Graph exploration by a single agent
One agent starts at a node of an unknown anonymous graph and must visit every node; how much memory does it need, which pebbles or knowledge make it possible, and how fast can it be?
Problem statement and measuresexpand
A single deterministic agent is placed at a node \(s\) of a connected graph \(G\) with \(n\) nodes, \(m\) edges and maximum degree \(\Delta\). Nodes are anonymous (they carry no names the agent can read), and the edges at each node are numbered locally; these numbers are the node's ports. The agent sees the degree of its node and the port by which it entered. It must visit every node of \(G\). Three termination requirements are standard:
- perpetual exploration: every node is eventually visited; the agent never has to stop;
- exploration with stop: every node is visited and the agent then stops, anywhere;
- exploration with return: every node is visited and the agent stops at \(s\).
Return implies stop, and stop implies perpetual exploration. Memory is the number of bits of the agent's state; when the agent carries pebbles (objects it can drop on a node and pick up again) their number is reported with the memory. Time is the number of moves (edge traversals) until exploration is complete; for periodic exploration it is the period. Unless a variant says otherwise the agent knows nothing about \(G\), not even \(n\).
Exploration is the oldest problem of the mobile-agent literature: it started as maze search by finite automata, and the question of how little memory suffices runs from Budach's impossibility (1978) to Reingold's log-space algorithm (2005). Several agents exploring together form the sibling problem, collaborative exploration.
Measures: Time, edge traversals until every node is visited (for periodic exploration: the period); Memory, bits of agent memory; pebbles, if any, are counted separately.
Exploration with return and pebbles
verified\(\Theta(\log\log n)\) pebbles are necessary and sufficient for a constant-memory agent, which then explores in polynomial time and returns to the start without knowing \(n\).
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.
- Node identifiers: anonymous nodes
- Nodes have no identifiers; an agent at \(v\) sees only \(\deg(v)\) and the port numbers \(1,\dots,\deg(v)\), which are assigned locally and independently at each endpoint of an edge.
- Knowledge: nothing known
- The agent knows nothing about \(G\): not \(n\), \(m\), \(\Delta\) or \(D\).
- Randomness: deterministic
- The next state and move of an agent are a function of its current state and what it observes.
- Markers: pebbles
- The agent carries \(p\) pebbles (identical or distinguishable, as stated); \(p\) is a resource reported with the memory.
- Preprocessing: adversarial ports
- Port numbers (and any node labels) are chosen by an adversary.
- Termination requirement: with return
- The agent visits every node and then halts at its start node \(s\).
- Topology: any topology
- \(G\) is any connected graph in the class allowed by the other axes.
What is known
- for any agent with \(O((\log n)^{1-\varepsilon})\) bits
- holds for every termination requirement
- deterministic
Results, in order of publication
Introductory and current best (SODA 2016 for bounded degree, JACM 2019 in general). Since \(p+1\) agents can simulate one agent with \(p\) pebbles, the same bounds hold for teams: \(\Theta(\log\log n)\) constant-memory agents.
Open questions
How much memory is needed to encode the agent's transition function? The naive table needs \((\log n)^{O(1)}\) bits (Remark 1 of Disser et al. 2019).
Notes
The trade-off reads: \(\Omega(\log n)\) bits make pebbles unnecessary; \(O((\log n)^{1-\varepsilon})\) bits force \(\Omega(\log\log n)\) pebbles; \(\Omega(\log\log n)\) pebbles make \(O(1)\) bits enough. The traps of Disser et al. 2019 have \(O(K^{2^{5k}})\) nodes for \(k\) agents with \(K\) states, improving Rollik's tower-type bound Rollik 1980.
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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