## 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 steps 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) and that the problem is NP-hard. Without this restriction, we show that the optimal assembly can be substantially smaller than the optimal context-free grammar, by a factor of \(\Omega(\sqrt{n/\log n})\) 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.

## Keywords

Context-free grammar Wang tile DNA computing Complexity## Notes

### Acknowledgments

We thank Martin Demaine, André Schulz, Diane Souvaine, and Hyunmin Yi for helpful discussions, and anonymous reviewers for helpful suggestions.

## References

- Harish C, Nikhil G, John R (2009) The tile complexity of linear assemblies. In: ICALP 2009, vol 5555 of Lecture notes in computer science, pp 235–253Google Scholar
- Charikarv M, Lehman E, Liu D, Panigrahy Rina, Prabhakaran M, Rasala A, Sahai A, Shelat A (2002) Approximating the smallest grammar: Kolmogorov complexity in natural models. In: Proceedings of the 34th Annual ACM Symposium on Theory of Computing. ACM, New York, NY, USA, pp 792–801Google Scholar
- Czeizler E, Popa A (2012) Synthesizing minimal tile sets for complex patterns in the framework of patterned dna self-assembly. In: DNA computing and molecular programming, vol 7433 of Lecture notes in computer science, pp 58–72Google Scholar
- Demaine E, Demaine M, Fekete S, Ishaque M, Rafalin E, Schweller R, Souvaine D (2008) Staged self-assembly: nanomanufacture of arbitrary shapes with
*O*(1) glues. Nat Comput 7:347–370MathSciNetMATHCrossRefGoogle Scholar - Farach M, Thorup M (1998) String matching in Lempel–Ziv compressed strings. Algorithmica 20:388–404MathSciNetMATHCrossRefGoogle Scholar
- Göös M, Orponen P (2011) Synthesizing minimal tile sets for patterned dna self-assembly. In: Yasubumi S, Yongli M (eds) DNA computing and molecular programming, vol 6518 of Lecture notes in computer science. Springer Berlin, pp 71–82 Google Scholar
- Kao M-Y, Schweller R (2008) Randomized self-assembly for approximate shapes. In: ICALP 2008, vol 5125 of Lecture notes in computer science, pp 370–384Google Scholar
- Kieffer J, Yang EH, Nelson G, Pamela C (2000) Universal lossless compression via multilevel pattern matching. IEEE Trans Inf Theory 46:1227–1245MATHCrossRefGoogle Scholar
- Lehman E (2002) Approximation algorithms for grammar-based data compression. PhD thesis, MITGoogle Scholar
- Ma X, Lombardi F (2008) Synthesis of tile sets for dna self-assembly. IEEE Trans Comput Aided Des Integr Circuits Syst 27(5):963–967CrossRefGoogle Scholar
- Rytter W (2002) Application of Lempel–Ziv factorization to the approximation of grammar-based compression. In: Alberto A, Masayuki T (eds) Combinatorial pattern matching, vol 2373 of Lecture notes in computer science. Springer, Berlin, pp 20–31. doi: 10.1007/3-540-45452-7_3
- Sakamoto H (2005) A fully linear-time approximation algorithm for grammar-based compression. J Discret Algorithms 3(2–4):416–430MathSciNetMATHCrossRefGoogle Scholar
- Soloveichik D, Winfree E (2005) Complexity of self-assembled shapes. In: Ferretti C et al. (eds) DNA computing, vol 3384 of Lecture notes in computer science, pp 344–354Google Scholar
- Winfree E (1998) Algorithmic self-assembly of DNA. PhD thesis, CaltechGoogle Scholar