Exploration by luminous robots
Myopic Look–Compute–Move robots with a light of a few colours must visit every node of a ring, grid or torus, and stop or keep visiting; how many robots and colours are needed?
Problem statement and measuresexpand
\(k\) anonymous robots, each with a light that shows one of \(c\) colours, are placed on an anonymous graph \(G=(V,E)\) (a ring, a finite grid or a torus) in an initial configuration designated by the algorithm (positions and colours). Each robot repeats Look–Compute–Move cycles. Look: it sees every node within distance \(\varphi\), with the multiset of colours of the robots on each (so it sees towers, nodes with several robots), and its own colour. Compute: a deterministic function of this snapshot, the same for all robots, gives a new colour and a destination, its own node or a neighbour; the robot sees no identifiers, no ports, and (without chirality) no orientation. Move: it moves there. A fair adversarial scheduler (FSYNC, SSYNC or ASYNC) activates the robots. Terminating exploration: every node is visited by some robot and eventually no robot is enabled. Perpetual exploration: every node is visited infinitely often. With finite \(\varphi\) no algorithm works from every initial configuration, so each result names its initial configurations; some papers also require exclusive executions (no two robots on one node or one edge) or opaque robots, recorded under each result. Measures: the number of robots \(k\) and the number of colours \(c\).
Measures: Robots, number of robots \(k\) needed; Colours, number of colours \(c\) of each light; \(c = 1\) means no usable light (oblivious robots).
Rings, FSYNC, perpetual, \(\varphi = 1\)
verifiedTwo robots with two colours explore a ring perpetually under FSYNC, which is optimal.
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.
- Network: ring
- An anonymous, unoriented ring of \(n\) nodes; \(n\) is unknown to the robots.
- Exploration goal: perpetual
- Every node is visited by at least one robot infinitely often; robots never stop.
- Lights (memory and messages): luminous
- LUMI (luminous robots): each robot has a light with \(c\) colours, set at the end of Compute, persistent across cycles, and visible to the robot itself and to every robot within distance \(\varphi\) Das et al. 2016. It is both memory and message. A Look returns, for every node within distance \(\varphi\), the multiset of colours of the robots on it.
- Scheduler: FSYNC
- FSYNC: in every round every robot executes one atomic Look–Compute–Move cycle.
- Visibility range: \(\varphi=1\)
- In a Look phase a robot at node \(v\) sees every node at distance at most \(\varphi = 1\) from \(v\) (in the graph metric), with the robots on it, their colours, and how many robots each node holds. It sees no node identifiers and, without chirality, no orientation.
- Chirality: no chirality
- Robots share no orientation: on a ring a robot cannot tell its two directions apart; on a grid or torus its snapshot is known only up to rotation and reflection, and the adversary may choose a different transformation in every cycle (self-inconsistent robots, as in Darwich et al. 2023 and Bramas et al. 2026).
- Faults: no faults
- All agents follow the algorithm forever.
- Identifiers: anonymous
- All agents run the same deterministic algorithm on the same input; no agent has a label. Formally the label function is constant: \(\mathrm{id}(a)=\mathrm{id}(b)\) for all agents \(a,b\).
What is known
- one robot sees nobody, so its snapshot is symmetric and the adversary chooses its direction (any \(\varphi\), any number of colours)
- stated in Section 3.1
- two robots, \(n \ge 6\) (Lemma 6)
- two robots with the same colour take the same snapshot and move symmetrically
Results, in order of publication
Introductory and current best. Universal for \(n \ge 6\): it works from every initial configuration from which two robots with colours \(\{G, W\}\) can explore (Theorem 7).
Open questions
None recorded for this variant.
Notes
The robot coloured G steps away from the other robot and the robot coloured W steps towards it; in FSYNC both move at once, so the pair keeps its shape and walks around the ring forever (Algorithm 1 of Ooshita & Tixeuil 2022). The colours give the pair a direction that the unoriented ring does not.
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.
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