A Lower Bound for Randomized Searching on m Rays

Abstract

We consider the problem of on-line searching on m rays. A point robot is assumed to stand at the origin of m concurrent rays one of which contains a goal g that the point robot has to find. Neither the ray containing g nor the distance to g are known to the robot. The only way the robot can detect g is by reaching its location. We use the competitive ratio as a measure of the performance of a search strategy, that is, the worst case ratio of the total distance D _R traveled by the robot to find g to the distance D from the origin to g. We present a new proof of a tight lower bound of the competitive ratio for randomized strategies to search on m rays. Our proof allows us to obtain a lower bound on the optimal competitive ratio for a fixed m even if the distance of the goal to the origin is bounded from above. Finally, we show that the optimal competitive ratio converges to 1+2(e ^α - 1)/α² m∼1+2·1.544m, for large m where á minimizes the function (e ^x-1)/x ².