Quadratic-Time Hardness of LCS and other Sequence Similarity Measures

Two important similarity measures between sequences are the longest common subsequence (LCS) and the dynamic time warping distance (DTWD). The computations of these measures for two given sequences are central tasks in a variety of applications. Simple dynamic programming algorithms solve these tasks in $O(n^2)$ time, and despite an extensive amount of research, no algorithms with significantly better worst case upper bounds are known. In this paper, we show that an $O(n^{2-\epsilon})$ time algorithm, for some $\epsilon>0$, for computing the LCS or the DTWD of two sequences of length $n$ over a constant size alphabet, refutes the popular Strong Exponential Time Hypothesis (SETH). Moreover, we show that computing the LCS of $k$ strings over an alphabet of size $O(k)$ cannot be done in $O(n^{k-\epsilon})$ time, for any $\epsilon>0$, under SETH. Finally, we also address the time complexity of approximating the DTWD of two strings in truly subquadratic time.