Subsequences in Bounded Ranges: Matching and Analysis Problems

TL;DR

This paper investigates the problem of finding subsequences within bounded ranges of words, establishing complexity bounds, analyzing related algorithmic problems, and demonstrating applications to circular words.

Contribution

It introduces the bounded-range subsequence problem, provides tight bounds, explores the complexity of related problems, and applies findings to circular word subsequence matching.

Findings

Matching bounds for bounded-range subsequence checking.

Most related problems are intractable in this setting.

Application to circular words subsequence matching.

Abstract

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