Optimal Staged Self-Assembly of General Shapes

Authors

Cameron Chalk, The University of Texas Rio Grande Valley
Eric M. Martinez, The University of Texas Rio Grande ValleyFollow
Robert Schweller, The University of Texas Rio Grande ValleyFollow
Luis Vega, The University of Texas Rio Grande Valley
Andrew Winslow, The University of Texas Rio Grande Valley
Tim Wylie, The University of Texas Rio Grande ValleyFollow

Document Type

Article

Publication Date

4-2018

Abstract

We analyze the number of tile types t, bins b, and stages necessary to assemble n \times n squares and scaled shapes in the staged tile assembly model. For n \times n squares, we prove \mathcal{O}(\frac{\log{n} - tb - t\log t}{b^2} + \frac{\log \log b}{\log t}) stages suffice and \Omega(\frac{\log{n} - tb - t\log t}{b^2}) are necessary for almost all n. For shapes S with Kolmogorov complexity K(S), we prove \mathcal{O}(\frac{K(S) - tb - t\log t}{b^2} + \frac{\log \log b}{\log t}) stages suffice and \Omega(\frac{K(S) - tb - t\log t}{b^2}) are necessary to assemble a scaled version of S, for almost all S. We obtain similarly tight bounds when the more powerful flexible glues are permitted.

Comments

Original published version available at https://doi.org/10.1007/s00453-017-0318-0

Recommended Citation

Chalk, Cameron, Eric Martinez, Robert Schweller, Luis Vega, Andrew Winslow, and Tim Wylie. "Optimal staged self-assembly of general shapes." Algorithmica 80, no. 4 (2018): 1383-1409. https://doi.org/10.1007/s00453-017-0318-0

Publication Title

Algorithmica

DOI

10.1007/s00453-017-0318-0