用于编辑距离的局部敏感桶函数
Locality-sensitive bucketing functions for the edit distance
摘要：
许多生物信息学应用涉及将一组序列分桶，其中允许将每个序列分配到多个桶中。为了实现高灵敏度和精确度，需要桶化方法将相似序列分配到相同桶中，同时将不相似序列分配到不同桶中。现有的基于$k$ -mer的桶化方法在处理具有低出错率的测序数据中是有效的，但是遇到对具有高出错率的数据的灵敏度大大降低。局部敏感哈希（LSH）方案能够通过容忍类似序列中的编辑来缓解这个问题，但最先进的方法仍然存在很大的差距。在这里，我们通过允许它将一个序列散列到多个桶中来概括LSH函数。形式上，如果在$d_1$的编辑距离内的任何两个序列被映射到至少一个共享桶中，并且距离至少为$d_2$的任何两个序列被映射到不相交的桶子集中，则将（固定长度的）序列映射到桶子集中的分桶函数被定义为是$(d_1, d_2)$敏感的。我们构造了具有各种$(d_1,d_2)$值的位置敏感的桶（LSB）函数，并分析了它们相对于所需桶的总数以及特定序列映射到的桶的数量的效率。我们还证明了这两个参数在不同的设置下的下界，并表明我们构造的LSB函数是最优的。这些结果提供了理论基础，为他们的实际应用中分析序列与高出错率，同时也提供了见解的难度设计无缺口LSH功能。
英文摘要：
Many bioinformatics applications involve bucketing a set of sequences where each sequence is allowed to be assigned into multiple buckets. To achieve both high sensitivity and precision, bucketing methods are desired to assign similar sequences into the same bucket while assigning dissimilar sequences into distinct buckets. Existing $k$-mer-based bucketing methods have been efficient in processing sequencing data with low error rate, but encounter much reduced sensitivity on data with high error rate. Locality-sensitive hashing (LSH) schemes are able to mitigate this issue through tolerating the edits in similar sequences, but state-of-the-art methods still have large gaps. Here we generalize the LSH function by allowing it to hash one sequence into multiple buckets. Formally, a bucketing function, which maps a sequence (of fixed length) into a subset of buckets, is defined to be $(d_1, d_2)$-sensitive if any two sequences within an edit distance of $d_1$ are mapped into at least one shared bucket, and any two sequences with distance at least $d_2$ are mapped into disjoint subsets of buckets. We construct locality-sensitive bucketing (LSB) functions with a variety of values of $(d_1,d_2)$ and analyze their efficiency with respect to the total number of buckets needed as well as the number of buckets that a specific sequence is mapped to. We also prove lower bounds of these two parameters in different settings and show that some of our constructed LSB functions are optimal. These results provide theoretical foundations for their practical use in analyzing sequences with high error rate while also providing insights for the hardness of designing ungapped LSH functions.
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