An Efficient Generalized Shift-Rule for the Prefer-Max De Bruijn Sequence

TL;DR

This paper introduces an efficient generalized shift-rule for De Bruijn sequences, enabling faster sequence construction by outputting multiple symbols after a given word, improving over traditional shift-rules.

Contribution

It proposes a generalized shift-rule with optimal time complexity for prefer-max and prefer-min De Bruijn sequences, enhancing sequence generation efficiency.

Findings

Achieves O(n+c) time complexity for the generalized shift-rule.

Enables efficient construction of entire De Bruijn sequences.

Provides a practical method for prefer-max and prefer-min sequences.

Abstract

One of the fundamental ways to construct De Bruijn sequences is by using a shift-rule. A shift-rule receives a word as an argument and computes the symbol that appears after it in the sequence. An optimal shift-rule for an $(n,k)$-De Bruijn sequence runs in time $O(n)$. We propose an extended notion we name a generalized-shift-rule, which receives a word, $w$, and an integer, $c$, and outputs the $c$ symbols that comes after $w$. An optimal generalized-shift-rule for an $(n,k)$-De Bruijn sequence runs in time $O(n+c)$. We show that, unlike in the case of a shift-rule, a time optimal generalized-shift-rule allows to construct the entire sequence efficiently. We provide a time optimal generalized-shift-rule for the well-known prefer-max and prefer-min De Bruijn sequences.