Bubble-Flip—A new generation algorithm for prefix normal words

We present a new recursive generation algorithm for prefix normal words. These are binary words with the property that no factor has more 1s than the prefix of the same length. The new algorithm uses two operations on binary words, 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 word, 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. 1

Authors

• Ferdinando Cicalese
• Zsuzsanna Lipták
• Massimiliano Rossi

Keywords

• Algorithms on automata and words
• Combinatorics on words
• Combinatorial generation
• Prefix normal words
• Infinite words
• Binary languages
• Combinatorial Gray code