## Abstract

In Mutual Search, recently introduced by Buhrman et al. (1998), static agents are searching for each other: each agent is assigned one of n locations, and the computations proceed by agents sending queries from their location to other locations, until one of the queries arrives at the other agent. The cost of a search is the number of queries made. The best known bounds for randomized protocols using private coins are (1) a protocol with worst-case expected cost of [(n + 1)/2], and (2) a lower bound of (n - 1)/8 queries for randomized protocols which make only a bounded number of coin-tosses. In this paper we strictly improve the lower bound, and present a new upper bound for shared random coins. Specifically, we first prove that the worst-case expected cost of any randomized protocol for two-agent mutual search is at least (n + 1)/3. This is an improvement both in terms of number of queries and in terms of applicability. We also give a randomized algorithm for mutual search with worst-case expected cost of (n + 1)/3. This algorithm works under the assumption that the agents share a random bit string. This bound shows that no better lower bound can be obtained using our technique.

Original language | English |
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Pages (from-to) | 187-191 |

Number of pages | 5 |

Journal | Information Processing Letters |

Volume | 71 |

Issue number | 5-6 |

DOIs | |

State | Published - 30 Sep 1999 |

Externally published | Yes |

### Bibliographical note

Funding Information:I Research partially supported by a grant from Israel Ministry of Science and Technology. ∗Corresponding author. Email: zvilo@eng.tau.ac.il. 1Email: boaz@eng.tau.ac.il. 2In [1], a general case of k agents operating in an asynchronous environment is defined too. In this note we focus on the basic synchronous two-agent case, which seems to represent the combinatorial difficulty of the problem.

## Keywords

- Algorithms
- Lower bound
- Randomized algorithms
- Two-agent mutual search
- Upper bound