Absent Subsequences in Words
This work provides foundational combinatorial and algorithmic results for absent subsequences, a new notion in stringology, but the results are primarily theoretical and incremental within the field.
The paper extends the concept of absent factors to absent subsequences in strings, defining minimal and shortest absent subsequences. It provides combinatorial characterizations, compact representations, and efficient algorithms for testing and computing these subsequences, including a data structure for shortest absent subsequence queries.
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 subsequence-queries for the factors of a given string can be efficiently computed.