Title: Fast Reconfiguration of Robot Swarms with Uniform Control Signals

Authors: David Caballero, Angel A. Cantu, Timothy Gomez, Austin Luchsinger, Robert Schweller, and Tim Wylie.

Published: Natural Computing, 2021

## Optimal Staged Self-Assembly of Linear Assemblies

Title: Optimal Staged Self-Assembly of Linear Assemblies

Authors: Cameron Chalk, Eric Martinez, Robert Schweller, Luis Vega, Andrew Winslow, Tim Wylie

Abstract:

We analyze the complexity of building linear assemblies, sets of linear assemblies, and $\mathcal{O}(1)$-scale general shapes in the staged tile assembly model. For systems with at most $b$ bins and $t$ tile types, we prove that the minimum number of stages to uniquely assemble a $1 \times n$ \emph{line} is $\Theta(\log_t{n} + \log_b{\frac{n}{t}} + 1)$. Generalizing to $\BO{1} \times n$ lines, we prove the minimum number of stages is $\BO{\frac{\log{n} – tb – t\log t}{b^2} + \frac{\log \log b}{\log t}}$ and $\Omega(\frac{\log{n} – tb – t\log t}{b^2})$. We also obtain similar upper and lower bounds in a model permitting \emph{flexible glues} using non-diagonal glue functions.

Next, we consider assembling sets of lines and general shapes using $t = \BO{1}$ tile types. We prove that the minimum number of stages needed to assemble a set of $k$ lines of size at most $\BO{1} \times n$ is $\BO{\frac{k\log n}{b^2}+\frac{k\sqrt{\log n}}{b}+\log\log n}$ and $\Omega(\frac{k\log n}{b^2})$. In the case that $b = \BO{\sqrt{k}}$, the minimum number of stages is $\Theta(\log{n})$. The upper bound in this special case is then used to assemble “hefty” shapes of at least logarithmic edge-length-to-edge-count ratio at $\BO{1}$-scale using $\BO{\sqrt{k}}$ bins and optimal $\BO{\log{n}}$ stages.

Citation: Proc. of 17th Inter. Conf. on Unconventional Computation and Natural Computation (UCNC’18)

Bibtex:

## Self-Assembly of Shapes at Constant Scale using Repulsive Forces

Self-Assembly of Shapes at Constant Scale using Repulsive Forces.

Austin Luchsinger, Robert Schweller, and Tim Wylie. In Natural Computing. 2018.

Link: https://link.springer.com/article/10.1007/s11047-018-9707-9

Abstract:

Bibtex:

Conference Version:

Self-Assembly of Shapes at Constant Scale using Repulsive Forces.

Austin Luchsinger, Robert Schweller, and Tim Wylie.

In Proc. of the 16th Inter. Conf. on Unconventional Computation and Natural Computation (UCNC’17), 2017.

Abstract:

Bibtex:

## Verification in Staged Tile Self-Assembly

Verification in Staged Tile Self-Assembly.

Robert Schweller, Andrew Winslow, and Tim Wylie.

In Proc. of the 16th Inter. Conf. on Unconventional Computation and Natural Computation (UCNC’17), 2017.