Near-gathering
Robots must get within a small distance of each other without ever colliding.
Problem statement and measuresexpand
Robots with limited visibility, each seeing only the robots within its visibility radius \(V\), start at distinct points whose visibility graph (two robots joined by an edge when they are within distance \(V\)) is connected. They must reach a configuration in which every robot sees all others, without a collision (two robots at the same point) ever happening.
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).
SSYNC, limited visibility, disoriented
verifiedDisoriented robots near-gather in \(O(\Delta_0^2)\) epochs in SSYNC.
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: 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: 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
- only for \(\lambda\)-contracting protocols
Results, in order of publication
Theorem 18; first near-gathering protocols for disoriented robots.
Open questions
Near-gathering in \(O(\Delta_0)\) epochs?
Know a better result for this setting, or found a mistake? Suggest a result · Report an error. Accepted contributions are acknowledged.