Natural Computing

, Volume 12, Issue 2, pp 247–258 | Cite as

One-dimensional staged self-assembly

  • Erik D. Demaine
  • Sarah Eisenstat
  • Mashhood Ishaque
  • Andrew Winslow


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.


Context-free grammar Wang tile DNA computing Complexity 



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


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Copyright information

© Springer Science+Business Media Dordrecht 2012

Authors and Affiliations

  • Erik D. Demaine
    • 1
  • Sarah Eisenstat
    • 1
  • Mashhood Ishaque
    • 2
  • Andrew Winslow
    • 2
  1. 1.MIT Computer Science and Artificial Intelligence LaboratoryCambridgeUSA
  2. 2.Department of Computer ScienceTufts UniversityMedfordUSA

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