Centre for Discrete and Applicable Mathematics

 CDAM Research Report, LSE-CDAM-2000-13

July 2000

Asymmetric Rendezvous Search on the Circle

Steve Alpern


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 arc-distance  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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