Combinatorics of minimal absent words for a sliding window

Abstract

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].