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

TL;DR

This paper introduces faster algorithms for approximate and k-approximated text-to-pattern L1 distance calculations, significantly improving upon previous methods in terms of computational complexity.

Contribution

The authors develop new algorithms that reduce the runtime for approximate L1 distance computation between text and pattern, surpassing prior approaches.

Findings

Achieved $ ilde{O}(rac{1}{ ext{epsilon}}) n$ complexity for $(1 ext{±} ext{epsilon})$-approximate distance.

Achieved $ ilde{O}((m + k\sqrt{m}) rac{n}{m})$ complexity for k-approximated distance.

Significant runtime improvements over previous algorithms by Lipsky and Porat, and Amir et al.

Abstract

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].