Combinatorics of minimal absent words for a sliding window

• Published in: Theoretical computer science Vol. 927; pp. 109 - 119
• Main Authors: Akagi, Tooru, Kuhara, Yuki, Mieno, Takuya, Nakashima, Yuto, Inenaga, Shunsuke, Bannai, Hideo, Takeda, Masayuki
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 26.08.2022
• Subjects: Combinatorics on words; Minimal absent words; Sliding window; Minimal absent words; Combinatorics on words; Sliding window
• ISSN: 0304-3975; 1879-2294
• DOI: 10.1016/j.tcs.2022.06.002

A string w is called a minimal absent word (MAW) for another string T if w does not occur in T but the proper substrings of w occur in T. For example, let Σ={a,b,c} be the alphabet. Then, the set of MAWs for string w=abaab is {aaa,aaba,bab,bb,c}. In this paper, we study combinatorial properties of MAWs in the sliding window model, namely, how the set of MAWs changes when a sliding window of fixed length d is shifted over the input string T of length n, where 1≤d<n. We present tight upper and lower bounds on the maximum number of changes in the set of MAWs for a sliding window over T, both in the cases of general alphabets and binary alphabets. Our bounds improve on the previously known best bounds [Crochemore et al., 2020].