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.
Disk search, \(k\) wireless robots, one Byzantine
verifiedWith one Byzantine robot, search takes exactly \(1+4\pi/k\), as with one crash fault.
Assumed (added on the last edge are outlined)
Not assumed
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: search
- The task ends as soon as a non-faulty robot has reached the exit and every non-faulty robot knows its position for certain; the robots need not go to 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: Byzantine
- Up to \(f\) robots may lie: stay silent when they find the target, or claim to have found it where it is not.
- 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
\(k \ge 2\) wireless robots start at the centre of the unit disk; one of them may be Byzantine (it may lie about the exit). Messages carry the sender's identifier. The task ends when a non-faulty robot has reached the exit and every non-faulty robot knows its position for certain.
What is known
- Corollary 1: the crash bound of Theorem 1 with \(f=1\)
Results, in order of publication
Introductory and current best (Theorem 3, with separate analyses for \(k = 3, 4\) in Lemma 1 and Section 4.2).
Open questions
Search of the disk with \(f \ge 2\) Byzantine robots: only the crash lower bound \(1+2\pi(f+1)/k\) and evacuation upper bounds are known Georgiou et al. 2023b Behrouz et al. 2025.
Notes
With one Byzantine and \(f-1\) crash-faulty robots, search takes at most \(1+2\pi f/k+2\sin(2\pi/k)\) (Theorem 4). Checked against the full text of 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
Lower bounds from stronger settings hold here
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