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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).

no assumptions›+ point, + unlimited visibility›+ transparent, + no faults

ASYNC, oblivious, disoriented

standard settingverified
Time: unknownColours: tight

Oblivious, disoriented, non-rigid robots form a regular \(n\)-gon from any configuration of distinct points, for every \(n\).

Assumed (added on the last edge are outlined)

+Bodypoint robots+Visibilityunlimited+Opacitytransparent: every robot sees every robot within range+Faultsno faults

Not assumed

Schedulerasynchronous (ASYNC)Lights (persistent memory)oblivious, no light (OBLOT)Agreement on coordinatesno agreement on local coordinate systemsMovementnon-rigid: the adversary may stop a move after some minimum distance \(\delta\)Multiplicity detectionnone: a point with several robots looks like one robotRandomnessdeterministicIdentifiersanonymous: all robots are identical
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Move phases take finite but unbounded, adversarially chosen time; a robot may look while others move, and compute on an outdated snapshot.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: 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

Time · rounds (FSYNC, SSYNC) or epochs (ASYNC) until the task is done; "finite" when only solvability is known
unknown
No time analysis of oblivious ASYNC uniform circle formation is recorded.
lower bound
\(\text{impossible}\) Flocchini et al. 2017
bottlenecksymmetry
holds when
  • if two robots start at the same point
introductory—
current best
\(\text{finite}\)
\(n \ne 4\)
Colours · colours of the light per robot (0 for oblivious robots)
tight
No colours.
lower boundnone known
introductory—
current best
\(0\)
\(n \ne 4\)

Results, in order of publication

Flocchini et al. 2017 current best
Time \(\text{finite}\)Colours \(0\)
\(n \ne 4\)

Introductory for ASYNC and current best. The case \(n = 4\) (square formation) is solved by Mamino & Viglietta 2016.

Mamino & Viglietta 2016 also
Time \(\text{finite}\)Colours \(0\)
\(n = 4\)

Completes the picture: synchrony, chirality and rigidity are computationally irrelevant for pattern formation.

Open questions

  1. Can oblivious robots without lights form a uniform circle in \(O(1)\) epochs, with rigid or non-rigid moves? Feletti et al. 2024

Notes

Uniform circle and point are the only patterns formable from every initial configuration Flocchini et al. 2017.

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

SSYNC, oblivious

Branches from here

SSYNC, oblivious

+SSYNC
The first deterministic uniform circle formation, for \(n \ge 5\) semi-synchronous robots.

Know a better result for this setting, or found a mistake? Suggest a result · Report an error. Accepted contributions are acknowledged.