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• W3169712904 abstract "In this work, we revisit the fundamental and well-studied problem of approximate pattern matching under edit distance. Given an integer $k$, a pattern $P$ of length $m$, and a text $T$ of length $n ge m$, the task is to find substrings of $T$ that are within edit distance $k$ from $P$. Our main result is a streaming algorithm that solves the problem in $tilde{O}(k^5)$ space and $tilde{O}(k^8)$ amortised time per character of the text, providing answers correct with high probability. (Hereafter, $tilde{O}(cdot)$ hides a $mathrm{poly}(log n)$ factor.) This answers a decade-old question: since the discovery of a $mathrm{poly}(klog n)$-space streaming algorithm for pattern matching under Hamming distance by Porat and Porat [FOCS 2009], the existence of an analogous result for edit distance remained open. Up to this work, no $mathrm{poly}(klog n)$-space algorithm was known even in the simpler semi-streaming model, where $T$ comes as a stream but $P$ is available for read-only access. In this model, we give a deterministic algorithm that achieves slightly better complexity. In order to develop the fully streaming algorithm, we introduce a new edit distance sketch parametrised by integers $nge k$. For any string of length at most $n$, the sketch is of size $tilde{O}(k^2)$ and it can be computed with an $tilde{O}(k^2)$-space streaming algorithm. Given the sketches of two strings, in $tilde{O}(k^3)$ time we can compute their edit distance or certify that it is larger than $k$. This result improves upon $tilde{O}(k^8)$-size sketches of Belazzougui and Zhu [FOCS 2016] and very recent $tilde{O}(k^3)$-size sketches of Jin, Nelson, and Wu [STACS 2021]." @default.
• W3169712904 created "2021-06-22" @default.
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• W3169712904 date "2021-06-10" @default.
• W3169712904 modified "2023-09-27" @default.
• W3169712904 title "Small space and streaming pattern matching with k edits." @default.
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