The Burrows-Wheeler Transform (BWT) of a string is an invertible permutation of the string, which can be used for data compression and compact indexes for string pattern matching. Ganguly et al. [SODA, 2017] introduced the parameterized BWT (pBWT) to design compact indexes for parameterized matching (p-matching), a variant of string pattern matching with parameter symbols introduced by Baker [STOC, 1993]. Although the pBWT was inspired by the BWT, it is not obvious whether the pBWT itself is invertible or not. In this paper we show that we can retrieve the original string (up to renaming of parameter symbols) from the pBWT of length $n$ in $O(n^2)$ time and $O(n)$ space.
A weight based dynamic compression method has recently been proposed, which is especially suitable for the encoding of files with locally skewed distributions. Its main idea is to assign larger weights to closer to be encoded symbols by means of an increasing weight function, rather than considering each position in the text evenly. A well known transformation that tends to convert input files into files with a more skewed distribution is the Burrows-Wheeler Transform. This paper employs the weighted approach on Burrows-Wheeler transformed files and provides empirical evidence of the efficienc…
The Burrows-Wheeler transform (BWT) is a permutation whose applications are prevalent in data compression and text indexing. The bijective BWT (BBWT) is a bijective variant of it. Although it is known that the BWT can be constructed in linear time for integer alphabets by using a linear time suffix array construction algorithm, it was up to now only conjectured that the BBWT can also be constructed in linear time. We confirm this conjecture by proposing a construction algorithm that is based on SAIS, improving the best known result of $O(n \lg n /\lg \lg n)$ time to linear.
The Burrows-Wheeler Transform (BWT) is often taught in undergraduate courses on algorithmic bioinformatics, because it underlies the FM-index and thus important tools such as Bowtie and BWA. Its admirers consider the BWT a thing of beauty but, despite thousands of pages being written about it over nearly thirty years, to undergraduates seeing it for the first time it still often seems like magic. Some who persevere are later shown the Positional BWT (PBWT), which was published twenty years after the BWT. In this paper we argue that the PBWT should be taught {\em before} the BWT. We first use t…
Parameterized strings are a generalization of strings in that their characters are drawn from two different alphabets, where one is considered to be the alphabet of static characters and the other to be the alphabet of parameter characters. Two parameterized strings are a parameterized match if there is a bijection over all characters such that the bijection transforms one string to the other while keeping the static characters (i.e., it behaves as the identity on the static alphabet). Ganguly et al. [SODA 2017] proposed the parameterized Burrows--Wheeler transform (pBWT) as a variant of the B…
On Extensions of Maximal Repeats in Compressed Strings
本文给出了压缩字符串中最大重复和最大对的几个子集的上界，并给出了最大对与游程Burrows-Wheeler变换之间的一个未知关系。这个关系被用来获得Burrows-Wheeler猜想的一个不同的证明，该猜想最近被Kempa和Kociumaka在“Resolution of the Burrows-Wheeler Transform Conjecture”中证明。本文证明了一个具有$z$ LZ ~（77）-因子而不具有$q$次幂的串$S$在游程长度Burrows-Wheeler变换中至多有$73(\log_2 |S|)(z+2)^2$游程，并且$S$的紧致有向无环字图中的弧数由$18q(1+\log_q |S|)(z+2)^2$自上而下有界.
With the rapid growth of Next Generation Sequencing (NGS) technologies, large amounts of "omics" data are daily collected and need to be processed. Indexing and compressing large sequences datasets are some of the most important tasks in this context. Here we propose algorithms for the computation of Burrows Wheeler transform relying on Big Data technologies, i.e., Apache Spark and Hadoop. Our algorithms are the first ones that distribute the index computation and not only the input dataset, allowing to fully benefit of the available cloud resources.
Indexing sequence data is important in the context of Precision Medicine, where large amounts of ``omics'' data have to be daily collected and analyzed in order to categorize patients and identify the most effective therapies. Here we propose an algorithm for the computation of Burrows Wheeler transform relying on Big Data technologies, i.e., Apache Spark and Hadoop. Our approach is the first that distributes the index computation and not only the input dataset, allowing to fully benefit of the available cloud resources.
