3 research outputs found
Dynamic Data Structures for Parameterized String Problems
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 and
, respectively, where is the alphabet and
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
, where is the parameter in question and 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 and ,
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
for Disjoint Factors and Edit Distance is unlikely already
for a constant value of the parameter .Comment: 28 page