Zusammenfassung
We revisit the classic combinatorial pattern matching problem of finding a
longest common subsequence (LCS). For strings $x$ and $y$ of length $n$, a
textbook algorithm solves LCS in time $O(n^2)$, but although much effort has
been spent, no $O(n^{2-\varepsilon})$-time algorithm is known. Recent work
indeed shows that such an algorithm would refute the Strong Exponential Time
Hypothesis (SETH) [Abboud, Backurs, Vassilevska Williams + Bringmann,
K\"unnemann FOCS'15].
Despite the quadratic-time barrier, for over 40 years an enduring scientific
interest continued to produce fast algorithms for LCS and its variations.
Particular attention was put into identifying and exploiting input parameters
that yield strongly subquadratic time algorithms for special cases of interest,
e.g., differential file comparison. This line of research was successfully
pursued until 1990, at which time significant improvements came to a halt. In
this paper, using the lens of fine-grained complexity, our goal is to (1)
justify the lack of further improvements and (2) determine whether some special
cases of LCS admit faster algorithms than currently known.
To this end, we provide a systematic study of the multivariate complexity of
LCS, taking into account all parameters previously discussed in the literature:
the input size $n:=\max\{|x|,|y|\}$, the length of the shorter string
$m:=\min\{|x|,|y|\}$, the length $L$ of an LCS of $x$ and $y$, the numbers of
deletions $\delta := m-L$ and $\Delta := n-L$, the alphabet size, as well as
the numbers of matching pairs $M$ and dominant pairs $d$. For any class of
instances defined by fixing each parameter individually to a polynomial in
terms of the input size, we prove a SETH-based lower bound matching one of
three known algorithms. Specifically, we determine the optimal running time for
LCS under SETH as $(n+\min\{d, \delta \Delta, \delta m\})^{1\pm o(1)}$.
[...]
