Title

Quadratic Conditional Lower Bounds for String Problems and Dynamic Time Warping

Abstract

We study similarity measures on strings, such as longest common subsequence and edit distance, and on curves, such as Fréchet distance and dynamic time warping. All of these measures have simple quadratic time dynamic programs (at least the decision versions). Recently quadratic-time lower bounds based on the Strong Exponential Time Hypothesis have been shown for Fréchet distance and edit distance. We extend these lower bounds by building a framework for proving quadratic-time hardness of similarity measures. We use this framework to obtain quadratic-time lower bounds for longest common subsequence (on binary strings) and dynamic time warping (on time-series). Moreover, we improve the hardness of edit distance by proving it on binary strings and for any non-trivial costs of the edit distance operations.