Computer Science in Perspective pp 264-277 | Cite as

# A Lower Bound for Randomized Searching on *m* Rays

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## 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)/α^{2} *m*∼1+2·1.544*m*, for large *m* where á minimizes the function (*e* ^{x}-1)/*x* ^{2}.

## Keywords

Step Length Competitive Ratio Simple Polygon Goal Position Hiding Strategy## Preview

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