Symmetric Rendezvous With Advice: How to Rendezvous in a Disk

Abstract

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.