Abstract
We study the online problem of evacuating k robots on m concurrent rays to a single unknown exit. All k robots start on the same point \(s\), not necessarily on the junction \(j\) of the m rays, move at unit speed, and can communicate wirelessly. The goal is to minimize the competitive ratio, i.e., the ratio between the time it takes to evacuate all robots to the exit and the time it would take if the location of the exit was known in advance, on a worst-case instance.
When \(k=m\), we show that a simple waiting strategy yields a competitive ratio of 4 and present a lower bound of \(2+\sqrt{7/3} \approx 3.52753\) for all \(k=m\ge 3\). For \(k=3\) robots on \(m=3\) rays, we give a class of parametrized algorithms with a nearly matching competitive ratio of \(2+\sqrt{3} \approx 3.73205\). We also present an algorithm for \(1<k<m\), achieving a competitive ratio of \(1 + 2 \cdot \frac{m - 1}{k} \cdot \left( 1 + \frac{k}{m - 1} \right) ^{1 + \frac{m-1}{k}}\), obtained by parameter optimization on a geometric search strategy. Interestingly, the robots can be initially oblivious to the value of \(m > 2\).
Lastly, by using a simple but fundamental argument, we show that for \(k<m\) robots, no algorithm can reach a competitive ratio better than \(3+2\left\lfloor (m-1)/k \right\rfloor \), for every k, m with \(k<m\).
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Notes
- 1.
If \(m=2\), then a competitive ratio of 3 can be reached again, as every point can be seen as the junction.
- 2.
Here we implicitly use that \(\alpha \beta ^{m-1} < 1\) which ensures that the ray on which \(R_h\) finds the exit, has been previously explored by some robot.
- 3.
For the following calculation of the upper bound, we assume for simplicity that if \(h = 0\), then \(R_{h-1 \pmod {m}}\) performs a 0th exploration step of length \(\alpha \beta ^{-1}\) before its 1st exploration step. Since this can only increase the upper bound, the given bound also holds if \(h = 0\).
- 4.
Here, a detail has to be mentioned: By changing the mapping of the m labels \(a'_0, \dots , a'_{m-1}\) to the m actual rays, we can change which robot is on which ray. We assume that the labels are changed in a way that ensures that \(R_0\) is actually on ray \(a_0\) in the third distribution.
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Acknowledgments
We would like to thank the anonymous reviewers for their helpful comments. Klaus-Tycho Foerster is supported by the Danish VILLUM FONDEN project ReNet.
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Brandt, S., Foerster, KT., Richner, B., Wattenhofer, R. (2017). Wireless Evacuation on m Rays with k Searchers. In: Das, S., Tixeuil, S. (eds) Structural Information and Communication Complexity. SIROCCO 2017. Lecture Notes in Computer Science(), vol 10641. Springer, Cham. https://doi.org/10.1007/978-3-319-72050-0_9
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