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Approximating Text-to-Pattern Distance via Dimensionality Reduction

Text-to-pattern distance is a fundamental problem in string matching, where given a pattern of length $m$ and a text of length $n$, over an integer alphabet, we are asked to compute the distance between pattern and the text at every location. The distance function can be e.g. Hamming distance or $\ell_p$ distance for some parameter $p > 0$. Almost all state-of-the-art exact and approximate algorithms developed in the past $\sim 40$ years were using FFT as a black-box. In this work we present $\widetilde{O}(n/\varepsilon^2)$ time algorithms for $(1\pm\varepsilon)$-approximation of $\ell_2$ distances, and $\widetilde{O}(n/\varepsilon^3)$ algorithm for approximation of Hamming and $\ell_1$ distances, all without use of FFT. This is independent to the very recent development by Chan et al. [STOC 2020], where $O(n/\varepsilon^2)$ algorithm for Hamming distances not using FFT was presented -- although their algorithm is much more "combinatorial", our techniques apply to other norms than Hamming.