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).
Crash faults, SSYNC
verifiedWith weak multiplicity detection and no common orientation, the correct robots gather despite up to \(n-1\) crashes.
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: unlimited visibility
- A robot sees all robots (subject to opacity).
- Opacity: transparent
- Robots do not obstruct each other's view.
- Faults: crash faults
- Up to \(f\) robots stop forever at adversarial times and remain visible at their last position.
- 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: weak multiplicity
- A snapshot marks each occupied point as holding one robot or more than one.
- 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
- if the capabilities are relaxed further (the paper shows its model is minimal)
Results, in order of publication
Theorem 3.8, SSYNC, non-rigid moves of at least a fixed distance \(S\); checked against the full text.
Wait-free; arbitrary initial configurations.
Wait-free, weak multiplicity detection, no agreement on coordinates. In ASYNC with instantaneous computation it tolerates \(\lfloor n/2\rfloor-2\) crashes for \(n \ge 7\).
Open questions
"The maximum number of faults under which a solution is still feasible, for gathering and other tasks, has yet to be established" Agmon & Peleg 2006, in particular in ASYNC.
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
Lower bounds from stronger settings hold here
No catalogued stronger setting.
Know a better result for this setting, or found a mistake? Suggest a result · Report an error. Accepted contributions are acknowledged.