基于字母重排的BWT游程最小化算法的复杂性
On the Complexity of BWT-runs Minimization via Alphabet Reordering
摘要：
Burrows-Wheeler变换（BWT）是文本压缩和索引的重要工具。它于1994年首次引入，为经典后缀树数据结构在接近基于熵的下界的空间中的第一个编码提供了主干。最近，已经有紧凑的后缀树在空间中成比例的“$r$“，在BWT中运行的数量，以及$r$的外观在新算法的时间复杂度的发展。与其他流行的压缩度量方法不同，参数$r$对文本字母表的词典顺序很敏感。尽管过去有几次尝试利用这一点，但多年来一直存在一种可证明有效的算法，用于找到或近似最小化$r$的字母顺序。我们提出的第一组结果的计算复杂性最小化BWT运行通过字母表重排。我们证明了这个问题的决策版本是NP完全的，不能解决的时间$2^{o(σ+ \sqrt{n})}$，除非指数时间假设失败，其中$σ$是字母表的大小和$n$是文本的长度。我们还表明，优化问题是APX困难的。在这样做的时候，我们涉及到两个以前不同的主题：最佳的旅行销售人员的路径和运行的BWT的文本的数量，提供了一个令人惊讶的图形和文本压缩的问题之间的连接。此外，有关最近的结果在字典压缩领域，我们说明，任意字母顺序提供了一个$O(\log^2 n)$ -近似。我们提供了一个最佳的线性时间算法的问题，找到一个运行最小化排序的符号（只发生一次）的子集的顺序约束下，并证明了这个问题的推广到一类图的BWT一样的属性称为惠勒图是NP-完全的。
英文摘要：
The Burrows-Wheeler Transform (BWT) has been an essential tool in text compression and indexing. First introduced in 1994, it went on to provide the backbone for the first encoding of the classic suffix tree data structure in space close to the entropy-based lower bound. Recently, there has been the development of compact suffix trees in space proportional to "$r$", the number of runs in the BWT, as well as the appearance of $r$ in the time complexity of new algorithms. Unlike other popular measures of compression, the parameter $r$ is sensitive to the lexicographic ordering given to the text's alphabet. Despite several past attempts to exploit this, a provably efficient algorithm for finding, or approximating, an alphabet ordering which minimizes $r$ has been open for years. We present the first set of results on the computational complexity of minimizing BWT-runs via alphabet reordering. We prove that the decision version of this problem is NP-complete and cannot be solved in time $2^{o(σ+ \sqrt{n})}$ unless the Exponential Time Hypothesis fails, where $σ$ is the size of the alphabet and $n$ is the length of the text. We also show that the optimization problem is APX-hard. In doing so, we relate two previously disparate topics: the optimal traveling salesperson path and the number of runs in the BWT of a text, providing a surprising connection between problems on graphs and text compression. Also, by relating recent results in the field of dictionary compression, we illustrate that an arbitrary alphabet ordering provides a $O(\log^2 n)$-approximation. We provide an optimal linear-time algorithm for the problem of finding a run minimizing ordering on a subset of symbols (occurring only once) under ordering constraints, and prove a generalization of this problem to a class of graphs with BWT like properties called Wheeler graphs is NP-complete.
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