Centre for Discrete and Applicable Mathematics 

CDAM Research Report, LSECDAM200013July 2000 
Steve Alpern
Abstract
The rendezvous search problem asks how two blind searchers in a known search region, having maximum speed one, can minimize the expected time needed to meet. Suppose that two players are placed an arcdistance x in [0,1/2] apart on a circle of circumference 1, and faced in random directions. If x has a continuous density function h which is either decreasing and satisfies h(1/2) > h(0)/2, or increasing, we determine an optimal rendezvous strategy. Furthermore if h is strictly monotone, this strategy (which depends in a simple manner on h ) is uniquely optimal. This work extends that of J.V. Howard, who showed for the uniform density that `search and wait' is optimal, with expected search time 1/2. We also show that the uniform density is the only counterexample on the circle to S. Gal's conjecture (which he proved for the line) on the nonoptimality of `search and wait'.
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