Mutual Search

We introduce a search problem called ``mutual search'' where $k$ \agents, arbitrarily distributed over $n$ sites, are required to locate one another by posing queries of the form ``Anybody at site $i$?''. We ask for the least number of queries that is necessary and sufficient. For the case of two \agents using deterministic protocols we obtain the following worst-case results: In an oblivious setting (where all pre-planned queries are executed) there is no savings: $n-1$ queries are required and are sufficient. In a nonoblivious setting we can exploit the paradigm of ``no news is also news'' to obtain significant savings: in the synchronous case $0.586n$ queries suffice and $0.536n$ queries are required; in the asynchronous case $0.896n$ queries suffice and a fortiori 0.536 queries are required; for $o(\sqrt{n})$ \agents using a deterministic protocol less than $n$ queries suffice; there is a simple randomized protocol for two \agents with worst-case expected $0.5n$ queries and all randomized protocols require at least $0.125n$ worst-case expected queries. The graph-theoretic framework we formulate for expressing and analyzing algorithms for this problem may be of independent interest.