The Dynamic Time Warping (DTW) distance is a popular similarity measure for
polygonal curves (i.e., sequences of points). It finds many theoretical and
practical applications, especially for temporal data, and is known to be a
robust, outlier-insensitive alternative to the \frechet distance. For static
curves of at most n points, the DTW distance can be computed in O(n2) time
in constant dimension. This tightly matches a SETH-based lower bound, even for
curves in R1.
In this work, we study \emph{dynamic} algorithms for the DTW distance. Here,
the goal is to design a data structure that can be efficiently updated to
accommodate local changes to one or both curves, such as inserting or deleting
vertices and, after each operation, reports the updated DTW distance. We give
such a data structure with update and query time O(n1.5logn), where n
is the maximum length of the curves.
As our main result, we prove that our data structure is conditionally
\emph{optimal}, up to subpolynomial factors. More precisely, we prove that,
already for curves in R1, there is no dynamic algorithm to maintain
the DTW distance with update and query time~\makebox{O(n1.5−δ)} for
any constant δ>0, unless the Negative-k-Clique Hypothesis fails. In
fact, we give matching upper and lower bounds for various trade-offs between
update and query time, even in cases where the lengths of the curves differ.Comment: To appear at SODA2