Sublinear-Time Algorithms for Computing & Embedding Gap Edit Distance
Sublinear-Time Algorithms for Computing & Embedding Gap Edit Distance
复制标题
用于计算的次线性时间算法
DOI：
10.1109/focs46700.2020.00112
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发表时间：
2020
期刊：
影响因子：
--
通讯作者：
B. Saha
中科院分区：
文献类型：
--
作者：
Tomasz Kociumaka;B. Saha
在本文中，我们设计了用于解决差距编辑距离问题的新sublinear-time算法，并将编辑距离嵌入到吊顶距离上。 \ Mathcal {o}}（\ frac {n} {k}+k^{2}）$ </tex>在最多<tex> $ k $ </tex>的长度n输入字符串之间用编辑距离大于<dex> $ 4K^{2} $ </tex>是对[Goldenberg，Krauthgamer，Saha，focs 2019]的主要结果的改进和简化，其中<dex> $ k $ </tex> vs vs <dex> $ \ theta（k^{2}）$ < /tex>差距编辑距离问题在<tex> $ \ tilde {\ mathcal {o}}}（\ frac {n} {k}+k^{3}）$ </tex>时，我们进一步推广我们的时间。解决<tex> $ k $ </tex> vs <dex> $ \ alpha k $ </tex>差距编辑距离问题的结果<tex> $ \ tilde {\ Mathcal {o}}（\ frac {n} {\ alpha}+k^{2}+\ frac {k} {\ alpha} {\ alpha} \ sqrt {nk} {nk}）$ </tex> ，严格改进先前已知的<dex> $ \ tilde {\ Mathcal {o}}（\ frac {n} {\ alpha}+k^{3}）$ </tex>。输入字符串没有很长的高周期子字符串，那么差距编辑距离问题可以在任何因素<tex> $ \ alpha>中以均匀时间解决1 $ </tex>。特别是，如果字符串不包含长度的子字符串<tex> $ \ ell $ </tex>，最短的长度最短<tex> $ 2K $ </tex>，则是< tex> $ k $ </tex> vs <dex> $（1+ \ varepsilon）k $ </tex>差距编辑距离问题可以在时间<tex> $ \ tilde {\ mathcal {o}}（\） frac {n} {\ varepsilon^{2} k}+k^{2} \ ell）$ </tex>。到锤距离的距离。 } p $ </tex>，其中<tex> $ k $ </tex>是原始字符串的编辑距离，最终字符串的锤子距离在<tex> $ \ frac之间} {p} $ </tex>和<dex> $ k^{2} $ </tex>概率很大。 [Chakraborty，Goldenberg，Koucký，Stoc 2016]，其中最终的锤击距离为<dex> $ k $ </tex>和<dex> $ k^{2} $ </tex>。随机浏览样品，我们认为这些样品将在均方根时期算法中找到其他应用。
In this paper, we design new sublinear-time algorithms for solving the gap edit distance problem and for embedding edit distance to Hamming distance. For the gap edit distance problem, we give a greedy algorithm that distinguishes in time <tex>$\tilde{\mathcal{O}}(\frac{n}{k}+k^{2})$</tex> between length-n input strings with edit distance at most <tex>$k$</tex> and those with edit distance more than <tex>$4k^{2}$</tex>. This is an improvement and a simplification upon the main result of [Goldenberg, Krauthgamer, Saha, FOCS 2019], where the <tex>$k$</tex> vs <tex>$\Theta(k^{2})$</tex> gap edit distance problem is solved in <tex>$\tilde{\mathcal{O}}(\frac{n}{k}+k^{3})$</tex> time. We further generalize our result to solve the <tex>$k$</tex> vs <tex>$\alpha k$</tex> gap edit distance problem in time <tex>$\tilde{\mathcal{O}}(\frac{n}{\alpha}+k^{2}+\frac{k}{\alpha}\sqrt{nk})$</tex>, strictly improving upon the previously known bound <tex>$\tilde{\mathcal{O}}(\frac{n}{\alpha}+k^{3})$</tex>. Finally, we show that if the input strings do not have long highly periodic substrings, then the gap edit distance problem can be solved in sublinear time within any factor <tex>$\alpha > 1$</tex>. Specifically, if the strings contain no substring of length <tex>$\ell$</tex> with the shortest period of length at most <tex>$2k$</tex>, then the <tex>$k$</tex> vs <tex>$(1+\varepsilon)k$</tex> gap edit distance problem can be solved in time <tex>$\tilde{\mathcal{O}}(\frac{n}{\varepsilon^{2}k}+k^{2}\ell)$</tex>. We further give the first sublinear-time algorithm for the probabilistic embedding of edit distance to Hamming distance. Our <tex>$\tilde{\mathcal{O}}(\frac{n}{p})$</tex>-time procedure yields an embedding with distortion <tex>$k^{2}p$</tex>, where <tex>$k$</tex> is the edit distance of the original strings. Specifically, the Hamming distance of the resultant strings is between <tex>$\frac{k-p+1}{p}$</tex> and <tex>$k^{2}$</tex> with good probability. This generalizes the linear-time embedding of [Chakraborty, Goldenberg, Koucký, STOC 2016], where the resultant Hamming distance is between <tex>$k$</tex> and <tex>$k^{2}$</tex>. Our algorithm is based on a random walk over samples, which we believe will find other applications in sublinear-time algorithms.
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影响因子：
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作者：
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DOI：
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DOI：
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发表时间：
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期刊：
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影响因子：
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DOI：
--
发表时间：
2020
期刊：
STOC 2020
影响因子：
--
作者：
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通讯作者：
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