Sarah Cannon

Two Hands Are Better Than One (up to constant factors): Self-Assembly In The 2HAM vs. aTAM

We study the dierence between the standard seeded model (aTAM) of tile self-assembly, and the “seedless” two-handed model of tile self-assembly (2HAM). Most of our results suggest that the two-handed model is more powerful. In particular, we show how to simulate any seeded system with a two-handed system that is essentially just a constant factor larger. We exhibit finite shapes with a busy-beaver separation in the number of distinct tiles required by seeded versus two-handed, and exhibit an infinite shape that can be constructed two-handed but not seeded. Finally, we show that verifying whether a given system uniquely assembles a desired supertile is co-NP-complete in the two-handed model, while it was known to be polynomially solvable in the seeded model. 1998 ACM Subject Classification F.1.2

A Markov Chain Algorithm for Compression in Self-Organizing Particle Systems

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 problems of movement, configuration, and coordination. Here, we focus on the compression problem, in which the particle 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 converge 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 systems 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 lattice triangles within the particle system. We show that for all λ > 5, there is a constant α > 1 such that at stationarity with all but exponentially small probability the particles are α-compressed, meaning the perimeter of the system configuration 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 β < 1 such that at stationarity, with all but an exponentially small probability, the perimeter will be at least β ⋅ pmax, where pmax is the maximum possible perimeter.

Combinatorics and complexity of guarding polygons with edge and point 2-transmitters

We consider a generalization of the classical Art Gallery Problem, where instead of a light source, the guards, called

Conflict-Free graph orientations with parity constraints

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A stochastic approach to shortcut bridging in programmable matter

-transmitters, model a wireless device with a signal that can pass through at most

Polynomial Mixing of the Edge-Flip Markov Chain for Unbiased Dyadic Tilings.

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Diffuse reflection diameter in simple polygons

walls. We show it is NP-hard to compute a minimum cover of point 2-transmitters, point

Two Hands Are Better Than One (up to constant factors)

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Phase transitions in random dyadic tilings and rectangular dissections

-transmitters, and edge 2-transmitters in a simple polygon. The point 2-transmitter result extends to orthogonal polygons. In addition, we give necessity and sufficiency results for the number of edge 2-transmitters in general, monotone, orthogonal monotone, and orthogonal polygons.

Hidden Mobile Guards in Simple Polygons

It is known that every multigraph with an even number of edges has an even orientation (i.e., all indegrees are even). We study parity constrained graph orientations under additional constraints. We consider two types of constraints for a multigraph G=(V,E): (1) an exact conflict constraint is an edge set C⊆E and a vertex v∈V such that C should not equal the set of incoming edges at v; (2) a subset conflict constraint is an edge set C⊆E and a vertex v∈V such that C should not be a subset of incoming edges at v. We show that it is NP-complete to decide whether G has an even orientation with exact or subset conflicts, for all conflict sets of size two or higher. We present efficient algorithms for computing parity constrained orientations with disjoint exact or subset conflict pairs.