Collaborative exploration
\(k\) agents start together at the root of an unknown graph and must visit all its nodes; how much faster than one agent can a team be?
Problem statement and measuresexpand
A team of \(k\) distinguishable agents starts at a node \(s\) (the root) of an unknown connected graph with \(n\) nodes; \(D\) is the largest distance from \(s\). In synchronous rounds every agent traverses at most one edge. When an agent reaches a node it sees the incident edges. The team must visit every node (some papers also require all agents to return to \(s\); the bounds coincide asymptotically).
Time is the number of rounds (in the asynchronous model: the total number of moves). The offline optimum is \(\Theta(n/k + D)\), so an algorithm that explores every tree in \(\mathrm{CR}(k)\cdot(n/k+D)\) rounds is called \(\mathrm{CR}(k)\)-competitive; a single depth-first search is \(O(k)\)-competitive. Memory is bits per agent; most of this literature lets it be unbounded.
Unless a variant says otherwise, nodes are labelled (an agent recognises a node it has seen), which is the standard assumption of collective tree exploration; on trees it does not matter for lower bounds Disser et al. 2020. With anonymous nodes and \(k = n\), the problem is solved by rooted dispersion.
Measures: Time, rounds until every node is visited (asynchronous: total moves); Memory, bits per agent.
Trees with \(k = n\) agents
verifiedWith as many agents as nodes, \(D^2\) rounds suffice, and some trees need \(D^2/3\); this is the bound dispersion inherits.
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: labelled nodes
- Each node \(v\) carries a distinct identifier that agents at \(v\) can read; hence an agent can recognise a node it has visited.
- Randomness: deterministic
- The next state and move of an agent are a function of its current state and what it observes.
- Communication: global communication
- At every round every agent can read the memory of every other agent, wherever it is.
- Node memory: no whiteboards
- Nodes have no storage: no information can be left at a node except by an agent staying there.
- Timing: synchronous
- Time proceeds in rounds \(t=1,2,\dots\); in every round every agent communicates, computes and moves (at most one edge) simultaneously.
- Team size: \(k = n\)
- Exactly \(k=n\) agents.
- Topology: tree
- \(G\) is connected and acyclic: \(|E|=n-1\).
What is known
- deterministic, global communication
- existential in \(D\): the hard tree has depth \(D=\lceil\sqrt{\log n}\rceil\) (Theorem 2.5)
- not shown for larger heights
Results, in order of publication
Introductory and current best: phase-wise breadth-first search, \(D\) phases of \(D\) rounds.
Open questions
For \(k=n\) and trees of depth \(D\) well above \(\sqrt{\log n}\) (e.g. \(D=\sqrt n\)), is \(\Omega(D^2)\) still necessary?
Notes
Dispersion papers cite this bound as "\(\Omega(D^2)\) for \(n\) agents on trees" Augustine & Moses Jr. 2018 Kshemkalyani et al. 2022; the theorem is existential in \(D\), see the overview From exploration to dispersion.
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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