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.
Hypercubes, round robin
verifiedOn hypercubes of dimension at least 3, robots that cannot see towers gather under round robin in \(O(\log n)\) epochs from every start except two adjacent occupied nodes, three occupied nodes forming a path, and every node 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: hypercube
- The hypercube of dimension \(\log n\): nodes are the \(n\) binary strings of length \(\log n\), neighbours when they differ in one bit. Its diameter is \(D = \log n\).
- 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
- starts whose occupied nodes are two adjacent nodes or all \(n\) nodes (Theorem 1, for every vertex- and edge-transitive graph)
- starts whose occupied nodes form a path of three nodes (Corollary 1): every algorithm must reach an empty node with all robots on its neighbours (Theorem 3) and then a path of three occupied nodes, and in a hypercube all such paths look alike
- the adversary chooses the hidden towers and the order
- two robots at distance \(D = \log n\) must meet, and each moves at most one edge per epoch (Lemma 1)
Results, in order of publication
Introductory and current best (Theorem 4): the algorithm shrinks the smallest subcube containing all robots by one dimension at a time, in \(O(1)\) epochs per dimension. Preprint. Read in the arXiv version (v1).
Open questions
None recorded for this variant.
Notes
Checked against the full text of Cicerone et al. 2025b (arXiv v1).
Know a better result for this setting, or found a mistake? Suggest a result · Report an error. Accepted contributions are acknowledged.