Mutual search

We introduce a search problem called “mutual search” where <italic>k</italic> agents, arbitrarily distributed over <italic>n</italic> sites, are required to locate one another by posing queries of the form “Anybody at site <italic>i</italic>?”. 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: <italic>n</italic>−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 0.586<italic>n</italic> queries are required; in the asynchronous case 0.896<italic>n</italic> queries suffice and a fortiori 0.536<italic>n</italic> queries are required; for <inline-equation> <f> o<fen lp="par"><rad><rcd>n</rcd></rad><rp post="par"></fen></f> </inline-equation> agents using a synchronous deterministic protocol less than <?Pub Fmt italic>n<?Pub Fmt /italic> queries suffice; there is a simple randomized protocol for two agents with worst-case expected 0.5<?Pub Fmt italic>n<?Pub Fmt /italic> queries and all radomized protocols require at least 0.25<italic>n</italic> worst-case expected queries. The graph-theoretic framework we formulate for expressing and analyzing algorithms for this problem may be of independent interest.