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.
Randomized exploration when \(n\) is known
verifiedA random walk covers any graph in expected \(2m(n-1)\) steps; with a step counter and a bound on \(n\) it stops after exploring with good probability.
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: 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: private coins
- Each agent additionally has access to independent fair coin flips; guarantees hold with probability 1, in expectation, or with high probability, as stated per result.
- 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
Results, in order of publication
Expected cover time at most \(2m(n-1)\) from any start (Theorem 4). Running the walk for a multiple of that many steps with an \(O(\log n)\)-bit counter gives the Monte Carlo agent.
Open questions
What is the memory needed by a randomized agent that must stop correctly with high probability, as a function of \(n\)?
Notes
The same paper proves by the probabilistic method that universal traversal sequences of length \(O(n^3\log n)\) exist for connected \(\Delta\)-regular graphs, the constant depending on \(\Delta\) (the proof uses length \(2\Delta n(n-1)(\Delta n+2)\lceil\log n\rceil\)) Aleliunas et al. 1979. For directed graphs the cover time can be exponential, while Eulerian digraphs behave like undirected graphs Aleliunas et al. 1979; Reingold later constructed universal exploration sequences in log space Reingold 2008.
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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