Approximating the smallest grammar: Kolmogorov complexity in natural models
STOC '02 Proceedings of the thiry-fourth annual ACM symposium on Theory of computing
Application of Lempel-Ziv Factorization to the Approximation of Grammar-Based Compression
CPM '02 Proceedings of the 13th Annual Symposium on Combinatorial Pattern Matching
Algorithmic self-assembly of dna
Algorithmic self-assembly of dna
Approximation algorithms for grammar-based data compression
Approximation algorithms for grammar-based data compression
Staged self-assembly: nanomanufacture of arbitrary shapes with O(1) glues
Natural Computing: an international journal
Randomized Self-assembly for Approximate Shapes
ICALP '08 Proceedings of the 35th international colloquium on Automata, Languages and Programming, Part I
The Tile Complexity of Linear Assemblies
ICALP '09 Proceedings of the 36th International Colloquium on Automata, Languages and Programming: Part I
Synthesizing minimal tile sets for patterned DNA self-assembly
DNA'10 Proceedings of the 16th international conference on DNA computing and molecular programming
Complexity of self-assembled shapes
DNA'04 Proceedings of the 10th international conference on DNA computing
Universal lossless compression via multilevel pattern matching
IEEE Transactions on Information Theory
Synthesis of Tile Sets for DNA Self-Assembly
IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems
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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.