De Bruijn Sequences: From Games to Shift-Rules to a Proof of the Fredricksen-Kessler-Maiorana Theorem

TL;DR

This paper introduces a combinatorial game and optimal strategies to derive efficient shift-rules for De Bruijn sequences, providing a new proof of a classic theorem connecting these sequences and Lyndon words.

Contribution

It presents a novel combinatorial game approach that yields efficient shift-rules and offers a new proof of the Fredricksen-Kessler-Maiorana theorem.

Findings

Efficient shift-rules for prefer-max and prefer-min De Bruijn sequences

A new combinatorial game framework for De Bruijn sequences

A novel proof of the Fredricksen-Kessler-Maiorana theorem

Abstract

We present a combinatorial game and propose efficiently computable optimal strategies. We then show how these strategies can be translated to efficiently computable shift-rules for the well known prefer-max and prefer-min De Bruijn sequences, in both forward and backward directions. Using these shift-rules, we provide a new proof of the well known theorem by Fredricksen, Kessler, and Maiorana on De Bruijn sequences and Lyndon words.