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).
Opaque fat robots with lights, ASYNC, one axis
verifiedOpaque fat robots with lights that agree on one axis form any pattern from every start not symmetric about a robot-free line parallel to that axis, with 10 colours.
Assumed (added on the last edge are outlined)
Not assumed
Formal definitions of this setting
- Body: fat
- Each robot is a closed disc of radius 1 (unit disc). Discs may not overlap, so robots cannot share a point and must avoid collisions; a disc hides what lies behind it from other robots. Gathering then asks for a configuration whose union of discs is connected and in which every robot sees all others Czyzowicz et al. 2009.
- 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Move phases take finite but unbounded, adversarially chosen time; a robot may look while others move, and compute on an outdated snapshot.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
Results, in order of publication
Introductory and current best. Theorem 1: solvable if and only if the start is not symmetric about a line parallel to the agreed axis through no robot centre. \(\Theta(n)\) moves in total (Theorem 8), analysed under SSYNC with rigid moves.
Open questions
None recorded for this variant.
Notes
Checked against the full text (arXiv 1910.02706).
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
No catalogued weaker setting.
Lower bounds from stronger settings hold here
Know a better result for this setting, or found a mistake? Suggest a result · Report an error. Accepted contributions are acknowledged.