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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).

no assumptions›+ point, + unlimited visibility›+ no faults›ASYNC, + lights (LUMI)›+ one axis

Lights, ASYNC, non-rigid, one axis

verified
Time: unknownColours: tight

With non-rigid moves, 3 colours suffice in ASYNC if robots agree on one axis; without axis agreement the problem is open.

Assumed (added on the last edge are outlined)

+Bodypoint robots+Visibilityunlimited+Faultsno faults+Lights (persistent memory)full light (LUMI): every robot sees all lights, including its own+Agreement on coordinatesagreement on one axis (no common chirality)

Not assumed

Opacityopaque: collinear robots hide each otherSchedulerasynchronous (ASYNC)Movementnon-rigid: the adversary may stop a move after some minimum distance \(\delta\)Multiplicity detectionnone: a point with several robots looks like one robotRandomnessdeterministicIdentifiersanonymous: all robots are identical
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: ASYNC
ASYNC: each robot's Look, Compute and Move phases take finite but unbounded, adversarially chosen time; a robot may look while others move, and compute on an outdated snapshot. Time is counted in epochs: an epoch is an interval of time in which every robot completes at least one full Look–Compute–Move cycle; it need not be the shortest such interval. The execution is divided into consecutive epochs, and a bound of \(X\) epochs holds for every such division. In FSYNC and SSYNC time is counted in rounds. Called Corda by Prencipe Prencipe 2007. Intermediate variants restrict interleavings: LC-atomic (no Look by another robot between a robot's Look and its Compute), Move-atomic (no Look during another robot's Move) and CM-atomic Flocchini et al. 2023.
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: one axis
All robots agree on the direction and orientation of one axis (e.g. North), but not necessarily on handedness: two robots may disagree on which side is East. This is how the papers use one axis agreement (e.g. robots "agree on only 'left' and 'right'" Bose et al. 2021; robots "do not necessarily have the same handedness" Pagli et al. 2015). One axis and chirality are alternatives on this axis; together they amount to a common compass.
Movement: non-rigid
In a Move phase the adversary may stop a robot before its destination, but only after it has travelled at least \(\delta>0\) (a constant unknown to the robots) or reached the 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

Time · rounds (FSYNC, SSYNC) or epochs (ASYNC) until the task is done; "finite" when only solvability is known
unknown
Not analysed.
lower boundnone known
introductory—
current best
\(\text{finite}\)
Colours · colours of the light per robot (0 for oblivious robots)
tight
2 or 3 colours.
lower bound
\(2\) Di Luna et al. 2017
bottleneckmemory
holds when
  • Section 7: one colour does not suffice when \(n\) is unknown and robots must terminate
introductory—
current best
\(3\)

Results, in order of publication

Di Luna et al. 2017 current best
Time \(\text{finite}\)Colours \(3\)

Introductory and current best (Theorem 1.2(d)).

Open questions

  1. Mutual visibility in non-rigid ASYNC without axis agreement, ideally with 2 colours Di Luna et al. 2017.

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

Fat robots, ASYNC, one axis
Time \(O(n)\)Colours \(O(1)\)

Lower bounds from stronger settings hold here

No catalogued stronger setting.

Know a better result for this setting, or found a mistake? Suggest a result · Report an error. Accepted contributions are acknowledged.