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Natural Computing

, Volume 14, Issue 2, pp 279–292 | Cite as

3-color bounded patterned self-assembly

  • Lila Kari
  • Steffen Kopecki
  • Shinnosuke Seki
Article

Abstract

The problem of patterned self-assembly tile set synthesis (Pats) is to find a minimal tile set which uniquely self-assembles into a given pattern. Czeizler and Popa proved the \(\mathrm {NP}\)-completeness of Pats and Seki showed that the Pats problem is already \(\mathrm {NP}\)-complete for patterns with 60 colors. In search for the minimal number of colors such that Pats remains \(\mathrm {NP}\)-complete, we introduce multiple bound Pats (mbPats) where we allow bounds for the numbers of tile types of each color. We show that mbPats is \(\mathrm {NP}\)-complete for patterns with just three colors and, as a byproduct of this result, we also obtain a novel proof for the \(\mathrm {NP}\)-completeness of Pats which is more concise than the previous proofs.

Keywords

Conjunctive Normal Form Boolean Formula Tile Type Unique Color Shaped Seed 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  1. 1.Department of Computer ScienceThe University of Western OntarioLondonCanada
  2. 2.Department of Information and Computer Science, Helsinki Institute for Information Technology (HIIT)Aalto UniversityAaltoFinland

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