Evacuation of Equilateral Triangles by Mobile Agents of Limited Communication Range

Abstract

We consider the problem of evacuating $k \ge 2$ mobile agents from a unit-sided equilateral triangle through an exit located at an unknown location on the perimeter of the triangle. The agents are initially located at the centroid of the triangle and they can communicate with other agents at distance at most r with $0\le r \le 1$. An agent can move at speed at most one, and finds the exit only when it reaches the point where the exit is located. The agents can collaborate in the search for the exit. The goal of the evacuation problem is to minimize the evacuation time, defined as the worst-case time for all the agents to reach the exit. We propose and analyze several algorithms for the problem of evacuation by $k \ge 2$ agents; our results indicate that the best strategy to be used varies depending on the values of r and k. For two agents, we give four algorithms, the last of which achieves the best performance for all sub-ranges of r in the range $0 < r \le 1$. We also show a lower bound on the evacuation time of two agents for any $r < 0.336$. For $k >2$ agents, we study three strategies for evacuation: in the first strategy, called X3C, agents explore all three sides of the triangle before connecting to exchange information; in the second strategy, called X1C, agents explore a single side of the triangle before connecting; in the third strategy, called CXP, the agents travel to the perimeter to locations in which they are connected, and explore it while always staying connected. For 3 or 4 agents, we show that X3C works better than X1C for small values of r, while X1C works better for larger values of r. Finally, we show that for any r, evacuation of $k=6 +2\lceil (\frac{1}{r}-1\rceil$ agents can be done using the CXP strategy in time $1+\sqrt{3}/3$, which is optimal in terms of time, and asymptotically optimal in terms of the number of agents.

This research was supported by NSERC, Canada.