Symmetric Rendezvous with Advice: How to Rendezvous in a Disk

In the classic Symmetric Rendezvous problem on a Line (SRL), two robots at known distance 2 but unknown direction execute the same randomized algorithm trying to minimize the expected rendezvous time. A long standing conjecture is that the best possible rendezvous time is 4.25 with known upper and lower bounds being very close to that value. We introduce and study a geometric variation of SRL that we call Symmetric Rendezvous in a Disk (SRD) where two robots at distance 2 have a common reference point at distance \(\rho \). We show that even when \(\rho \) is not too small, the two robots can meet in expected time that is less than 4.25. Part of our contribution is that we demonstrate how to adjust known, even simple and provably non-optimal, algorithms for SRL, effectively improving their performance in the presence of a reference point. Special to our algorithms for SRD is that, unlike in SRL, for every fixed \(\rho \) the worst case distance traveled, i.e. energy that is used, in our algorithms is finite. In particular, we show that the energy of our algorithms is \(O\left( \rho ^2\right) \), while we also explore time-energy tradeoffs, concluding that one may be efficient both with respect to time and energy, with only a minor compromise on the optimal termination time.