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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›+ triangle, boundary start›any \(k\), + wireless, + no faults, + equal speeds

Triangle, any number of wireless robots starting on the boundary

verified

Evacuation time: tight

From a point of the boundary at distance \(y\) from the nearest midpoint of a side, any number \(k \ge 2\) of wireless robots needs exactly \(3/2+y\): extra robots do not help.

Assumed (added on the last edge are outlined)

+Domaintriangle, start on the boundary+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: triangle, boundary start: A triangle (the equilateral triangle of unit side unless the setting names another family); the robots start together at a point of its boundary, either given (at distance \(y\) from the nearest midpoint of a side) or chosen by the algorithm, as the setting says; the exit is a point of the boundary.
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, or the weighted average or search time where the goal says so; unit disk, unit speed

tight

Exact for every \(k \ge 2\).

lower bound

\(3/2+y\) Czyzowicz et al. 2015c

bottleneckunknown exit

holds when

Theorems 1 and 3
unit equilateral triangle, given start on the boundary
the argument does not depend on the number of robots

introductory—

current best

\(3/2+y\)

Czyzowicz et al. 2015c

Results, in order of publication

Czyzowicz et al. 2015c current best

Evacuation time \(3/2+y\)

Introductory and current best: two robots walk the boundary in opposite directions (Theorem 1); more robots do no better (Theorem 3).

Open questions

None recorded for this variant.

Notes

Checked against the full text.

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

No catalogued weaker setting.

Lower bounds from stronger settings hold here

Triangle, two wireless robots starting on the boundary

Evacuation time \((a+b+c)/2\)

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