Absent Subsequences in Words

m-Nearly k-Universal Words -- Investigating Simon Congruence

“…These are words, where in comparison to Σ k , m words of length k are absent from the scattered factor set. A special case of these words was investigated in [21] where the shortest absent scattered factors of a word are determined. In the unary alphabet ε is the only word which has |Σ| k − 1 = 0 scattered factors and the notion is not well-defined for m > 1.…”

“…The latter one leads immediately to the index of the Simon congruence restricted to nearly k-universal words. Notice that for the first problem, a linear time algorithm is implicitly given in [21]: if w is a word of length n, the SAS tree can be constructed in time O(n) and in time O(k) the lexicographically smallest shortest absent scattered factors can be determined; if there is only one shortest absent scattered factor, we have w ∈ NUniv Σ,1,k . The following algorithm can only check whether a word is nearly k-universal but therefore does not need any additional data structures.…”

“…Implicitly, a subset of these words was investigated in [21]. There, the authors determine all shortest absent scattered factors, i.e.…”

“…For instance, the word aabb is 1-nearly 2-universal since ba is absent and aab is 2-nearly 2-universal since ba and bb are absent. A special subclass of m-nearly k-universal words has recently been studied from an algorithmic point of view in [21]. There the authors investigated shortest absent scattered factors of words, i.e., for a given (k − 1)-universal word w the set of words with length k that are not scattered factors of w. If this set has cardinality m, we obtain a subset of m-nearly k-universal words.…”

“…Afterwards, we give some first insights into mnearly k-universal words for m > 1. Our main result in this part is built on the algorithm in [21] by putting this algorithmic result into a combinatorial context, i.e. we are able to determine the number of absent scattered factors and giving the congruence classes w.r.t.…”