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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, + wireless, + no faults›+ equal speeds

Two wireless robots, disk

standard settingverified

Evacuation time: tight

Two wireless robots evacuate the unit disk in exactly \(1+2\pi/3+\sqrt3 \approx 4.826\).

Assumed (added on the last edge are outlined)

+Domainunit disk, start at the centre+Number of robotstwo robots+Communicationwireless: instant communication at any distance+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: 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

Evacuation time · worst-case time until the last robot (or the queen) reaches the exit; unit disk, unit speed

tight

\(\approx 4.826\), exact.

lower bound

\(1+2\pi/3+\sqrt{3}\) Czyzowicz et al. 2014

bottleneckunknown exit

holds when

deterministic
start at the centre

introductory—

current best

\(1+2\pi/3+\sqrt{3}\)

Czyzowicz et al. 2014

Results, in order of publication

Czyzowicz et al. 2014 current best

Evacuation time \(1+2\pi/3+\sqrt{3}\)

Introductory and current best.

Open questions

None recorded for this variant.

Notes

With one robot that has crashed and must be carried out, see Chauffeuring. In the unit disk of an \(\ell_p\) norm, two wireless robots evacuate optimally for every \(p \ge 1\), in time \(5\) for \(p = 1\) and \(p = \infty\) and about \(4.7544\) at best Georgiou et al. 2023. If every robot must visit every one of \(k\) treasures, \(k = 1\) is this problem Kaminski et al. 2026, a preprint.

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

Many wireless robots, disk

Evacuation time \(3+\pi/k+O(k^{-4/3})\)

Two robots with asymmetric communication, disk

Evacuation time \(\pi+2\)

Disk, two wireless robots of different speeds

Evacuation time \(\tfrac{1+2\arccos(-2/v)}{v}+\sqrt{1-4/v^2}\)

Lower bounds from stronger settings hold here

Two robots, two exits, wireless

Evacuation time \(2\)

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