Pattern formation
Robots start at distinct points and must form a given geometric pattern, up to similarity; which patterns are formable, and from which initial configurations?
Problem statement and measuresexpand
\(n\) robots start at distinct points of the plane. Each is given the same pattern \(P\), a set (or multiset) of \(n\) points in its own local coordinate system. The robots must reach a configuration similar to \(P\) (equal up to translation, rotation, uniform scaling and, without chirality (a common clockwise direction), reflection) and stay there. The question is for which pairs (initial configuration, pattern) this is possible under the given assumptions, and at what cost in time and colours.
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).
Randomized, ASYNC, no agreement
verifiedRandomness breaks symmetry: with one random bit per cycle, \(n \ge 5\) robots form any pattern except gathering.
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: transparent
- Robots do not obstruct each other's view.
- 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Move phases take finite but unbounded, adversarially chosen time; a robot may look while others move, and compute on an outdated snapshot.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): oblivious
- OBLOT: a robot's computation depends only on the current snapshot; nothing persists across cycles.
- 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: 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: randomized
- Robots may use random bits in Compute; guarantees hold with probability 1 or in expectation, as stated per result.
- Identifiers: anonymous
- Robots are indistinguishable and run the same algorithm.
What is known
Results, in order of publication
Patterns with multiplicity points are allowed, except gathering. Its proof is reported as not correct by Cicerone et al. 2019.
Open questions
None recorded for this variant.
Notes
Yamauchi and Yamashita gave a randomized algorithm with chirality (DISC 2014, not yet in the bibliography). Disputed: the proof of Bramas & Tixeuil 2016 is questioned by Cicerone et al. 2019; no undisputed result is recorded for this setting.
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.
Know a better result for this setting, or found a mistake? Suggest a result · Report an error. Accepted contributions are acknowledged.