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

Line, any number of wireless robots

draft
Evacuation time: tight

With wireless communication, two or more robots evacuate the line with competitive ratio exactly 3.

Assumed (added on the last edge are outlined)

+Domainthe infinite line+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: line
The robots start at the origin of the real line; the exit is at an unknown point at distance at least \(1\); the measure is the competitive ratio, the evacuation time divided by that distance.
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
\(3\) Chrobak et al. 2015
bottleneckunknown distance
holds when
introductory—
current best
\(3\)

Results, in order of publication

Chrobak et al. 2015 current best
Evacuation time \(3\)

Introductory and current best: two robots search in opposite directions and call the others; as cited by Czyzowicz et al. 2021.

Open questions

None recorded for this variant.

Notes

Unverified: Chrobak et al. 2015 was not read; the bounds are as cited by later papers.

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

Line, any number of face-to-face robots
Evacuation time \(9\)
Line, \(2f+1\) wireless robots, \(f\) crash faults
Evacuation time \(\approx 7.437\)

Lower bounds from stronger settings hold here

No catalogued stronger setting.

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