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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›+ three robots›all robots, + wireless›+ no faults, + equal speeds

Three wireless robots, disk

verified
Evacuation time: polynomial gap

Three wireless robots: between \(\approx 4.159\) and \(4.22\).

Assumed (added on the last edge are outlined)

+Domainunit disk, start at the centre+Number of robotsthree 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: three robots
Three 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
polynomial gap
A small constant gap.
lower bound
\(\approx 4.159\) Czyzowicz et al. 2014
bottleneckunknown exit
holds when
  • three robots, wireless
introductory—
current best
\(< 4.22\)

Results, in order of publication

Czyzowicz et al. 2014 current best
Evacuation time \(< 4.22\)

Introductory and current best.

Open questions

  1. Find the optimal evacuation time for three wireless robots.

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

Three face-to-face robots, disk
Evacuation time \(3+2\pi/3\)
Many wireless robots, disk
Evacuation time \(3+\pi/k+O(k^{-4/3})\)
Three wireless robots, one crash fault
Evacuation time \(\approx 6.309\)

Lower bounds from stronger settings hold here

No catalogued stronger setting.

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