An Elegant Algorithm for the Construction of Suffix Arrays

The suffix array is a data structure that finds numerous applications in string processing problems for both linguistic texts and biological data. It has been introduced as a memory efficient alternative for suffix trees. The suffix array consists of the sorted suffixes of a string. There are several linear time suffix array construction algorithms (SACAs) known in the literature. However, one of the fastest algorithms in practice has a worst case run time of $O(n^2)$. The problem of designing practically and theoretically efficient techniques remains open. In this paper we present an elegant algorithm for suffix array construction which takes linear time with high probability; the probability is on the space of all possible inputs. Our algorithm is one of the simplest of the known SACAs and it opens up a new dimension of suffix array construction that has not been explored until now. Our algorithm is easily parallelizable. We offer parallel implementations on various parallel models of computing. We prove a lemma on the $\ell$-mers of a random string which might find independent applications. We also present another algorithm that utilizes the above algorithm. This algorithm is called RadixSA and has a worst case run time of $O(n\log{n})$. RadixSA introduces an idea that may find independent applications as a speedup technique for other SACAs. An empirical comparison of RadixSA with other algorithms on various datasets reveals that our algorithm is one of the fastest algorithms to date. The C++ source code is freely available at http://www.engr.uconn.edu/~man09004/radixSA.zi

Handling Massive N-Gram Datasets Efficiently

This paper deals with the two fundamental problems concerning the handling of large n-gram language models: indexing, that is compressing the n-gram strings and associated satellite data without compromising their retrieval speed; and estimation, that is computing the probability distribution of the strings from a large textual source. Regarding the problem of indexing, we describe compressed, exact and lossless data structures that achieve, at the same time, high space reductions and no time degradation with respect to state-of-the-art solutions and related software packages. In particular, we present a compressed trie data structure in which each word following a context of fixed length k, i.e., its preceding k words, is encoded as an integer whose value is proportional to the number of words that follow such context. Since the number of words following a given context is typically very small in natural languages, we lower the space of representation to compression levels that were never achieved before. Despite the significant savings in space, our technique introduces a negligible penalty at query time. Regarding the problem of estimation, we present a novel algorithm for estimating modified Kneser-Ney language models, that have emerged as the de-facto choice for language modeling in both academia and industry, thanks to their relatively low perplexity performance. Estimating such models from large textual sources poses the challenge of devising algorithms that make a parsimonious use of the disk. The state-of-the-art algorithm uses three sorting steps in external memory: we show an improved construction that requires only one sorting step thanks to exploiting the properties of the extracted n-gram strings. With an extensive experimental analysis performed on billions of n-grams, we show an average improvement of 4.5X on the total running time of the state-of-the-art approach.Comment: Published in ACM Transactions on Information Systems (TOIS), February 2019, Article No: 2

Lyndon Array Construction during Burrows-Wheeler Inversion

In this paper we present an algorithm to compute the Lyndon array of a string $T$ of length $n$ as a byproduct of the inversion of the Burrows-Wheeler transform of $T$. Our algorithm runs in linear time using only a stack in addition to the data structures used for Burrows-Wheeler inversion. We compare our algorithm with two other linear-time algorithms for Lyndon array construction and show that computing the Burrows-Wheeler transform and then constructing the Lyndon array is competitive compared to the known approaches. We also propose a new balanced parenthesis representation for the Lyndon array that uses $2n+o(n)$ bits of space and supports constant time access. This representation can be built in linear time using $O(n)$ words of space, or in $O(n\log n/\log\log n)$ time using asymptotically the same space as $T$

Universal lossless source coding with the Burrows Wheeler transform

The Burrows Wheeler transform (1994) is a reversible sequence transformation used in a variety of practical lossless source-coding algorithms. In each, the BWT is followed by a lossless source code that attempts to exploit the natural ordering of the BWT coefficients. BWT-based compression schemes are widely touted as low-complexity algorithms giving lossless coding rates better than those of the Ziv-Lempel codes (commonly known as LZ'77 and LZ'78) and almost as good as those achieved by prediction by partial matching (PPM) algorithms. To date, the coding performance claims have been made primarily on the basis of experimental results. This work gives a theoretical evaluation of BWT-based coding. The main results of this theoretical evaluation include: (1) statistical characterizations of the BWT output on both finite strings and sequences of length n → ∞, (2) a variety of very simple new techniques for BWT-based lossless source coding, and (3) proofs of the universality and bounds on the rates of convergence of both new and existing BWT-based codes for finite-memory and stationary ergodic sources. The end result is a theoretical justification and validation of the experimentally derived conclusions: BWT-based lossless source codes achieve universal lossless coding performance that converges to the optimal coding performance more quickly than the rate of convergence observed in Ziv-Lempel style codes and, for some BWT-based codes, within a constant factor of the optimal rate of convergence for finite-memory source

The Parallelism Motifs of Genomic Data Analysis

Genomic data sets are growing dramatically as the cost of sequencing continues to decline and small sequencing devices become available. Enormous community databases store and share this data with the research community, but some of these genomic data analysis problems require large scale computational platforms to meet both the memory and computational requirements. These applications differ from scientific simulations that dominate the workload on high end parallel systems today and place different requirements on programming support, software libraries, and parallel architectural design. For example, they involve irregular communication patterns such as asynchronous updates to shared data structures. We consider several problems in high performance genomics analysis, including alignment, profiling, clustering, and assembly for both single genomes and metagenomes. We identify some of the common computational patterns or motifs that help inform parallelization strategies and compare our motifs to some of the established lists, arguing that at least two key patterns, sorting and hashing, are missing