Running time and program size for self-assembled squares

  • Authors:

  • Leonard Adleman;Qi Cheng;Ashish Goel;Ming-Deh Huang

  • Affiliations:

  • Professor of Computer Science and Molecular Biology, University of Southern California;Department of Computer Science, University of Southern California;Assistant Professor of Computer Science, University of Southern California;Professor of Computer Science, University of Southern California

  • Venue:
  • STOC '01 Proceedings of the thirty-third annual ACM symposium on Theory of computing
  • Year:
  • 2001

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Abstract

Recently Rothemund and Winfree [6] have considered the program size complexity of constructing squares by self-assembly. Here, we consider the time complexity of such constructions using a natural generalization of the Tile Assembly Model defined in [6]. In the generalized model, the Rothemund-Winfree construction of n \times n squares requires time &THgr;(n log n) and program size &THgr;(log n). We present a new construction for assembling n \times n squares which uses optimal time &THgr;(n) and program size &THgr;(\frac{log n}{log log n}). This program size is also optimal since it matches the bound dictated by Kolmogorov complexity. Our improved time is achieved by demonstrating a set of tiles for parallel self-assembly of binary counters. Our improved program size is achieved by demonstrating that self-assembling systems can compute changes in the base representation of numbers. Self-assembly is emerging as a useful paradigm for computation. In addition the development of a computational theory of self-assembly promises to provide a new conduit by which results and methods of theoretical computer science might be applied to problems of interest in biology and the physical sciences.