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).
Limited visibility, FSYNC, disoriented
verifiedDisoriented robots with limited visibility gather in \(O(\Delta_0^2)\) synchronous rounds, \(\Delta_0\) the initial diameter.
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: limited visibility
- A robot sees exactly the robots within Euclidean distance \(V\) of its position, for a fixed radius \(V>0\).
- Opacity: transparent
- Robots do not obstruct each other's view.
- Faults: no faults
- All robots follow the algorithm forever.
- Scheduler: FSYNC
- FSYNC: every robot executes one atomic cycle in every round.
- 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.
- Number of robots: any \(n\)
- Any number \(n\ge 2\) of robots.
What is known
- only for protocols that move inside the local convex hull of the neighbourhood
- \(\Delta_0\) = largest initial distance between two robots
Results, in order of publication
Go-To-The-Center; \(\Theta(n^2)\) for that algorithm (as restated by Castenow et al. 2022).
Every \(\lambda\)-contracting protocol, including Go-To-The-Center; works in two and three dimensions.
Open questions
Is \(\Omega(\Delta_0^2)\) necessary for every protocol, not only local convex-hull ones?
Notes
\(\Delta_0\) here is a distance, not the maximum degree.
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