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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.

no assumptions›+ disk›+ two robots›all robots, one exit, + sender/receiver, + no faults, + equal speeds

Two robots with asymmetric communication, disk

verified
Evacuation time: polynomial gap

When one robot can only send and the other only receive, evacuation lies strictly between the wireless and face-to-face optima; \(\pi+2\) suffices.

Assumed (added on the last edge are outlined)

+Domainunit disk, start at the centre+Number of robotstwo robots+Communicationasymmetric: senders and receivers+Faultsno faults+Speedsall robots have the same unit speed

Not assumed

Goalevery robot must reach the exitExitsone exitRandomnessdeterministic
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: 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: sender/receiver
Sender/receiver: one robot can only transmit and the other only receive, 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

Evacuation time · worst-case time until the last robot (or the queen) reaches the exit; unit disk, unit speed
polynomial gap
No explicit lower bound.
lower bound
\(> 1+2\pi/3+\sqrt3\) Georgiou et al. 2022
bottleneckone-way communication
holds when
  • strictly above the wireless optimum; no explicit value
introductory—
current best
\(\pi+2\)

Results, in order of publication

Georgiou et al. 2022 current best
Evacuation time \(\pi+2\)

Introductory and current best.

Open questions

  1. Give an explicit lower bound above \(\approx 4.826\) for sender/receiver evacuation.

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

Two face-to-face robots, disk
Evacuation time \(\approx 5.6234\)

Lower bounds from stronger settings hold here

Two wireless robots, disk
Evacuation time \(1+2\pi/3+\sqrt{3}\)

Know a better result for this setting, or found a mistake? Suggest a result · Report an error. Accepted contributions are acknowledged.