Wireless Evacuation on m Rays with k Searchers

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 $k<m$.