Rendezvous Search on the Interval and the Circle

Abstract

Two people are placed randomly and independently on a street of unit length. They attempt to find each other in the shortest possible expected time. We solve this problem, assuming each searcher knows where he or she is on the street, for monotonic density functions for the initial placement (this includes the uniform pdf as a special case). This gives an example of a rendezvous search problem where there is no advantage in being allowed to use asymmetric strategies. We also solve some corresponding problems for the circle when asymmetric strategies are permitted: One of these shows that it can sometimes be optimal for one player to wait for the other to find him.

Optimal Trade-Off Between Speed and Acuity When Searching for a Small Object

“…One might require that one or both of the searchers are in their slow mode to enable rendezvous. In particular the case of rendezvous on the line (as in Gal (1999), Howard (1999); Chester and Tutuncu (2004) and Han et al (2008)) should be re-evaluated in the context in which the agents must slow down to enable mutual detection. We also believe that the problem of search in a maze (unknown network), as studied in Anderson and Gal (1990) could usefully be attacked from this direction.…”

Rendezvous Search: A Personal Perspective

“…We note that Howard (1999) has shown that this result does not hold for asymmetric rendezvous on the circle, although Alpern (2000) has shown that it almost holds there. See §4.3 for these results.…”

“…Then we discuss the solution of Howard (1999) to the rendezvous problem on the closed interval, where the two players are initially placed randomly and independently on the interval, at which time they know their own position but not their partner's. Finally, we discuss the case, mentioned as an example in §2, where rendezvous takes place on the circle.…”

“…The successful work presented in this subsection is due to Howard (1999). We assume that the two players have a common labeling of [0, 1] so that after their initial placement, they know exactly where they are.…”

Rendezvous on a Planar Lattice

“…In every time period each player must move to an adjacent node or stay still, although we show in Theorem 10 staying still is never optimal. This 'even distance'initial placement (originating in the interval network of Howard (1999)) ensures that the two optimizing players will always have the same parity, and cannot pass each other on an edge without meeting at a node. The players both wish to minimize the expected number of periods required for them to be at the same node.…”