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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›no orientation, stationary, + no holes

Stationary, fair scheduler, no orientation, no holes

verified
Time: unknownMoves: tight

Stationary constant-memory particles with no orientation and no chirality elect a unique leader from any initial memory: silent self-stabilising election on simply connected systems.

Assumed (added on the last edge are outlined)

+Memoryconstant memory and local communication+Schedulerstrong: atomic activations under a fair adversary+Initial shapesimply connected (no holes)

Not assumed

Orientationno common orientation and no chiralityMovementstationary: particles never moveRandomnessdeterministicConnectivity during executionthe system must stay connected
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: no orientation
Each particle labels its six incident directions in its own clockwise or counter-clockwise order, starting anywhere.
Movement: stationary
No particle ever expands or contracts; the configuration is fixed and only states change.
Randomness: deterministic
No random choices.
Initial shape: no holes
The initial configuration is connected and encloses no empty node. Also called contractible Briones et al. 2023 or hole-free Gastineau et al. 2019.
Connectivity during execution: stays connected
At every time the occupied nodes induce a connected subgraph.

What is known

Time · asynchronous rounds
unknown
—
lower boundnone known
introductory—
current best
\(\text{finite}\)
Gouda-fair schedulerself-stabilising: any initial memory
Moves · total expansions and contractions (work)
tight
No moves.
lower boundnone known
introductory—
current best—

Results, in order of publication

Chalopin et al. 2025 current best
Time \(\text{finite}\)
Gouda-fair schedulerself-stabilising: any initial memory

Introductory and current best. The algorithm fails under an unfair scheduler (Section 6). Constant memory suffices although silent self-stabilising election in general networks needs \(\Omega(\log n)\) bits, even on rings.

Open questions

None recorded for this variant.

Notes

Self-stabilisation is a stronger requirement than the problem statement and has no axis here. Checked against the full text.

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

Stationary, asynchronous, no orientation, no holes
Time \(\text{finite}\)

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