Hardness Results on Characteristics for Elastic-Degenerated Strings

Abstract

Generalizations of plain strings have been proposed as a compact way to represent a collection of nearly identical sequences or to express uncertainty at specific text positions by enumerating all possibilities. While a plain string stores a character at each of its positions, generalizations consider a set of characters (indeterminate strings), a set of strings of equal length (generalized degenerate strings, or shortly GD strings), or a set of strings of arbitrary lengths (elastic-degenerate strings, or shortly ED strings). These generalizations are of importance to compactly represent such type of data, and find applications in bioinformatics for representing and maintaining a set of genetic sequences of the same taxonomy or a multiple sequence alignment. To be of use, attention has been drawn to answering various query types such as pattern matching or measuring similarity of ED strings by generalizing techniques known to plain strings. However, for some types of queries, it has been shown that a generalization of a polynomial-time solvable query on classic strings becomes NP-hard on ED strings, e.g. [Russo et al.,2022]. In that light, we wonder about other types of queries, which are of particular interest to bioinformatics: the search for the longest repeating factor, unique substrings, absent words, anti-powers, and longest previous factors. While we obtain a polynomial time algorithm for the first problem on ED strings, we show that all others are NP-hard to compute, some of them even under the restriction that the input can be modelled as an indeterminate or GD string.