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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 at the centroid, or at a given point of the boundary, 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; in weighted evacuation of two robots, a weighted average of their arrival times; in search, until a non-faulty robot has reached the exit and every non-faulty robot knows where it is).

Measures: Evacuation time, worst-case time until the last robot (or the queen) reaches the exit, or the weighted average or search time where the goal says so; unit disk, unit speed.

no assumptions›+ disk›any \(k\), all robots, face-to-face, + crash faults, + equal speeds

Disk, \(k\) face-to-face robots, \(f\) crash faults

verified

Evacuation time: polynomial gap

Face-to-face with \(f\) crash faults, \(k\) robots evacuate in at most \(3+2\pi(f+1)/k\).

Assumed (added on the last edge are outlined)

+Domainunit disk, start at the centre+Faultsup to \(f\) crash-faulty robots (fail to report the target)+Speedsall robots have the same unit speed

Not assumed

Number of robotsany number \(k\) of robotsGoalevery robot must reach the exitExitsone exitCommunicationface-to-face: robots exchange information only when they meetRandomnessdeterministic

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: face-to-face: Face-to-face: two robots exchange information only when they are at the same point at the same time.
Faults: crash faults: Up to \(f\) robots may fail to detect or to report the target when they pass over it.
Speeds: equal speeds: All robots move at most at unit speed.
Randomness: deterministic: Trajectories are fixed functions of the information received.

Statement for this variant

\(k\) robots start at the centre of the unit disk and exchange information only when they meet; up to \(f < k\) of them may crash. Every non-faulty robot must reach the exit.

What is known

Evacuation time · worst-case time until the last robot (or the queen) reaches the exit, or the weighted average or search time where the goal says so; unit disk, unit speed

polynomial gap

Additive gap at most \(2\).

lower bounds

\(1+2\pi(f+1)/k\) Georgiou et al. 2023b

bottleneckfaulty reports

holds when

Theorem 1: search bound with \(f\) crash-faulty robots, hence with Byzantine ones
every point of the boundary must be visited by \(f+1\) robots
wireless search bound; it holds face-to-face and for evacuation

\(3+2\pi/k-O(k^{-2})\) Czyzowicz et al. 2014

bottleneckunknown exit

holds when

fault-free bound of \(k\) face-to-face robots (setting with many face-to-face robots in the disk)

introductory—

current best

\(3+2\pi(f+1)/k\)

Leonardos et al. 2023

Results, in order of publication

Leonardos et al. 2023 current best

Evacuation time \(3+2\pi(f+1)/k\)

Introductory and current best (Theorem 5).

Open questions

Tight bounds for face-to-face evacuation of the disk with crash faults.

Notes

Checked against the full texts of Leonardos et al. 2023 and Georgiou et al. 2023b.

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

Disk, \(k\) face-to-face robots, \(f\) Byzantine

Evacuation time \(3+2\pi(f+1)/k+\max_{2\le j\le k} 2(j-1)\sin\bigl(\tfrac{f-j+2}{j-1}\cdot\tfrac{\pi}{k}\bigr)\)

Lower bounds from stronger settings hold here

Many face-to-face robots, disk

Evacuation time \(3+2\pi/k-O(k^{-2})\)

Three wireless robots, one crash fault

Evacuation time \(1+4\pi/3 \approx 5.188\)

Disk search, \(k\) wireless robots, \(f\) crash faults

Evacuation time \(1+2\pi(f+1)/k\)

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