Paper by Erik D. Demaine

Reference:

Erik D. Demaine, Sarah Eisenstat, Mashhood Ishaque, and Andrew Winslow, “One-Dimensional Staged Self-Assembly”, in Proceedings of the 17th International Conference on DNA Computing and Molecular Programming (DNA 2011), Lecture Notes in Computer Science, volume 6937, Pasadena, California, September 19–23, 2011, pages 100–114.

Abstract:

We introduce the problem of staged self-assembly of one-dimensional nanostructures, which becomes interesting when the elements are labeled (e.g., representing functional units that must be placed at specific locations). In a restricted model in which each operation has a single terminal assembly, we prove that assembling a given string of labels with the fewest stages is equivalent, up to constant factors, to compressing the string to be uniquely derived from the smallest possible context-free grammar (a well-studied O(log n)-approximable problem). Without this restriction, we show that the optimal assembly can be substantially smaller than the optimal context-free grammar, by a factor of Ω(√n/log n) even for binary strings of length n. Fortunately, we can bound this separation in model power by a quadratic function in the number of distinct glues or tiles allowed in the assembly, which is typically small in practice.

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