On Longest Repeat Queries Using GPU

Repeat finding in strings has important applications in subfields such as computational biology. The challenge of finding the longest repeats covering particular string positions was recently proposed and solved by \.{I}leri et al., using a total of the optimal $O(n)$ time and space, where $n$ is the string size. However, their solution can only find the \emph{leftmost} longest repeat for each of the $n$ string position. It is also not known how to parallelize their solution. In this paper, we propose a new solution for longest repeat finding, which although is theoretically suboptimal in time but is conceptually simpler and works faster and uses less memory space in practice than the optimal solution. Further, our solution can find \emph{all} longest repeats of every string position, while still maintaining a faster processing speed and less memory space usage. Moreover, our solution is \emph{parallelizable} in the shared memory architecture (SMA), enabling it to take advantage of the modern multi-processor computing platforms such as the general-purpose graphics processing units (GPU). We have implemented both the sequential and parallel versions of our solution. Experiments with both biological and non-biological data show that our sequential and parallel solutions are faster than the optimal solution by a factor of 2--3.5 and 6--14, respectively, and use less memory space.Comment: 14 page

A framework for space-efficient string kernels

String kernels are typically used to compare genome-scale sequences whose length makes alignment impractical, yet their computation is based on data structures that are either space-inefficient, or incur large slowdowns. We show that a number of exact string kernels, like the $k$-mer kernel, the substrings kernels, a number of length-weighted kernels, the minimal absent words kernel, and kernels with Markovian corrections, can all be computed in $O(nd)$ time and in $o(n)$ bits of space in addition to the input, using just a $\mathtt{rangeDistinct}$ data structure on the Burrows-Wheeler transform of the input strings, which takes $O(d)$ time per element in its output. The same bounds hold for a number of measures of compositional complexity based on multiple value of $k$, like the $k$-mer profile and the $k$-th order empirical entropy, and for calibrating the value of $k$ using the data

Space-efficient detection of unusual words

Detecting all the strings that occur in a text more frequently or less frequently than expected according to an IID or a Markov model is a basic problem in string mining, yet current algorithms are based on data structures that are either space-inefficient or incur large slowdowns, and current implementations cannot scale to genomes or metagenomes in practice. In this paper we engineer an algorithm based on the suffix tree of a string to use just a small data structure built on the Burrows-Wheeler transform, and a stack of $O(\sigma^2\log^2 n)$ bits, where $n$ is the length of the string and $\sigma$ is the size of the alphabet. The size of the stack is $o(n)$ except for very large values of $\sigma$. We further improve the algorithm by removing its time dependency on $\sigma$, by reporting only a subset of the maximal repeats and of the minimal rare words of the string, and by detecting and scoring candidate under-represented strings that $\textit{do not occur}$ in the string. Our algorithms are practical and work directly on the BWT, thus they can be immediately applied to a number of existing datasets that are available in this form, returning this string mining problem to a manageable scale.Comment: arXiv admin note: text overlap with arXiv:1502.0637

Composite repetition-aware data structures

In highly repetitive strings, like collections of genomes from the same species, distinct measures of repetition all grow sublinearly in the length of the text, and indexes targeted to such strings typically depend only on one of these measures. We describe two data structures whose size depends on multiple measures of repetition at once, and that provide competitive tradeoffs between the time for counting and reporting all the exact occurrences of a pattern, and the space taken by the structure. The key component of our constructions is the run-length encoded BWT (RLBWT), which takes space proportional to the number of BWT runs: rather than augmenting RLBWT with suffix array samples, we combine it with data structures from LZ77 indexes, which take space proportional to the number of LZ77 factors, and with the compact directed acyclic word graph (CDAWG), which takes space proportional to the number of extensions of maximal repeats. The combination of CDAWG and RLBWT enables also a new representation of the suffix tree, whose size depends again on the number of extensions of maximal repeats, and that is powerful enough to support matching statistics and constant-space traversal.Comment: (the name of the third co-author was inadvertently omitted from previous version

Lempel-Ziv Parsing in External Memory

For decades, computing the LZ factorization (or LZ77 parsing) of a string has been a requisite and computationally intensive step in many diverse applications, including text indexing and data compression. Many algorithms for LZ77 parsing have been discovered over the years; however, despite the increasing need to apply LZ77 to massive data sets, no algorithm to date scales to inputs that exceed the size of internal memory. In this paper we describe the first algorithm for computing the LZ77 parsing in external memory. Our algorithm is fast in practice and will allow the next generation of text indexes to be realised for massive strings and string collections.Comment: 10 page