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 oblivious robots, ASYNC, compass
verifiedWithout lights, opaque robots cannot form arbitrary patterns in ASYNC, even with a common compass.
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: 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Move phases take finite but unbounded, adversarially chosen time; a robot may look while others move, and compute on an outdated snapshot.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: 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: 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
- Theorem 1: oblivious opaque point robots, ASYNC
- holds even with a common compass
Results, in order of publication
No result recorded yet.
Open questions
None recorded for this variant.
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
No catalogued weaker setting.
Lower bounds from stronger settings hold here
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