Uniform circle formation
Robots must form a regular \(n\)-gon: a circle with equal spacing.
Problem statement and measuresexpand
\(n\) robots start at distinct points and must reach the vertices of a regular \(n\)-gon (in any position and size) and stay there. Time is in epochs (ASYNC: intervals in which every robot completes at least one full Look–Compute–Move cycle) or rounds; colours count light colours in the luminous model.
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).
Transparent fat robots, SSYNC
verifiedTransparent fat robots that agree on one axis form a regular \(n\)-gon of given side in SSYNC.
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: transparent
- Robots do not obstruct each other's view.
- 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): oblivious
- OBLOT: a robot's computation depends only on the current snapshot; nothing persists across cycles.
- 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: 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 (Theorem 1). A preprint.
Open questions
None recorded for this variant.
Notes
The paper assumes agreement on one axis without chirality, which is the catalogue's one-axis value. Checked against the full text.
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