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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›+ knows a bound on \(n\)›deterministic, + with stop

Exploration with stop when \(n\) is known

verified
Time: unknownMemory: near-optimal

Reingold's log-space universal exploration sequences give an \(O(\log n)\)-bit agent that explores and stops in polynomial time.

Assumed (added on the last edge are outlined)

+Graph dynamicsstatic graph+Edge directionundirected+Knowledgean upper bound on \(n\) is known+Termination requirementexplore, then stop anywhere

Not assumed

Node identifiersanonymous nodesRandomnessdeterministicMarkersno 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: knows a bound on \(n\)
The agent is given an integer \(N\ge n\) (an upper bound on the number of nodes) as input.
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)
unknown
Polynomial; the exponent is not tracked and no non-trivial lower bound is recorded.
lower boundnone known
introductory—
current best
\(n^{O(1)}\)
Memory · bits of agent memory; pebbles, if any, are counted separately
near-optimal
\(O(\log n)\) versus \(\omega((\log n)^{1-\varepsilon})\): a \((\log n)^{\varepsilon}\) factor.
lower bound
\(\omega((\log n)^{1-\varepsilon})\) Disser et al. 2019
bottlenecktrap graphs
holds when
  • for every constant \(\varepsilon>0\)
  • the agent may depend on \(n\): it must explore all graphs with at most \(n\) nodes
  • no pebbles; deterministic
Corollary 25 of Disser et al. 2019: with \(O((\log n)^{1-\varepsilon})\) bits an agent needs \(\Omega(\log\log n)\) pebbles.
introductory—
current best
\(O(\log n)\)

Results, in order of publication

Reingold 2008 current best
Time \(n^{O(1)}\)Memory \(O(\log n)\)

Introductory and current best. A log-space constructible universal exploration sequence Koucký 2002 guides a walk through all nodes; the consequence for agents is stated by Disser et al. 2019.

Open questions

  1. Is \(\Omega(\log n)\) bits necessary when \(n\) is known? The trap arguments Fraigniaud et al. 2005 are stated for agents that know nothing.

Notes

Whether the same agent can also return to \(s\) (by reversing the sequence) is plausible but not checked against a source.

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

Exploration with stop, nothing known

Lower bounds from stronger settings hold here

Randomized exploration when \(n\) is known

Know a better result for this setting, or found a mistake? Suggest a result · Report an error. Accepted contributions are acknowledged.