Faster Approximate(d) Text-to-Pattern L1 Distance

The problem of finding \emph{distance} between \emph{pattern} of length $m$ and \emph{text} of length $n$ is a typical way of generalizing pattern matching to incorporate dissimilarity score. For both Hamming and $L_1$ distances only a super linear upper bound $\widetilde{O}(n\sqrt{m})$ are known, which prompts the question of relaxing the problem: either by asking for $1 \pm \varepsilon$ approximate distance (every distance is reported up to a multiplicative factor), or $k$-approximated distance (distances exceeding $k$ are reported as $\infty$). We focus on $L_1$ distance, for which we show new algorithms achieving complexities respectively $\widetilde{O}(\varepsilon^{-1} n)$ and $\widetilde{O}((m+k\sqrt{m}) \cdot n/m)$. This is a significant improvement upon previous algorithms with runtime $\widetilde{O}(\varepsilon^{-2} n)$ of Lipsky and Porat (Algorithmica 2011) and $\widetilde{O}(n\sqrt{k})$ of Amir, Lipsky, Porat and Umanski (CPM 2005). We also provide a series of reductions, showing that if our upper bound for approximate $L_1$ distance is tight, then so is our upper bound for $k$-approximated $L_1$ distance, and if the latter is tight then so is $k$-approximated Hamming distance upper bound due to the result of Gawrychowski and Uzna\'nski (arXiv 2017).