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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›ASYNC, + lights (LUMI)›+ chirality, + \(n \ge 3\)

Gathering, full lights, ASYNC, no multiplicity detection

verified
Time: unknownColours: tight

With lights and chirality, \(n \ge 3\) robots gather in ASYNC with non-rigid moves and no multiplicity detection using 3 colours; whether 2 suffice is open.

Assumed (added on the last edge are outlined)

+Bodypoint robots+Visibilityunlimited+Opacitytransparent: every robot sees every robot within range+Faultsno faults+Lights (persistent memory)full light (LUMI): every robot sees all lights, including its own+Agreement on coordinatescommon chirality (same clockwise orientation)+Number of robots\(n \ge 3\)

Not assumed

Schedulerasynchronous (ASYNC)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: 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Move phases take finite but unbounded, adversarially chosen time; a robot may look while others move, and compute on an outdated snapshot.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: 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
\(\text{finite}\)
current best
\(\text{finite}\)
Colours · colours of the light per robot (0 for oblivious robots)
tight
\(\Theta(1)\); the exact number is 2 or 3.
lower bound
\(2\) Prencipe 2007
bottleneckindistinguishable multiplicities
holds when
  • inherited: one colour is an oblivious robot
introductory
\(10\)
current best
\(3\)

Results, in order of publication

Terai et al. 2023 introductory
Time \(\text{finite}\)Colours \(10\)

Concluding remarks: the SSYNC algorithm through the \(5c\)-colour simulation of a \(c\)-colour algorithm Das et al. 2016.

Nakai et al. 2021 current best
Time \(\text{finite}\)Colours \(3\)

Theorem 6; 6 colours by a new \(3c\)-colour simulation (for a \(c\)-colour algorithm) of unfair SSYNC (Theorem 4). The authors conjecture that 3 is optimal.

Open questions

  1. Is 3 the minimum number of colours for ASYNC gathering with lights, and can chirality be dropped Nakai et al. 2021?

  2. Self-stabilizing gathering of \(n\) robots with lights Heriban et al. 2018.

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

Fat robots, chirality, any number
Time \(\text{finite}\)Colours \(0\)
Transparent fat robots, no chirality
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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