Evacuating Equilateral Triangles and Squares in the Face-to-Face Model
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Consider k robots initially located at a point inside a region T. Each robot can move anywhere in T independently of other robots with maximum speed one. The goal of the robots is to evacuate T through an exit at an unknown location on the boundary of T. The objective is to minimize the evacuation time, which is defined as the time the last robot reaches the exit. We consider the face-to-face communication model for the robots: a robot can communicate with another robot only when they meet in T. In this paper, we give upper and lower bounds for the face-to-face evacuation time by k robots that are initially located at the centroid of a unit-sided equilateral triangle or square. For the case of a triangle with k=2 robots, we give a lower bound of 1+2/√(3)≈ 2.154, and an algorithm with upper bound of 2.3367 on the worst-case evacuation time. We show that for any k, any algorithm for evacuating k≥ 2 robots requires at least √(3) time. This bound is asymptotically optimal, as we show that even a straightforward strategy of evacuation by k robots gives an upper bound of √(3) + 3/k. For k=3 and 4, we give better algorithms with evacuation times of 2.0887 and 1.9816, respectively. For the case of the square and k=2, we give an algorithm with evacuation time of 3.4645 and show that any algorithm requires time at least 3.118 to evacuate in the worst-case. Moreover, for k=3, and 4, we give algorithms with evacuation times 3.1786 and 2.6646, respectively. The algorithms given for k=3 and 4 for evacuation in the triangle or the square can be easily generalized for larger values of k.
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