Running time and program size for self-assembled squares

Recently Rothemund and Winfree 6] have considered the program size complexity of constructing squares by selfassembly. Here, we consider the time complexity of such constructions using a natural generalization of the Tile Assembly Model de ned in 6]. In the generalized model, the Rothemund-Winfree construction of n n squares requires time (nlogn) and program size (log n). We present a new construction for assembling n n squares which uses optimal time (n) and program size ( 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 selfassembly 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 ...

• Type: Conference
• Year: 2001
• Where: STOC
• Authors: Leonard M. Adleman, Qi Cheng, Ashish Goel, Ming-Deh A. Huang
• Keywords: Algorithms, Log N, Program Size, STOC 2001, Tile Assembly Model