Combinatorics of minimal absent words for a sliding window

05/18/2021

∙

by   Tooru Akagi, et al.

∙

0

∙

share

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].