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, terminating, \(\varphi = 1\)
standard settingverifiedThree robots with two colours explore a ring and stop under FSYNC; two robots cannot, whatever the number of colours.
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: terminating
- Every node is visited by at least one robot, and from some time on no robot is enabled (no robot changes its position or colour again).
- 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
- two robots, any number of colours (Theorem 8)
- while two robots stay within distance one they cannot tell a ring of \(n_1\) nodes from one of \(n_2 > 2(t+1)\) nodes, so they stop too early in the larger ring; once they separate, each is confined to two nodes (Lemma 4)
- three or four oblivious robots on rings with \(n > 7\) (Theorem 3.2)
Results, in order of publication
Introductory and current best (Theorem 9). No algorithm with three robots and two colours is universal (Theorem 11).
Open questions
Characterise the initial configurations of three robots with \(c \ge 3\) colours from which terminating exploration of a ring is solvable under FSYNC with \(\varphi = 1\) (with two colours no universal algorithm exists Ooshita & Tixeuil 2022).
Notes
From GWG the middle robot joins one end robot, forming a pair GW that walks away from the third robot, which waits; when the pair arrives at the waiting robot from the other side every node has been visited, and one more move forms a tower that no rule enables (Algorithm 2 of Ooshita & Tixeuil 2022). Without lights five robots are necessary and sufficient Datta et al. 2013 (variant Rings, oblivious, FSYNC, terminating).
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.