Rendezvous search on the interval and the circle

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. W e solve this problem, assuming each searcher knows where he or she is on th e street, for monotonic density functions for the initial placement (this i ncludes the uniform pdf as a special case). This gives an example of a rend ezvous search problem where there is no advantage in being allowed to use a symmetric strategies. We also solve some corresponding problems for the cir cle when asymmetric strategies are permitted: One of these shows that it ca n sometimes be optimal for one player to wait for the other to find him.