Evacuation
Robots start at the centre of a disk (or a polygon) and must all leave through an exit at an unknown point of the boundary, as fast as possible.
Problem statement and measuresexpand
\(k\) unit-speed robots start together at the centre of a unit disk (or at the centroid, or at a given point of the boundary, of a unit equilateral triangle or square). An exit lies at an unknown point of the boundary; a robot recognises it only on reaching it. Time is the worst case, over exit positions, of the time until the last robot reaches the exit (in priority evacuation, where one robot is the queen: until the queen does; in weighted evacuation of two robots, a weighted average of their arrival times; in search, until a non-faulty robot has reached the exit and every non-faulty robot knows where it is).
Measures: Evacuation time, worst-case time until the last robot (or the queen) reaches the exit, or the weighted average or search time where the goal says so; unit disk, unit speed.
Triangle, any number of wireless robots at the centroid
verified\(k\) wireless robots at the centroid evacuate in \(1+\sqrt3/3+3/k\), and no number of robots beats \(1+\sqrt3/3\), which 8 robots attain.
Assumed (added on the last edge are outlined)
Not assumed
Formal definitions of this setting
- Domain: triangle
- The equilateral triangle of unit side; robots start at its centroid; the exit is a point of the boundary.
- Number of robots: any \(k\)
- \(k\ge1\) robots.
- Goal: all robots
- The task ends when the last robot reaches the exit.
- Exits: one exit
- One exit, at an unknown point of the boundary.
- Communication: wireless
- Wireless: every robot can send messages to every other robot instantly at any distance.
- Faults: no faults
- Every robot detects and reports the target.
- Speeds: equal speeds
- All robots move at most at unit speed.
- Randomness: deterministic
- Trajectories are fixed functions of the information received.
What is known
Results, in order of publication
Theorem 5.2: \(6+2\lceil 1/r-1\rceil\) robots of communication range \(r < 1\) suffice, hence 8 wireless robots; they report that 6 suffice with range 1, citing an extended version of Czyzowicz et al. 2015c.
Open questions
The optimal evacuation time of \(k = 3, \dots, 7\) wireless robots from the centroid of the triangle.
Notes
For range-limited robots, \(\Omega(1/r)\) robots are needed to reach \(1+\sqrt3/3\) (Theorem 5.1 of Bagheri et al. 2019); their Table 1 gives about \(1.667\) and \(1.6105\) for three and four robots with range 1, citing the extended version. Checked against the full texts of Czyzowicz et al. 2015c and Bagheri et al. 2019.
What carries over from neighbouring settings
Read off the order of assumptions, not checked case by case: results may carry extra assumptions (shown on their pages), and asynchronous time is measured in epochs.
Algorithms from weaker settings work here
Lower bounds from stronger settings hold here
Know a better result for this setting, or found a mistake? Suggest a result · Report an error. Accepted contributions are acknowledged.