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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›any \(k\), all robots, + wireless, + no faults, + equal speeds

Many wireless robots, disk

verified

Evacuation time: near-optimal

With \(k\) wireless robots the time approaches \(3+\pi/k\).

Assumed (added on the last edge are outlined)

+Domainunit disk, start at the centre+Communicationwireless: instant communication at any distance+Faultsno faults+Speedsall robots have the same unit speed

Not assumed

Number of robotsany number \(k\) of robotsGoalevery 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: 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: 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

near-optimal

Matching up to \(O(k^{-4/3})\).

lower bound

\(3+\pi/k\) Czyzowicz et al. 2014

bottleneckunknown exit

holds when

\(k\) robots, wireless

introductory—

current best

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

Czyzowicz et al. 2014

Results, in order of publication

Czyzowicz et al. 2014 current best

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

Introductory and current best.

Open questions

None recorded for this variant.

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 face-to-face robots, disk

Evacuation time \(3+2\pi/k\)

Lower bounds from stronger settings hold here

Two wireless robots, disk

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

Three wireless robots, disk

Evacuation time \(\approx 4.159\)

Priority evacuation (queen and servants), disk

Evacuation time \(1+\tfrac{2}{\sigma}\arccos\left(-\tfrac{2}{\sigma}\right)+\sqrt{1-\tfrac{4}{\sigma^2}}\)

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