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.

/pdf/combinatorial-algorithms-for-subsequence-matching-a-survey-34jkz1sl.pdf

Combinatorial Algorithms for Subsequence Matching: A Survey

Using arch-jumping functions and properties of the arch factorization of words, we propose a new algorithm for computing the subword circular universality index of words. We also introduce the subword universality signature for words, that leads to simple algorithms for the universality indexes of SLP-compressed words.

On arch factorization and subword universality for words and compressed words

We consider the longest common subsequence problem in the context of subsequences with gap constraints. In particular, following Day et al. 2022, we consider the setting when the distance (i. e., the gap) between two consecutive symbols of the subsequence has to be between a lower and an upper bound (which may depend on the position of those symbols in the subsequence or on the symbols bordering the gap) as well as the case where the entire subsequence is found in a bounded range (defined by a single upper bound), considered by Kosche et al. 2022. In all these cases, we present effcient algorithms for determining the length of the longest common constrained subsequence between two given strings.

pdf/longest-common-subsequence-with-gap-constraints-1mfioelr.pdf

Longest Common Subsequence with Gap Constraints

A query model for sequence data was introduced in [11] in the form of subsequence-queries with wildcards and gap-size constraints (swg-queries, for short). These queries consist of a pattern over an alphabet of variables and types, as well as a global window size and a number of local gap-size constraints. We propose two new extensions of swg-queries, which both enrich the expressive power of swg-queries in different ways: subsequence-queries with generalised gap-size constraints (swgg-queries, for short) and disjunctive subsequence-queries (dswg-queries, for short). We discuss a suitable characterisation of containment, a classical property considered in database theory, and adapt results concerning the discovery of swg- queries to both, swgg-queries and dswg-queries.

Puzzling over Subsequence-Query Extensions: Disjunction and Generalised Gaps

For two strings u, v over some alphabet A, we investigate the problem of embedding u into w as a subsequence under the presence of generalised gap constraints. A generalised gap constraint is a triple (i, j, C_{i, j}), where 1<= i<j<= |u| and C_{i, j} is a subset of A^*. Embedding u as a subsequence into v such that (i, j, C_{i, j}) is satisfied means that if u[i] and u[j] are mapped to v[k] and v[l], respectively, then the induced gap v[k + 1..l - 1] must be a string from C_{i, j}. This generalises the setting recently investigated in [Day et al., ISAAC 2022], where only gap constraints of the form C_{i, i + 1} are considered, as well as the setting from [Kosche et al., RP 2022], where only gap constraints of the form C_{1, |u|} are considered. We show that subsequence matching under generalised gap constraints is NP-hard, and we complement this general lower bound with a thorough (parameterised) complexity analysis. Moreover, we identify several efficiently solvable subclasses that result from restricting the interval structure induced by the generalised gap constraints.

Subsequences With Generalised Gap Constraints: Upper and Lower Complexity Bounds

. 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. Fi-nally, we provide an example of how some of our results can be applied to subsequence matching problems for circular

Subsequences in Bounded Ranges: Matching and Analysis Problems

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Subsequences in Bounded Ranges: Matching and Analysis Problems