Information Processing Letters 71 (1999) 187–191 A note on randomized mutual search ✩ Zvi Lotker ∗ , Boaz Patt-Shamir 1 Department ofElectrical Engineering, Tel Aviv University, Tel Aviv 69978, Israel Received 14 July 1999; received in revised form 20 September 1999 Communicated by P.M.B. Vitányi 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.  1999 Published by Elsevier Science B.V. All rights reserved. Keywords: Algorithms; Two-agent mutual search; Randomized algorithms; Upper bound; Lower bound 1. Introduction Background The concept of Mutual Search, recently introduced by Buhrman et al. [1], is an extension of classical search problems to a distributed environment. In this problem, two “agents” are searching for each other. 2 Intuitively, each agent is assigned one of n memory ✩ Research partially supported by a grant from Israel Ministry of Science and Technology. ∗ Corresponding author. Email: zvilo@eng.tau.ac.il. 1 Email: boaz@eng.tau.ac.il. 2 In [1], a general case of k agents operating in an asynchronous environment is defined too. In this note we focus on the basic syn- chronous two-agent case, which seems to represent the combinator- ial difficulty of the problem. locations, and computation proceeds by the agents sending queries to locations. The agents are static during the computation, and a query which is sent to a vacant location is useless: that is, the result of a query is merely “success” if the other agent is at the queried location, and “failure” otherwise. The computation ends when a successful query is made. The cost of a search is just the number of queries made until termination. A few real-life situations motivating Mutual Search are presented Buhrman et al. [1]. In their paper, upper and lower bounds are given for the cost of deterministic and randomized algorithms for Mutual Search, mostly for two agents. In particular, for randomized algorithms they present an upper bound (i.e., a protocol with worst-case expected cost) of ⌈(n + 1)/2⌉, and a lower bound of (n − 1)/8 queries. 0020-0190/99/$ – see front matter  1999 Published by Elsevier Science B.V. All rights reserved. PII:S0020-0190(99)00112-X