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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.

no assumptions›+ static graph›+ undirected›nothing known, no markers, adversarial ports, + with stop

Exploration with stop, nothing known

verified

Time: impossibleMemory: impossible

Without a bound on \(n\) and without a way to mark a node, an agent can never be sure it has seen everything.

Assumed (added on the last edge are outlined)

+Graph dynamicsstatic graph+Edge directionundirected+Termination requirementexplore, then stop anywhere

Not assumed

Node identifiersanonymous nodesKnowledgenothing known about the graphRandomnessdeterministicMarkersno markersPreprocessingports and nodes as given by an adversaryTopologyarbitrary connected graph

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: no markers: The agent cannot mark nodes.
Preprocessing: adversarial ports: Port numbers (and any node labels) are chosen by an adversary.
Termination requirement: with stop: The agent visits every node and then halts, at any node. Also called terminating exploration.
Topology: any topology: \(G\) is any connected graph in the class allowed by the other axes.

What is known

Time · edge traversals until every node is visited (for periodic exploration: the period)

impossible

Feasible with one unmovable token at the start node Dieudonné & Pelc 2012.

lower bound

\(\text{impossible}\) Dieudonné & Pelc 2012

bottleneckno way to recognise the start

holds when

deterministic agent, no upper bound on \(n\), no marking of nodes
unbounded memory does not help
even cycles cannot be explored with stop without a way to mark a node Fraigniaud et al. 2006

introductory—

current best—

Memory · bits of agent memory; pebbles, if any, are counted separately

impossible

Impossible for any memory size.

lower bound

\(\text{impossible}\) Dieudonné & Pelc 2012

bottleneckno way to recognise the start

holds when

same conditions as for time

introductory—

current best—

Results, in order of publication

No result recorded yet.

Open questions

None recorded for this variant.

Notes

The impossibility is why every stopping variant below adds either knowledge (a bound on \(n\)) or a way to mark a node (a pebble or a token). Dieudonné and Pelc also study Byzantine tokens: identical unmovable tokens, some of which may be faulty; exploration is feasible in polynomial cost if a bound on the number of tokens is known and at least one is fault-free Dieudonné & Pelc 2012.

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.