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 ρ. We show that even when ρ 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 ρ 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 ρ 2 , 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. This is the full version of the paper with the same title which will appear in the The rendezvous problem was first proposed informally by Alpern [1] in 1976, and received attention due to the seminal works of Anderson and Weber [12] for discrete domains and of Alpern [2] for continuous domains. Our work is a direct generalization of the special and so-called Symmetric Rendezvous Search Problem on a Line (SRL) proposed by Alpern [2] in 1995. In that problem, two blind agents are at known distance 2 on a line, and they can perform the same synchronized randomized algorithm (with no shared randomness). The original algorithm of Alpern [2] had performance (expected rendezvous time) 5, which was later improved to 4.5678 [13], then to 4.4182 [15], then to 4.3931 [35], and finally to the best performance known of 4.2574 [28] by Han et al. Similarly, a series of proven lower bounds [8], [35] have lead to the currently best value known of 4.1520 [28].A number of variations of SRL have been exhaustively studied, and below we mention just a few. The symmetric rendezvous problem with unknown initial distance or with partial information about it has been considered in [17] and [16]. A number of different topologies have been considered including labeled network [10], labeled line [18], ring [31], [27] (see survey monograph [30]), torus [29], planar lattice [5], and high dimensional host spaces [7]. We note here that the topology we consider in this work follows a long list studies of relevant search/rendezvous-type problems in the disk. The rendezvous problem with faulty components has been studied in [24] and [25]. Asynchronous strategies have been explored in [34] and [33]. Studied variations of robots capabilities include sense of direction [6], [14], memory [20], visibility [22], speed [26], pow...

No abstract

No abstract