25 Additional Problems -- Extension to the Book "125 Problems in Text Algorithms"

TL;DR

The paper introduces the notion of subsequence s-covers, defining when a word $x$ of length $m$ can cover a longer word $y$ of length $n$ via subsequences. It presents a linear-time decision algorithm, $\text{s-Covers-Cover}$, built from left/right subsequence position lists $\mathbf{L},\mathbf{R}$, prefix/suffix analyses ($LEFT$, $RIGHT$), a forward table $F$ that yields $P$, and a predicate $\Psi(x,y)$ that certifies coverage. Key theoretical results include Observation 1, which localizes coverage to a stratified interplay of $L$, $R$, and character matches, and Observation 2, which reduces coverage to $\Psi(x,y)$; the algorithm thereby runs in $O(n)$ time given $x$ and $y$. The notes also discuss gaps with standard covers, including that the shortest s-cover length is NP-hard to optimize, and summarize bounds $\gamma(k)$ for fixed alphabet size $k$, e.g., $\gamma(3)=8$, $\gamma(4)=19$, with $\gamma(5)$ unknown. Placed in the broader context of extending the book "125 Problems in Text Algorithms" and related works in Algorithmic Stringology, the work contributes a concise, actionable linear-time solution for a nontrivial subsequence-cover problem and highlights interesting complexity boundaries for synthesis of covers.

Abstract

This very preliminary text is related to ``Algorithms on Texts'', also called ``Algorithmic Stringology''. It is an extension of the book ``125 Problems in Text Algorithms'' providing, in the same compact style, more problems with solutions. We refer also to the companions to ``Text algorithms'' available at http://monge.univ-mlv.fr/~mac/CLR/clr1-20.pdf and at the web page http://125-problems.univ-mlv.fr, where all 150 problems (including the ones presented here) are briefly announced. The selected problems satisfy three criteria: challenging, having short tricky solutions and solvable with only very basic background in stringology. For the basics in stringology we refer to http://monge.univ-mlv.fr/~mac/CLR/clr1-20.pdf.