A substring u of a string T is called a minimal unique substring (MUS) of T if u occurs exactly once in T and any proper substring of u occurs at least twice in T . A string w is called a minimal absent word (MAW) of T if w does not occur in T and any proper substring of w occurs in T . In this paper, we study the problems of computing MUSs and MAWs in a sliding window over a given string T . We first show how the set of MUSs can change in a sliding window over T , and present an O(n log σ)-time and O(d)-space algorithm to compute MUSs in a sliding window of width d over T , where σ is the maximum number of distinct characters in every window. We then give tight upper and lower bounds on the maximum number of changes in the set of MAWs in a sliding window over T . Our bounds improve on the previous results in [Crochemore et al., 2017].

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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].

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