A Note on Randomized Mutual Search

In Mutual Search, recently introduced by Buhrman et al. in [1], 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.