Leader election by agents
\(n\) agents in an anonymous graph must elect one leader; how fast, and with how little memory?
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 elect exactly one leader: 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.
Knowing \(n\) and \(\Delta\)
verifiedWith \(n\) and \(\Delta\) known, leader election takes \(O(n\log^2 n+D\Delta\log n)\) 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: knows \(n\), \(\Delta\)
- Every agent knows \(n\) and \(\Delta\) (upper bounds suffice where stated).
- Initial configuration: any start
- Agents start at arbitrary nodes; \(\ell\le k\) nodes are initially occupied, possibly with several agents each.
- Topology: any topology
- \(G\) is any connected graph in the class allowed by the other axes.
What is known
Results, in order of publication
The bound with the \(\log n\) factor is the correct one (confirmed by the catalogue owner, a co-author); the abstract omits it.
Open questions
None recorded for this variant.
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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