TY - GEN

T1 - A markov chain algorithm for compression in self-organizing particle systems

AU - Cannon, Sarah

AU - Daymude, Joshua J.

AU - Randall, Dana

AU - Richa, Andrea

N1 - Funding Information:
Supported in part by NSF DGE-1148903 and a grant from the Simons Foundation (#361047 to Sarah Cannon) Supported in part by NSF awards CCF-1353089, CCF-1422603, and REU-026935. Supported in part by NSF grant CCF-1526900.

PY - 2016/7/25

Y1 - 2016/7/25

N2 - We consider programmable matter as a collection of simple computational elements (or particles) with limited (constant- size) memory that self-organize to solve system-wide prob- lems of movement, configuration, and coordination. Here, we focus on the compression problem, in which the parti- cle system gathers as tightly together as possible, as in a sphere or its equivalent in the presence of some underlying geometry. More specifically, we seek fully distributed, local, and asynchronous algorithms that lead the system to con- verge to a configuration with small perimeter. We present a Markov chain based algorithm that solves the compression problem under the geometric amoebot model, for particle sys- tems that begin in a connected configuration with no holes. The algorithm takes as input a bias parameter λ, where λ > 1 corresponds to particles favoring inducing more lat- tice triangles within the particle system. We show that for all λ > 5, there is a constant α > 1 such that at stationar- ity with all but exponentially small probability the particles are α-compressed, meaning the perimeter of the system con- figuration is at most α · pmin, where pmin is the minimum possible perimeter of the particle system. We additionally prove that the same algorithm can be used for expansion for small values of λ; in particular, for all 0 < λ < √2; there is a constant fi < 1 such that at stationarity, with all but an ex-ponentially small probability, the perimeter will be at least β · pmax, where pmax is the maximum possible perimeter.

AB - We consider programmable matter as a collection of simple computational elements (or particles) with limited (constant- size) memory that self-organize to solve system-wide prob- lems of movement, configuration, and coordination. Here, we focus on the compression problem, in which the parti- cle system gathers as tightly together as possible, as in a sphere or its equivalent in the presence of some underlying geometry. More specifically, we seek fully distributed, local, and asynchronous algorithms that lead the system to con- verge to a configuration with small perimeter. We present a Markov chain based algorithm that solves the compression problem under the geometric amoebot model, for particle sys- tems that begin in a connected configuration with no holes. The algorithm takes as input a bias parameter λ, where λ > 1 corresponds to particles favoring inducing more lat- tice triangles within the particle system. We show that for all λ > 5, there is a constant α > 1 such that at stationar- ity with all but exponentially small probability the particles are α-compressed, meaning the perimeter of the system con- figuration is at most α · pmin, where pmin is the minimum possible perimeter of the particle system. We additionally prove that the same algorithm can be used for expansion for small values of λ; in particular, for all 0 < λ < √2; there is a constant fi < 1 such that at stationarity, with all but an ex-ponentially small probability, the perimeter will be at least β · pmax, where pmax is the maximum possible perimeter.

KW - Compression

KW - Markov Chains

KW - Self-organizing Particles

UR - http://www.scopus.com/inward/record.url?scp=84984678041&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84984678041&partnerID=8YFLogxK

U2 - 10.1145/2933057.2933107

DO - 10.1145/2933057.2933107

M3 - Conference contribution

AN - SCOPUS:84984678041

T3 - Proceedings of the Annual ACM Symposium on Principles of Distributed Computing

SP - 279

EP - 288

BT - PODC 2016 - Proceedings of the 2016 ACM Symposium on Principles of Distributed Computing

PB - Association for Computing Machinery

T2 - 35th ACM Symposium on Principles of Distributed Computing, PODC 2016

Y2 - 25 July 2016 through 28 July 2016

ER -