Loopless Gray Code Enumeration and the Tower of Bucharest

TL;DR

This paper introduces new loopless algorithms for generating Gray codes for n-tuples over an m-letter alphabet, inspired by Towers of Hanoi variations, enabling efficient hardware implementation.

Contribution

It presents novel loopless algorithms for Gray code generation based on Towers of Hanoi, with implementations suitable for hardware and work-ahead strategies.

Findings

Algorithms are loopless with constant-time next change determination.

Applicable to hardware implementations for efficient Gray code generation.

Provides two families of algorithms, including work-ahead buffering methods.

Abstract

We give new algorithms for generating all n-tuples over an alphabet of m letters, changing only one letter at a time (Gray codes). These algorithms are based on the connection with variations of the Towers of Hanoi game. Our algorithms are loopless, in the sense that the next change can be determined in a constant number of steps, and they can be implemented in hardware. We also give another family of loopless algorithms that is based on the idea of working ahead and saving the work in a buffer.