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Leader election in programmable matter

Anonymous constant-memory particles must elect exactly one leader, which most shape-formation algorithms need first.

Problem statement and measuresexpand

A connected configuration (the set of nodes occupied by particles) of \(n\) anonymous particles (initially contracted) must reach a state in which exactly one particle is the leader and all others know they are not. Time is in asynchronous rounds; \(L_{out}\) is the length of the outer boundary and \(D\) the diameter of the configuration.

Measures: Time, asynchronous rounds; Moves, total expansions and contractions (work).

no assumptions›+ constant memory›+ strong scheduler›+ chirality›+ may move›deterministic, + may disconnect

Deterministic, with chirality

standard settingverified

Time: unknownMoves: unknown

Deterministic leader election in \(O(L_{out}+D)\) rounds, as fast as the best randomized algorithm, holes allowed.

Assumed (added on the last edge are outlined)

+Memoryconstant memory and local communication+Schedulerstrong: atomic activations under a fair adversary+Orientationcommon chirality+Movementparticles may move+Connectivity during executionthe system may disconnect temporarily

Not assumed

RandomnessdeterministicInitial shapeconnected, holes allowed

Formal definitions of this setting

Memory: constant memory: Each particle has \(O(1)\) bits of persistent memory and reads or writes neighbours' memory.
Scheduler: strong scheduler: Each activation is executed atomically; the adversary chooses the order, fairly.
Orientation: chirality: All particles agree on clockwise order, not on a common direction.
Movement: may move: Particles may expand and contract during the election.
Randomness: deterministic: No random choices.
Initial shape: holes allowed: The initial set of occupied nodes induces a connected subgraph; it may enclose empty regions.
Connectivity during execution: may disconnect: Connectivity is required only initially and at the end.

What is known

Time · asynchronous rounds

unknown

Only an informal \(\Omega(D)\) remark; no proven lower bound.

lower boundnone known

introductory—

current best

\(O(L_{out}+D)\)

Dufoulon et al. 2021

Moves · total expansions and contractions (work)

unknown

—

lower boundnone known

introductory—

current best—

Results, in order of publication

Dufoulon et al. 2021 current best

Time \(O(L_{out}+D)\)

Introductory for linear time and current best; \(O(D_A)\), where \(D_A\) is the diameter of the area enclosed by the particles (holes included), if particles know whether their boundary is the outer one. The system may temporarily disconnect.

Open questions

Fast deterministic leader election under weaker (non-atomic) schedulers, and without letting the system disconnect Dufoulon et al. 2021.

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

Deterministic, no orientation

Time \(O(L_{out}n^2)\)

Stationary, chirality, weak scheduler

Time \(O(n^2)\)

Lower bounds from stronger settings hold here

No catalogued stronger setting.

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