BFS tree by agents
\(n\) agents in an anonymous graph must build a breadth-first spanning tree, without being told the root.
Problem statement and measuresexpand
\(n\) agents with distinct identifiers (IDs) are placed on the nodes of an anonymous port-labelled graph with \(n\) nodes, \(m\) edges and maximum degree \(\Delta\). The start is rooted (all agents on one node), dispersed (one agent on each node) or arbitrary. Agents compute only by moving and by talking with agents on the same node. The agents must build a BFS tree from a root they choose, each agent knowing its parent and children: every agent (or node) learns its part of the output. Time is in synchronous rounds; memory in bits per agent.
Measures: Time, rounds; Memory, bits per agent.
Dispersed start, root unknown
standard settingverifiedA BFS tree in \(O(\min(D\Delta, m\log n)+n\log n+\Delta\log^2 n)\) rounds with \(O(\log n)\) bits.
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.
- Identifiers: distinct IDs (eponymous)
- Distinct identifiers (eponymous agents): the label function \(\mathrm{id}:\{1,\dots,k\}\to\{1,\dots,L\}\) is injective; usually \(L=k^{O(1)}\), so a label takes \(O(\log k)\) bits.
- Communication: local communication
- Agents at the same node at the same time can read each other's entire memory (face-to-face communication). Agents crossing on an edge do not meet.
- Timing: synchronous
- Time proceeds in rounds \(t=1,2,\dots\); in every round every agent communicates, computes and moves (at most one edge) simultaneously.
- Knowledge: no knowledge
- Agents know no graph parameter (\(n\), \(m\), \(\Delta\), \(D\)).
- Initial configuration: dispersed start
- Every initially occupied node holds exactly one agent (\(\ell=k\)). Also called a scattered start, notably in black hole search.
- Topology: any topology
- \(G\) is any connected graph in the class allowed by the other axes.
What is known
Results, in order of publication
Introductory and current best without knowledge of the root; \(O(D\Delta)\) was known with a known root.
Open questions
None recorded for this variant.
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