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.
Disk, search-and-fetch, two face-to-face robots
verifiedFace-to-face, two robots fetch the treasure in \(1+\pi-\alpha/2+3\sin(\alpha/2)\), against \(1+\pi/3+4\sin(\alpha/2)\) for \(\alpha \le 2\pi/3\).
Assumed (added on the last edge are outlined)
Not assumed
Formal definitions of this setting
- Domain: disk
- The unit disk; robots start at its centre; the exit is a point of the boundary circle.
- Number of robots: two robots
- Two robots.
- Goal: search-and-fetch
- A treasure lies at an unknown point of the boundary, and the exit at a known arc distance \(\alpha\) from it; the task ends when some robot has carried the treasure to 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: distinct speeds
- Each robot \(i\) has its own maximum speed \(v_i\), known to all.
- Randomness: deterministic
- Trajectories are fixed functions of the information received.
What is known
- Theorem 4.1
- \(\alpha \le 2\pi/3\); \(1+\pi/3+2\sin\alpha+2\sin(\alpha/2)\) for larger \(\alpha\)
Results, in order of publication
Introductory and current best (Theorem 3.1).
Open questions
None recorded for this variant.
Notes
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
No catalogued weaker setting.
Lower bounds from stronger settings hold here
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