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, SSYNC, rigid
verifiedOpaque luminous robots reach mutual visibility in rigid SSYNC with 2 colours, which is optimal; oblivious robots suffice if \(n\) is known.
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: SSYNC
- SSYNC: time is divided into rounds; in each round a non-empty subset of robots, chosen by a fair adversary, executes one atomic Look–Compute–Move cycle. Called Atom by Prencipe Prencipe 2007.
- 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: disoriented
- Each robot \(r\) observes in its own coordinate system, obtained from the global one by an unknown similarity (translation, rotation, uniform scaling, and possibly reflection); systems may differ between robots and, in some models, between cycles. Papers that call robots disoriented do not always mean this: Défago and Souissi use the word while assuming chirality Défago & Souissi 2008. Some papers also let the adversary change a robot's rotation, reflection and unit at every Look (variable disorientation) Flocchini et al. 2023 Masuzawa et al. 2026.
- 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
- Section 7: one colour does not suffice when \(n\) is unknown and robots must terminate
Results, in order of publication
Introductory and current best (Theorem 1.1(b)); first announced at SSS 2014 Di Luna et al. 2014. With \(n\) known, oblivious robots (no colours) suffice.
\(O(1)\) rounds; as described by later papers, not read.
Open questions
None recorded for this variant.
Notes
Checked against the full text of Di Luna et al. 2017.
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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