Home›Programmable matter›Self-organising particle systems›Leader election in programmable matter

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, + may move

Deterministic, no orientation

verified

Time: polynomial gapMoves: unknown

Without chirality, a unique leader can still be elected deterministically, in \(O(L_{out}n^2)\) rounds.

Assumed (added on the last edge are outlined)

+Memoryconstant memory and local communication+Schedulerstrong: atomic activations under a fair adversary+Movementparticles may move

Not assumed

Orientationno common orientation and no chiralityRandomnessdeterministicInitial shapeconnected, holes allowedConnectivity 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: 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: stays connected: At every time the occupied nodes induce a connected subgraph.

What is known

Time · asynchronous rounds

polynomial gap

Far from the linear bound known with chirality.

lower boundnone known

introductory—

current best

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

Emek et al. 2019

Moves · total expansions and contractions (work)

unknown

—

lower boundnone known

introductory—

current best—

Results, in order of publication

Emek et al. 2019 current best

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

Introductory and current best; also yields agreement on chirality.

Open questions

Deterministic leader election without chirality in linear time.

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

SILBOT with holes

Time \(O(n^2)\)

Lower bounds from stronger settings hold here

Deterministic, with chirality

Randomized, with chirality

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