Fast Arithmetic in Algorithmic Self-Assembly

TL;DR

This paper analyzes the time complexity of arithmetic operations in the tile self-assembly model, providing new algorithms that outperform previous bounds for addition and multiplication in 2D and 3D systems.

Contribution

It introduces optimal and sublinear time algorithms for addition and multiplication in the tile self-assembly model, improving upon prior upper bounds.

Findings

Addition algorithms match the lower bounds of ( ext{ } ext{ }) in 2D and 3D.

Average case addition achieves ( ext{ } ext{ }) time complexity.

Multiplication in 3D runs in (n^{5/6}) time, beating previous bounds.

Abstract

In this paper we consider the time complexity of computing the sum and product of two $n$-bit numbers within the tile self-assembly model. The (abstract) tile assembly model is a mathematical model of self-assembly in which system components are square tiles with different glue types assigned to tile edges. Assembly is driven by the attachment of singleton tiles to a growing seed assembly when the net force of glue attraction for a tile exceeds some fixed threshold. Within this frame work, we examine the time complexity of computing the sum or product of 2 n-bit numbers, where the input numbers are encoded in an initial seed assembly, and the output is encoded in the final, terminal assembly of the system. We show that the problems of addition and multiplication have worst case lower bounds of $\Omega(\sqrt{n})$ in 2D assembly, and $\Omega(\sqrt[3]{n})$ in 3D assembly. In the case of addition, we design algorithms for both 2D and 3D that meet this bound with worst case run times of $O(\sqrt{n})$ and $O(\sqrt[3]{n})$ respectively, which beats the previous best known upper bound of O(n). Further, we consider average case complexity of addition over uniformly distributed n-bit strings and show how to achieve $O(\log n)$ average case time with a simultaneous $O(\sqrt{n})$ worst case run time in 2D. For multiplication, we present an $O(n^{5/6})$ time multiplication algorithm which works in 3D, which beats the previous best known upper bound of O(n). As additional evidence for the speed of our algorithms, we implement our addition algorithms, along with the simpler O(n) time addition algorithm, into a probabilistic run-time simulator and compare the timing results.