Gathering by oblivious robots on graphs
Oblivious Look–Compute–Move robots must all reach one node of a graph and stay there; which starts can be gathered depends on the symmetries of the graph, on multiplicity detection and on the scheduler.
Problem statement and measuresexpand
\(k \ge 2\) anonymous, oblivious robots stand on the nodes of a connected anonymous graph \(G\) with \(n\) nodes, which every robot sees whole. A robot runs Look–Compute–Move cycles: in a Look it takes a snapshot of the graph with its occupied nodes, it computes from that snapshot only, and it stays or moves to a neighbour; moves are instantaneous, so robots are always seen on nodes. A node with two or more robots is a tower. With multiplicity detection a snapshot shows which nodes are towers (weak) or how many robots each node holds (strong); without it a tower looks like one robot. The robots must reach a configuration in which all of them are on one node and none moves again. A setting fixes the scheduler, the multiplicity detection, whether towers may exist at the start, and the graph class; an algorithm must gather from every start of the setting except the starts listed as unsolvable. Under round robin the scheduler activates one robot at a time in a fixed cyclic order. Time is counted in epochs: an epoch is an interval in which every robot completes at least one cycle (under round robin, \(k\) consecutive activations).
Measures: Time, epochs until all robots are on one node and no robot moves again.
Rings, sequential scheduler, no multiplicity detection
verifiedIf the scheduler may activate the robots one at a time in any fair order, \(k \ge 3\) robots that cannot see towers never gather on a ring.
Assumed (added on the last edge are outlined)
Not assumed
Formal definitions of this setting
- Graph dynamics: static graph
- \(G_t=G\) for all \(t\): the edge set never changes.
- Edge direction: undirected
- \(G=(V,E)\) with \(E\subseteq\binom V2\); every edge can be traversed in both directions.
- Faults: no faults
- All agents follow the algorithm forever.
- Network: ring
- An anonymous, unoriented ring of \(n\) nodes.
- Multiplicity detection: no multiplicity detection
- A snapshot shows which nodes are occupied, not how many robots each holds.
- Initial configuration: any start
- Agents start at arbitrary nodes; \(\ell\le k\) nodes are initially occupied, possibly with several agents each.
- Scheduler: sequential
- Sequential: an SSYNC scheduler that activates exactly one robot per round, fairly; which robot is the adversary's choice unless the activation order is fixed (see the order axis).
- Activation order: any order
- The adversary chooses which robots to activate, subject only to the scheduler's fairness.
What is known
- \(k \ge 3\) (Theorem III.1)
- just before gathering, all robots are on two adjacent nodes and one of them is a tower; every robot must move to the other node (Lemma III.1), and the adversary alternates between the two nodes
Results, in order of publication
No result recorded yet.
Open questions
None recorded for this variant.
Notes
The proof uses only the last step of an execution, which also arises from starts with one robot per node; the paper states the result for starts that may contain towers. Round robin, which fixes the order, makes the problem solvable. Checked against the full text of Navarra & Piselli 2025 (arXiv v2).
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
No catalogued weaker setting.
Lower bounds from stronger settings hold here
Branches from here
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