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In this paper, we consider a variant of the classical algorithmic problem of checking whether a given word v is a subsequence of another word w. More precisely, we consider the problem of deciding, given a number p (defining a range-bound) and two words v and w, whether there exists a factor w[i : i + p − 1] (or, in other words, a range of length p) of w having v as subsequence (i. e., v occurs as a subsequence in the bounded range w[i : i + p − 1]). We give matching upper and lower quadratic bounds for the time complexity of this problem. Further, we consider a series of algorithmic problems in this setting, in which, for given integers k, p and a word w, we analyse the set p-Subseq k (w) of all words of length k which occur as subsequence of some factor of length p of w. Among these, we consider the k-universality problem, the k-equivalence problem, as well as problems related to absent subsequences. Surprisingly, unlike the case of the classical model of subsequences in words where such problems have efficient solutions in general, we show that most of these problems become intractable in the new setting when subsequences in bounded ranges are considered. Finally, we provide an example of how some of our results can be applied to subsequence matching problems for circular words.

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In this paper we provide an overview of a series of recent results regarding algorithms for searching for subsequences in words or for the analysis of the sets of subsequences occurring in a word.

An absent factor of a string w is a string u which does not occur as a contiguous substring (a.k.a. factor) inside w. We extend this well-studied notion and define absent subsequences: a string u is an absent subsequence of a string w if u does not occur as subsequence (a.k.a. scattered factor) inside w. Of particular interest to us are minimal absent subsequences, i.e., absent subsequences whose every subsequence is not absent, and shortest absent subsequences, i.e., absent subsequences of minimal length. We show a series of combinatorial and algorithmic results regarding these two notions. For instance: we give combinatorial characterisations of the sets of minimal and, respectively, shortest absent subsequences in a word, as well as compact representations of these sets; we show how we can test efficiently if a string is a shortest or minimal absent subsequence in a word, and we give efficient algorithms computing the lexicographically smallest absent subsequence of each kind; also, we show how a data structure for answering shortest absent subsequencequeries for the factors of a given string can be efficiently computed.