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 the centroid 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).
Measures: Evacuation time, worst-case time until the last robot (or the queen) reaches the exit; unit disk, unit speed.
Line, any number of face-to-face robots
verifiedOn the line, face-to-face robots evacuate with competitive ratio exactly 9: extra robots do not help.
Assumed (added on the last edge are outlined)
Not assumed
Formal definitions of this setting
- Domain: line
- The robots start at the origin of the real line; the exit is at an unknown point at distance at least \(1\); the measure is the competitive ratio, the evacuation time divided by that distance.
- 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: face-to-face
- Face-to-face: two robots exchange information only when they are at the same point at the same time.
- 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
- any number of robots of maximum speed at most 1
- full proof of \(9d-o(d)\)
Results, in order of publication
Introductory and current best: the robots move together and run the doubling strategy; as described by later papers.
Open questions
None recorded for this variant.
Notes
For two face-to-face robots the energy of group search can be traded against time; with maximum speed \(v\) and time at most \(9d/v\) the energy is about \(8.42588\,v^2 d\) (the paper writes \(b\) for \(v\); Theorem 16) Czyzowicz et al. 2019. Checked against the full text of Bampas et al. 2019; Chrobak et al. 2015 was not read.
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
No catalogued weaker setting.
Lower bounds from stronger settings hold here
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