Mutual visibility
Opaque robots must reach positions where no three are collinear, so that every robot sees every other.
Problem statement and measuresexpand
\(n\) opaque point robots (or unit discs) start at distinct points: a robot hides the robots behind it on the same line of sight. They must reach a configuration in which every robot sees every other (no three collinear), without collisions.
Measures: Time, rounds (FSYNC, SSYNC) or epochs (ASYNC) until the task is done; "finite" when only solvability is known; Colours, colours of the light per robot (0 for oblivious robots).
Lights, FSYNC, rigid, chirality
verifiedIn FSYNC with rigid moves and chirality, mutual visibility takes \(O(\log n)\) rounds with 12 colours.
Assumed (added on the last edge are outlined)
Not assumed
Formal definitions of this setting
- Body: point
- Each robot is a point of the plane: several robots may share a point (multiplicity), and a robot hides only what lies exactly behind it on a line of sight.
- Visibility: unlimited visibility
- A robot sees all robots (subject to opacity).
- Opacity: opaque
- Robots are points that obstruct sight: \(r\) sees \(s\) iff no third robot lies on the open segment \(\overline{rs}\).
- Faults: no faults
- All robots follow the algorithm forever.
- Scheduler: FSYNC
- FSYNC: every robot executes one atomic cycle in every round.
- Lights (persistent memory): lights (LUMI)
- LUMI: each robot has a light with \(c\) colours, set at the end of Compute, visible to all robots including itself; it is both memory and communication. Also called luminous robots, or a full light.
- Agreement on coordinates: chirality
- All local coordinate systems have the same handedness (clockwise orientation): no reflection. Also called handedness. Prencipe uses chirality for agreement on North and East, i.e. a compass Prencipe 2007.
- Movement: rigid
- Every Move phase ends at the computed destination.
- Multiplicity detection: no multiplicity detection
- A snapshot is a set of points: a point occupied by several robots looks like a single robot.
- Randomness: deterministic
- The destination computed in a cycle is a function of the snapshot (and the light, if any).
- Identifiers: anonymous
- Robots are indistinguishable and run the same algorithm.
What is known
Results, in order of publication
Introductory and current best (Theorem 11): \(n \ge 4\) robots, \(O(\log n)\) rounds, 12 colours, \(n\) unknown. Robot paths may cross; no two robots head for the same point.
Open questions
What is the lower bound on the number of rounds, and can chirality and crossing paths be dropped Vaidyanathan et al. 2015?
Notes
Vaidyanathan, Busch, Trahan, Sharma and Rai also show that the algorithm of Di Luna et al. 2014 takes \(\Theta(n)\) rounds in the worst case even in FSYNC. Checked against the full text of Vaidyanathan et al. 2015 (model in Section IV, Theorem 11). The variant was previously recorded without rigid moves; the paper assumes that "a robot in motion cannot be stopped".
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
No catalogued stronger setting.
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