Elastic-Degenerate String Matching via Fast Matrix Multiplication
نویسندگان
چکیده
An elastic-degenerate (ED) string is a sequence of $n$ sets strings total length $N$ which was recently proposed to model set similar sequences. The ED matching (EDSM) problem find all occurrences pattern $m$ in an text. EDSM has received some attention the combinatorial community, and $\mathcal{O}(nm^{1.5}\sqrt{\log m} + N)$-time algorithm known [Aoyama et al., CPM 2018]. standard assumption prior work on this question that substantially larger than both $m$, thus we would like have linear dependency former. Under assumption, natural open whether can decrease 1.5 exponent time complexity, similarly as related (but, best our knowledge, not equivalent) word break [Backurs Indyk, FOCS 2016]. Our starting point conditional lower bound for problem. We use popular Boolean matrix multiplication (BMM) conjecture stating there no truly subcubic BMM [Abboud Williams, 2014]. By designing appropriate reduction, show solving $\mathcal{O}(nm^{1.5-\epsilon} N)$ time, any $\epsilon>0$, refutes conjecture. reduction should be understood indication decreasing requires fast multiplication. String periodicity Fourier transform are two tools algorithms. main technical contribution successfully combine these with design noncombinatorial $\tilde{\mathcal{O}}(nm^{\omega-1}+N)$-time EDSM, where $\omega$ denotes $\tilde{\mathcal{O}}(\cdot)$ notation suppresses polylog factors. To first tools. In particular, using fact $\omega<2.373$ [Alman SODA 2021; Le Gall, ISSAC 2014; STOC 2012], obtain $\mathcal{O}(nm^{1.373} EDSM. important building block solution might applications other problems method selecting small length-$\ell$ substrings pattern, called anchors, so occurrence from text contains at least one but too many (on average) such anchors inside.
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ژورنال
عنوان ژورنال: SIAM Journal on Computing
سال: 2022
ISSN: ['1095-7111', '0097-5397']
DOI: https://doi.org/10.1137/20m1368033