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, two wireless robots at the centroid
verifiedTwo wireless robots at the centroid of the unit equilateral triangle evacuate in exactly \(3/2+\sqrt3/6 \approx 1.789\).
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: two robots
- Two 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
- Theorem 2 with the start at the centroid, at distance \(\sqrt3/6\) from the midpoints of the sides
Results, in order of publication
Introductory and current best (Theorem 2: from an interior point at distance \(y\) from the nearest midpoint of a side, \(3/2+y\) is tight).
Open questions
None recorded for this variant.
Notes
Three robots at the centroid do better than two (Theorem 4). Checked against the full text.
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
No catalogued stronger setting.
Know a better result for this setting, or found a mistake? Suggest a result · Report an error. Accepted contributions are acknowledged.