Bubble-Flip -- A New Generation Algorithm for Prefix Normal Words

TL;DR

This paper introduces Bubble-Flip, a recursive algorithm for generating prefix normal words, leveraging string operations to efficiently produce and analyze infinite prefix normal words with predictable periodic properties.

Contribution

It presents a novel recursive generation algorithm for prefix normal words and explores the properties of infinite prefix normal words generated by this method.

Findings

The algorithm efficiently generates prefix normal words.

Infinite prefix normal words can be produced with predictable periodicity.

The method reveals structural properties of prefix normal words.

Abstract

We present a new recursive generation algorithm for prefix normal words. These are binary strings with the property that no substring has more 1s than the prefix of the same length. The new algorithm uses two operations on binary strings, which exploit certain properties of prefix normal words in a smart way. We introduce infinite prefix normal words and show that one of the operations used by the algorithm, if applied repeatedly to extend the string, produces an ultimately periodic infinite word, which is prefix normal. Moreover, based on the original finite word, we can predict both the length and the density of an ultimate period of this infinite word.