Counting Permutations Modulo Pattern-Replacement Equivalences for Three-Letter Patterns

We study a family of equivalence relations on $S_n$, the group of permutations on $n$ letters, created in a manner similar to that of the Knuth relation and the forgotten relation. For our purposes, two permutations are in the same equivalence class if one can be reached from the other through a series of pattern-replacements using patterns whose order permutations are in the same part of a predetermined partition of $S_c$. When the partition is of $S_3$ and has one nontrivial part and that part is of size greater than two, we provide formulas for the number of classes created in each previously unsolved case. When the partition is of $S_3$ and has two nontrivial parts, each of size two (as do the Knuth and forgotten relations), we enumerate the classes for $13$ of the $14$ unresolved cases. In two of these cases, enumerations arise which are the same as those yielded by the Knuth and forgotten relations. The reasons for this phenomenon are still largely a mystery

Dynamic Time Warping in Strongly Subquadratic Time: Algorithms for the Low-Distance Regime and Approximate Evaluation

Dynamic time warping distance (DTW) is a widely used distance measure between time series. The best known algorithms for computing DTW run in near quadratic time, and conditional lower bounds prohibit the existence of significantly faster algorithms. The lower bounds do not prevent a faster algorithm for the special case in which the DTW is small, however. For an arbitrary metric space $\Sigma$ with distances normalized so that the smallest non-zero distance is one, we present an algorithm which computes $\operatorname{dtw}(x, y)$ for two strings $x$ and $y$ over $\Sigma$ in time $O(n \cdot \operatorname{dtw}(x, y))$. We also present an approximation algorithm which computes $\operatorname{dtw}(x, y)$ within a factor of $O(n^\epsilon)$ in time $\tilde{O}(n^{2 - \epsilon})$ for $0 < \epsilon < 1$. The algorithm allows for the strings $x$ and $y$ to be taken over an arbitrary well-separated tree metric with logarithmic depth and at most exponential aspect ratio. Extending our techniques further, we also obtain the first approximation algorithm for edit distance to work with characters taken from an arbitrary metric space, providing an $n^\epsilon$-approximation in time $\tilde{O}(n^{2 - \epsilon})$, with high probability. Additionally, we present a simple reduction from computing edit distance to computing DTW. Applying our reduction to a conditional lower bound of Bringmann and K\"unnemann pertaining to edit distance over $\{0, 1\}$, we obtain a conditional lower bound for computing DTW over a three letter alphabet (with distances of zero and one). This improves on a previous result of Abboud, Backurs, and Williams. With a similar approach, we prove a reduction from computing edit distance to computing longest LCS length. This means that one can recover conditional lower bounds for LCS directly from those for edit distance, which was not previously thought to be the case