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Gathering

Robots must meet at a single point that is not fixed in advance, and stay there.

Problem statement and measuresexpand

\(n\) robots start at distinct points and must all reach the same point in finite time and remain there. For \(n = 2\) the problem is called rendezvous. Time is counted in rounds (FSYNC, SSYNC) or in epochs (ASYNC: intervals in which every robot completes at least one full Look–Compute–Move cycle); colours count light 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).

no assumptions›+ point›+ unlimited visibility, + transparent›+ no faults›+ SSYNC›+ internal light (FSTA)›+ chirality, + \(n \ge 3\)

Gathering, internal light (FSTA), SSYNC, non-rigid

verified
Time: unknownColours: unknown

With internal lights, 2 colours gather \(n \ge 3\) robots in SSYNC only from restricted starts; 2 colours do not suffice in general, and whether more do is open.

Assumed (added on the last edge are outlined)

+Bodypoint robots+Visibilityunlimited+Opacitytransparent: every robot sees every robot within range+Faultsno faults+Schedulersemi-synchronous (SSYNC)+Lights (persistent memory)internal light only (FSTA): a robot sees its own light, not the others'+Agreement on coordinatescommon chirality (same clockwise orientation)+Number of robots\(n \ge 3\)

Not assumed

Movementnon-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: 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): internal light (FSTA)
FSTA: each robot has a light with \(c\) colours that only it can see; it persists between cycles (constant persistent memory, no communication). Also called an internal light.
Agreement on coordinates: chirality
All local coordinate systems have the same handedness (clockwise orientation): no reflection. Also called handedness. Prencipe uses chirality for agreement on North and East, i.e. a compass Prencipe 2007.
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.
Number of robots: \(n \ge 3\)
\(n\ge 3\) robots.

What is known

Time · rounds (FSYNC, SSYNC) or epochs (ASYNC) until the task is done; "finite" when only solvability is known
unknown
Time is not analysed; only solvability.
lower boundnone known
introductory
\(\text{finite}\)
\(\delta\) known and commonrestricted initial configurations
current best
\(\text{finite}\)
\(\delta\) known and commonsome two-point starts excluded
Colours · colours of the light per robot (0 for oblivious robots)
unknown
Unconditional gathering with more colours is open.
lower bound
\(3\) Otaka et al. 2026
bottleneckmemory
holds when
  • Theorem 1: 2 colours are impossible without restrictions on the start
introductory
\(2\)
\(\delta\) known and commonrestricted initial configurations
current best
\(2\)
\(\delta\) known and commonsome two-point starts excluded

Results, in order of publication

Terai et al. 2023 introductory
Time \(\text{finite}\)Colours \(2\)
\(\delta\) known and commonrestricted initial configurations

Theorem 8, from starts in which robots are at most \(2\delta\) apart.

Otaka et al. 2026 current best
Time \(\text{finite}\)Colours \(2\)
\(\delta\) known and commonsome two-point starts excluded

Theorem 3: far fewer excluded starts; the remaining exclusions cannot all be removed (Theorem 2(3)).

Open questions

  1. Gathering with internal lights in SSYNC from every start, with more than 2 colours, and in ASYNC Otaka et al. 2026.

Notes

Checked against the full texts.

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 multiplicity detection, SSYNC
Fat robots, chirality, any number
Time \(\text{finite}\)Colours \(0\)

Lower bounds from stronger settings hold here

Gathering, full lights, SSYNC, no multiplicity detection
Colours \(2\)

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