Dynamic data structures for parameterized string problems

TL;DR

This paper develops dynamic data structures for classic parameterized string problems, enabling efficient updates and reporting solutions or their absence, with new algorithms and complexity bounds for problems like Closest String, Disjoint Factors, and Edit Distance.

Contribution

It introduces randomized dynamic data structures for Closest String and applies a meta-theorem to create efficient structures for Disjoint Factors and Edit Distance, along with lower bound insights.

Findings

Dynamic data structures for Closest String with amortized $d^{O(d)}$ and $|A|^{O(d)}$ update times.

Meta-theorem providing $O(\u2212 ext{log} ext{log} n)$ worst-case update time for parameterized string problems.

Explicit data structures for Disjoint Factors and Edit Distance with $O(k2^{k} ext{log} ext{log} n)$ and $O(k^2 ext{log} ext{log} n)$ update times.

Abstract

We revisit classic string problems considered in the area of parameterized complexity, and study them through the lens of dynamic data structures. That is, instead of asking for a static algorithm that solves the given instance efficiently, our goal is to design a data structure that efficiently maintains a solution, or reports a lack thereof, upon updates in the instance. We first consider the Closest String problem, for which we design randomized dynamic data structures with amortized update times $d^{\mathcal{O}(d)}$ and $|\Sigma|^{\mathcal{O}(d)}$, respectively, where $\Sigma$ is the alphabet and $d$ is the assumed bound on the maximum distance. These are obtained by combining known static approaches to Closest String with color-coding. Next, we note that from a result of Frandsen et al.~[J. ACM'97] one can easily infer a meta-theorem that provides dynamic data structures for parameterized string problems with worst-case update time of the form $\mathcal{O}(\log \log n)$, where $k$ is the parameter in question and $n$ is the length of the string. We showcase the utility of this meta-theorem by giving such data structures for problems Disjoint Factors and Edit Distance. We also give explicit data structures for these problems, with worst-case update times $\mathcal{O}(k2^{k}\log \log n)$ and $\mathcal{O}(k^2\log \log n)$, respectively. Finally, we discuss how a lower bound methodology introduced by Amarilli et al.~[ICALP'21] can be used to show that obtaining update time $\mathcal{O}(f(k))$ for Disjoint Factors and Edit Distance is unlikely already for a constant value of the parameter $k$.