Vigemers: 关于共享相同基于异或的最小化器的 $k$-mers 的数量
Vigemers: on the number of $k$-mers sharing the same XOR-based minimizer
AI总结：
本文研究了基于异或运算的最小化器，提出了一种计算 $k$-mers 分区质量的动态规划方法。
摘要：
在生物信息学中，最小化器已成为处理从DNA或RNA测序中提取的固定大小 $k$-mers（即长度为 $k$ 的单词）的不可避免方法，无论是在采样、存储、查询还是分区中。根据某些固定的顺序对 $m$-mers（$m < k$）进行排序，$k$-mers 的最小化器被定义为其最小的 $m$-mer，并用作其指纹。尽管最小化器被广泛用于分区目的，但几乎没有任何理论工作探讨由此产生的分区质量。例如，已有数十年研究表明，字典序在实践中会导致高度不平衡的分区，无法使用，但直到最近才在理论上得到了证实。字典序的拒绝促使社区转向使用哈希函数的（伪）随机顺序。在本工作中，我们扩展了与字典序所得分区相关的理论结果，从字典序转向一个（指数级）大的哈希函数家族，即其中的 $m$-mers 与固定密钥进行异或运算。更具体地说，给定一个密钥 $γ$ 和一个 $m$-mer $w$，我们研究一个函数，该函数计算有多少 $k$-mers 以 $w$ 作为其最小化器（即其中 $w\oplusγ$ 是所述 $k$-mers 中所有 $m$-mers 的最小值）。这个数字，记为 $π_k^γ(w)$，代表如果所有可能的 $k$-mers 都被看到并分区时，与 $w$ 关联的桶的最大大小。我们适应文献中字典序的方法，提出了允许通过动态规划计算 $π_k^γ(w)$ 的组合方程，时间复杂度为 $O(km^2)$，空间复杂度为 $O(km)$。
英文摘要：
In bioinformatics, minimizers have become an inescapable method for handling $k$-mers (words of fixed size $k$) extracted from DNA or RNA sequencing, whether for sampling, storage, querying or partitioning. According to some fixed order on $m$-mers ($m<k$), the minimizer of a $k$-mer is defined as its smallest $m$-mer -- and acts as its fingerprint. Although minimizers are widely used for partitioning purposes, there is almost no theoretical work on the quality of the resulting partitions. For instance, it has been known for decades that the lexicographic order empirically leads to highly unbalanced partitions that are unusable in practice, but it was not until very recently that this observation was theoretically substantiated. The rejection of the lexicographic order has led the community to resort to (pseudo-)random orders using hash functions. In this work, we extend the theoretical results relating to the partitions obtained by the lexicographical order, departing from it to a (exponentially) large family of hash functions, namely where the $m$-mers are XORed against a fixed key. More precisely, provided a key $γ$ and a $m$-mer $w$, we investigate the function that counts how many $k$-mers admit $w$ as their minimizer (i.e. where $w\oplusγ$ is minimal among all $m$-mers of said $k$-mers). This number, denoted by $π_k^γ(w)$, represents the maximum size of the bucket associated with $w$, if all possible $k$-mers were to be seen and partitioned. We adapt the (lexicographical order) method of the literature to our framework and propose combinatorial equations that allow to compute, using dynamic programming, $π_k^γ(w)$ in $O(km^2)$ time and $O(km)$ space.
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