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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›stationary, + no holes

Stationary, chirality, no holes

verified
Time: unknownMoves: tight

On hole-free configurations with chirality, stationary constant-memory particles elect a leader deterministically in \(O(n)\) rounds.

Assumed (added on the last edge are outlined)

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

Not assumed

Movementstationary: 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: chirality
All particles agree on clockwise order, not on a common direction.
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
No lower bound recorded.
lower boundnone known
introductory—
current best
\(O(n)\)
Moves · total expansions and contractions (work)
tight
No moves.
lower boundnone known
introductory—
current best—

Results, in order of publication

Gastineau et al. 2019 current best
Time \(O(n)\)

Introductory and current best: \(O(r + m_{tree})\) rounds, fewer than \(2n\) (Proposition 1). Also computes k-local identifiers; works on the square, triangular and king grids.

Open questions

None recorded for this variant.

Notes

\(r\) and \(m_{tree}\) are parameters of the configuration defined in Gastineau et al. 2019: \(r\) is the radius of the configuration with respect to its border particles (the least, over particles \(u\), of the largest distance from \(u\) to a border particle), and \(m_{tree}\) is the largest height of an induced subgraph of the configuration that is a tree. The scheduler is strong, as listed by Dufoulon et al. 2021. 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, chirality, weak scheduler
Time \(O(n^2)\)
Stationary, fair scheduler, no orientation, no holes
Time \(\text{finite}\)

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.