## Abstract

We consider a parallel version of a classical Bayesian search problem. k agents are looking for a treasure that is placed in one of the boxes indexed by ℕ^{+} according to a known distribution p. The aim is to minimize the expected time until the first agent finds it. Searchers run in parallel where at each time step each searcher can “peek” into a box. A basic family of algorithms which are inherently robust is non-coordinating algorithms. Such algorithms act independently at each searcher, differing only by their probabilistic choices. We are interested in the price incurred by employing such algorithms when compared with the case of full coordination. We first show that there exists a non-coordination algorithm, that knowing only the relative likelihood of boxes according to p, has expected running time of at most 10 + 4(1 + 1/k)^{2}T,, where T is the expected running time of the best fully coordinated algorithm. This result is obtained by applying a refined version of the main algorithm suggested by Fraigniaud, Korman and Rodeh in STOC’16, which was designed for the context of linear parallel search. We then describe an optimal non-coordinating algorithm for the case where the distribution p is known. The running time of this algorithm is difficult to analyse in general, but we calculate it for several examples. In the case where p is uniform over a finite set of boxes, then the algorithm just checks boxes uniformly at random among all non-checked boxes and is essentially 2 times worse than the coordinating algorithm. We also show simple algorithms for Pareto distributions over M boxes. That is, in the case where p(x) ~ 1/x^{b} for 0 < b < 1, we suggest the following algorithm: at step t choose uniformly from the boxes unchecked in {1, …, min(M,⌊t/σ⌋)},, where σ = b/(b + k − 1). It turns out this algorithm is asymptotically optimal, and runs about 2 + b times worse than the case of full coordination.

Original language | English |
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Title of host publication | Structural Information and Communication Complexity - 24th International Colloquium, SIROCCO 2017, Revised Selected Papers |

Editors | Shantanu Das, Sebastien Tixeuil |

Publisher | Springer Verlag |

Pages | 195-211 |

Number of pages | 17 |

ISBN (Print) | 9783319720494 |

DOIs | |

State | Published - 2017 |

Externally published | Yes |

Event | 24th International Colloquium on Structural Information and Communication Complexity, SIROCCO 2017 - Porquerolles, France Duration: 19 Jun 2017 → 22 Jun 2017 |

### 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 | 10641 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Conference

Conference | 24th International Colloquium on Structural Information and Communication Complexity, SIROCCO 2017 |
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Country/Territory | France |

City | Porquerolles |

Period | 19/06/17 → 22/06/17 |

### Bibliographical note

Publisher Copyright:© Springer International Publishing AG 2017.

## ASJC Scopus subject areas

- Theoretical Computer Science
- General Computer Science