We characterize those strings whose suffix arrays are based on arithmetic progressions, in particular, arithmetically progressed permutations where all pairs of successive entries of the permutation have the same difference modulo the respective string length. We show that an arithmetically progressed permutation $P$ coincides with the suffix array of a unary, binary, or ternary string. We further analyze the conditions of a given $P$ under which we can find a uniquely defined string over either a binary or ternary alphabet having $P$ as its suffix array. For the binary case, we show its conne…
The Burrows-Wheeler Transform (BWT) is an invertible text transformation that permutes symbols of a text according to the lexicographical order of its suffixes. BWT is the main component of popular lossless compression programs (such as bzip2) as well as recent powerful compressed indexes (such as $r$-index [Gagie et al., J. ACM, 2020]), central in modern bioinformatics. The compression ratio of BWT is quantified by the number $r$ of equal-letter runs. Despite the practical significance of BWT, no non-trivial bound on the value of $r$ is known. This is in contrast to nearly all other known com…
The Burrows-Wheeler Transform is a string transformation that plays a fundamental role for the design of self-indexing compressed data structures. Over the years, researchers have successfully extended this transformation outside the domains of strings. However, efforts to find non-trivial alternatives of the original, now 25 years old, Burrows-Wheeler string transformation have met limited success. In this paper we bring new lymph to this area by introducing a whole new family of transformations that have all the myriad virtues of the BWT: they can be computed and inverted in linear time, the…
Burrows-Wheeler变换（BWT）是1994年为数据压缩引入的一种字变换。它已成为设计自索引数据结构的基本工具，在科学和工程的许多领域有着重要的应用。交替Burrows-Wheeler变换（ABWT）是最近在[Gessel et al. 2012]中引入的另一种变换，并在单词组合学领域进行了研究。它类似于BWT，除了它使用交替的字典顺序而不是通常的顺序。基于[Giancarlo et al. 2018]中的结果，我们已经证明BWT和ABWT是更大一类可逆变换的一部分，在这里我们提供了新变换ABWT的组合和算法研究。我们建立了一个深刻的类比BWT和ABWT证明他们是唯一的在上述类是秩可逆的，一个新的概念，保证有效的可逆性。此外，我们表明，向后搜索过程可以有效地推广到ABWT，这一结果意味着ABWT也可以作为一个基础，高效的压缩全文索引。最后，我们证明了ABWT可以通过使用差分覆盖后缀排序算法的组合来有效地计算[Kärkkäinen等人，2006]与线性时间算法，用于找到一个词的最小循环旋转相对于交替字典顺序。
It is known that the exact form of the Burrows-Wheeler-Transform (BWT) of a string collection depends, in most implementations, on the input order of the strings in the collection. Reordering strings of an input collection affects the number of equal-letter runs $r$, arguably the most important parameter of BWT-based data structures, such as the FM-index or the $r$-index. Bentley, Gibney, and Thankachan [ESA 2020] introduced a linear-time algorithm for computing the permutation of the input collection which yields the minimum number of runs of the resulting BWT. In this paper, we present the f…
The run-length compressed Burrows-Wheeler transform (RLBWT) used in conjunction with the backward search introduced in the FM index is the centerpiece of most compressed indexes working on highly-repetitive data sets like biological sequences. Compared to grammar indexes, the size of the RLBWT is often much bigger, but queries like counting the occurrences of long patterns can be done much faster than on any existing grammar index so far. In this paper, we combine the virtues of a grammar with the RLBWT by building the RLBWT on top of a special grammar based on induced suffix sorting. Our expe…
Indexing highly repetitive strings (i.e., strings with many repetitions) for fast queries has become a central research topic in string processing, because it has a wide variety of applications in bioinformatics and natural language processing. Although a substantial number of indexes for highly repetitive strings have been proposed thus far, developing compressed indexes that support various queries remains a challenge. The run-length Burrows-Wheeler transform (RLBWT) is a lossless data compression by a reversible permutation of an input string and run-length encoding, and it has received int…
Indexing very large collections of strings, such as those produced by the widespread next generation sequencing technologies, heavily relies on multistring generalization of the Burrows-Wheeler Transform (BWT): large requirements of in-memory approaches have stimulated recent developments on external memory algorithms. The related problem of computing the Longest Common Prefix (LCP) array of a set of strings is instrumental to compute the suffix-prefix overlaps among strings, which is an essential step for many genome assembly algorithms. In a previous paper, we presented an in-memory divide-a…
The compression of highly repetitive strings (i.e., strings with many repetitions) has been a central research topic in string processing, and quite a few compression methods for these strings have been proposed thus far. Among them, an efficient compression format gathering increasing attention is the run-length Burrows--Wheeler transform (RLBWT), which is a run-length encoded BWT as a reversible permutation of an input string on the lexicographical order of suffixes. State-of-the-art construction algorithms of RLBWT have a serious issue with respect to (i) non-optimal computation time or (ii…
Let $G = (V, E)$ be a digraph where each vertex is unlabeled, each edge is labeled by a character in some alphabet $Ω$, and any two edges with both the same head and the same tail have different labels. The powerset construction gives a transform of $G$ into a weakly connected digraph $G' = (V', E')$ that enables solving the decision problem of whether there is a walk in $G$ matching an arbitrarily long query string $q$ in time linear in $|q|$ and independent of $|E|$ and $|V|$. We show $G$ is uniquely determined by $G'$ when for every $v_\ell \in V$, there is some distinct string $s_\ell$ on …
The Burrows-Wheeler-Transform (BWT) is a reversible string transformation which plays a central role in text compression and is fundamental in many modern bioinformatics applications. The BWT is a permutation of the characters, which is in general better compressible and allows to answer several different query types more efficiently than the original string. It is easy to see that not every string is a BWT image, and exact characterizations of BWT images are known. We investigate a related combinatorial question. In many applications, a sentinel character dollar is added to mark the end of th…
