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, one crash, SSYNC, rigid, compass
verifiedWith one crash fault, robots that share a compass reach mutual visibility in rigid SSYNC with 3 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: crash faults
- Up to \(f\) robots stop forever at adversarial times and remain visible at their last position.
- 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: compass
- All robots agree on the directions and orientations of both axes (a common compass); they may still disagree on origin and unit. Called full compass by Efrima and Peleg, who also use half compass (both axis directions, the positive sense of one axis, no chirality), direction-only and axes-only Efrima & Peleg 2009. Agreement on one axis together with common chirality is equivalent: the orientation of the second axis then follows from handedness.
- 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; one crash fault.
Open questions
Tolerating three or more crashes, and using 2 colours Aljohani & Sharma 2018.
Notes
With \(f\) crashes, robots that agree on one axis reach complete visibility in ASYNC in \(O(n)\) epochs with 3 colours Poudel et al. 2021 (abstract only). Checked against the full text of Aljohani & Sharma 2018.
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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