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.
Complete graphs, round robin
verifiedOn a complete graph, robots that cannot see towers never gather under round robin once two nodes are occupied.
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: complete graph
- \(K_n\): every two nodes are neighbours.
- 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: round robin
- With the sequential scheduler: the robots are activated one per round in a cyclic order that the adversary fixes at the start, that the robots do not know, and that repeats forever; an epoch is \(k\) consecutive rounds, one activation per robot.
What is known
- every start with at least two occupied nodes (Theorem 2)
- a robot can only move to an occupied or to an empty node, all alike; moving to empty nodes eventually occupies every node (unsolvable by Theorem 1), and with a hidden tower on every occupied node the adversary's order never empties a node
- the same holds on complete bipartite graphs with equal sides when both sides are occupied (Theorem 2)
Results, in order of publication
No result recorded yet.
Open questions
None recorded for this variant.
Notes
With all robots on one side of a complete bipartite graph, one epoch suffices: the first robot crosses and every later one joins it Cicerone et al. 2025b. Checked against the full text of Cicerone et al. 2025b (arXiv v1).
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