20/08/18 → 22/08/18 |

## Keywords

- Average case analysis
- Data structures
- Graph search

## All Science Journal Classification (ASJC) codes

- Software