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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›+ rays, + wireless, + no faults, + equal speeds

\(m\) rays, \(k = m\) wireless robots at the junction

verified
Evacuation time: tight

With one wireless robot per ray, starting at the junction, the competitive ratio is exactly 3.

Assumed (added on the last edge are outlined)

+Domain\(m\) rays from a junction (a star)+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: rays
\(m \ge 2\) half-lines share an endpoint, the junction; the exit lies on one ray at distance at least \(1\) from the start; the measure is the competitive ratio.
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
tight
Exactly 3.
lower bound
\(3\) Brandt et al. 2020
bottleneckunknown exit
holds when
  • Observation 2
  • every \(2 \le k \le m\)
introductory—
current best
\(3\)

Results, in order of publication

Brandt et al. 2020 current best
Evacuation time \(3\)

Introductory and current best (Observation 1): each robot explores its own ray.

Open questions

  1. Starting off the junction with \(k = m\): close \(2+\sqrt{7/3} \approx 3.528\) against \(4\) Brandt et al. 2020.

  2. Fewer robots than rays (\(1 < k < m\)): the bounds \(3+2\lfloor (m-1)/k\rfloor\) and \(1+2\tfrac{m-1}{k}(1+\tfrac{k}{m-1})^{1+(m-1)/k}\) differ Brandt et al. 2020.

Notes

Checked against the full text (conference version).

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