Matching Patterns with Variables Under Simon’s Congruence

Matching Patterns with Variables in Approximate Settings

“…At last, these problems were solved in optimal linear time in [17,76]. In recent research, an interesting algorithmic problem was tackled in the article [104], which is related to the problem of matching patterns with variables under Simon's congruence [65] ( Chapter 5 of this thesis). In that paper, the authors present an efficient solution for the following problem: given two strings w, u and a natural number k, decide whether there exists a factor w[i : j] with arbitrary i, j of w which is k-Simon congruent to u.…”

“…More precisely, one is interested in finding, for a given integer ℓ, a word w over an alphabet Σ, such that each word of length ℓ over Σ occurs exactly once as a contiguous factor of w. The De Bruijn sequences [52] fulfill this property and have many applications in computer science or combinatorics [45,82]. The notion of k-subsequence universal words is going to play an important role in proving a lower bound for matching patterns with variables under Simon's congruence in [65], as well as being the obvious central notion of the problem the edit distance to k-subsequence universality from [50] (Chapter 5 and 6 from this thesis).…”

“…In the upcoming chapters of this thesis, I am going to discuss and present our results on an extended setting of the matching problem for patterns with variables: given a pattern α and a word w, decide if there exists a substitution h such that h(α) is similar to w, with respect to a similarity measure (Hamming resp. edit distance in [78,79], and Simon's congruence [65], as well as a comparison to the original problem, string equality for (exact) Match). This modification to the problem is called (approximate) pattern matching.…”

“…The first problem involves finding a substitution h, such as h(α) ∼ k w. Alternatively, the problem of finding a substitution h to reach a target universality index k, i.e., ι(h(α)) = k is considered. Lastly, in the case of word equations, it is examined how to find a substitution h for two patterns α and β, such that h(α) ∼ k h(β) (published at the 17th International Conference on Reachability Problems) [65].…”

“…First of all, note that the core of this thesis are the Chapters 3, 4, 5 and 6, which are slightly modified versions, but essentially unchanged core content from our published papers [78,79,65,50], respectively, to all of which I contributed as a co-author. In most parts, I altered some of the notations and merged all their respective preliminaries and introductions into one section to provide a concise presentation of all those thematically well-connected results in one thesis.…”