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 with a compass, rigid, ASYNC
verifiedWith a common compass and rigid moves, asynchronous robots with limited visibility gather at a point in time linear in the initial diameter, which is optimal.
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: 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Move phases take finite but unbounded, adversarially chosen time; a robot may look while others move, and compute on an outdated snapshot.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): oblivious
- OBLOT: a robot's computation depends only on the current snapshot; nothing persists across cycles.
- Agreement on coordinates: compass
- All robots agree on the directions and orientations of both axes (a common compass); they may still disagree on origin and unit. Called full compass by Efrima and Peleg, who also use half compass (both axis directions, the positive sense of one axis, no chirality), direction-only and axes-only Efrima & Peleg 2009. Agreement on one axis together with common chirality is equivalent: the orientation of the second axis then follows from handedness.
- Movement: rigid
- Every Move phase ends at the computed 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.
- Initial configurations: any start
- The algorithm must gather from every initial configuration from which gathering is possible at all in the setting; configurations proved unsolvable there (for example bivalent ones, or several multiplicities with weak detection and crashes) are excluded by the problem, not by the algorithm.
What is known
- \(\Delta_0\) = Euclidean diameter of the initial configuration
- called the trivial lower bound in Poudel & Sharma 2021: a robot moves at most a constant distance per cycle, inside its axis-aligned \(2 \times 2\) square
Results, in order of publication
Current best (Theorem 3): all robots gather at one point in \(O(\Delta_0)\) epochs; robots that occupy one point are not merged. Finite-time gathering in this setting already follows from the non-rigid algorithm of Flocchini et al. 2005.
Open questions
Does \(O(\Delta_0)\)-epoch gathering need rigid moves, or a visibility radius larger than the connectivity range Poudel & Sharma 2021?
Notes
Poudel and Sharma conjecture that with viewing and connectivity ranges of 1 no \(O(\Delta_0)\) algorithm exists, even with a compass. Checked against the full text of Poudel & Sharma 2021 (Information 2021; Theorems 2 and 3, model in Section 2).
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.