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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 matchingto incorporate dissimilarity score. For both Hamming and $L_1$ distances only asuper linear upper bound $\widetilde{O}(n\sqrt{m})$ are known, which promptsthe question of relaxing the problem: either by asking for $(1 \pm\varepsilon)$ approximate distance (every distance is reported up to amultiplicative factor), or $k$-approximated distance (distances exceeding $k$are reported as $\infty$). We focus on $L_1$ distance, for which we show newalgorithms achieving complexities respectively $\widetilde{O}(\varepsilon^{-1}n)$ and $\widetilde{O}((m+k\sqrt{m}) \cdot n/m)$. This is a significantimprovement 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].

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