Combinatorics of unique maximal factorization families (UMFFs)

Suppose a set W of strings contains exactly one rotation (cyclic shift) of every primitive string on some alphabet Σ. Then W is a circ-UMFF if and only if every word in Σ+ has a unique maximal factorization over W. The classic circ-UMFF is the set of Lyndon words based on lexicographic ordering (1958). Duval (1983) designed a linear sequential Lyndon factorization algorithm; a corresponding PRAM parallel algorithm was described by J. Daykin, Iliopoulos and Smyth (1994). Daykin and Daykin defined new circ-UMFFs based on various methods for totally ordering sets of strings (2003), and further described the structure of all circ-UMFFs (2008). Here we prove new combinatorial results for circ-UMFFs, and in particular for the case of Lyndon words. We introduce Acrobat and Flight Deck circ-UMFFs, and describe some of our results in terms of dictionaries. Applications of circ-UMFFs pertain to structured methods for concatenating and factoring strings over ordered alphabets, and those of Lyndon words are wide ranging and multidisciplinary

Sensitivity of the Burrows-Wheeler Transform to small modifications, and other problems on string compressors in Bioinformatics

Extensive amount of data is produced in textual form nowadays, especially in bioinformatics. Several algorithms exist to store and process this data efficiently in compressed space. In this thesis, we focus on both combinatorial and practical aspects of two of the most widely used algorithms for compressing text in bioinformatics: the Burrows-Wheeler Transform (BWT) and Lempel-Ziv compression (LZ77). In the first part, we focus on combinatorial aspects of the BWT. Given a word v, r = r(v) denotes the number of maximal equal-letter runs in BWT(v). First, we investigate the relationship between r of a word and r of its reverse. We prove that there exist words for which these two values differ by a logarithmic factor in the length of the word. In other words, although the repetitiveness in the two words is preserved, the number of runs can change by a non-constant factor. This suggests that the number of runs may not be an ideal repetitiveness measure. The second combinatorial aspect we are interested in is how small alterations in a word may affect its BWT in a relevant way. We prove that the number of runs of the BWT of a word can change (increase or decrease) by up to a logarithmic factor in the length of the word by just adding, removing, or substituting a single character. We then consider the special character $used in real-life applications to mark the end of a word. We investigate the impact of this character on words with respect to the BWT. We characterize positions in a word where$ can be inserted in order to turn it into the BWT of a $-terminated word over the same alphabet. We show that, whether and where$ is allowed, depends entirely on the structure of a specific permutation of the indices of the word, which is called the standard permutation of the word. The final part of this thesis treats more applied aspects of text compressors. In bioinformatics, BWT-based compressed data structures are widely used for pattern matching. We give an algorithm based on the BWT to find Maximal Unique Matches (MUMs) of a pattern with respect to a reference text in compressed space, extending an existing tool called PHONI [Boucher et. al, DCC 2021]. Finally, we study some aspects of the Lempel-Ziv 77 (LZ77) factorization of a word. Modeling DNA short reads, we provide a bound on the compression size of the concatenation of regular samples of a word

Burrows–Wheeler compression: Principles and reflections

AbstractAfter a general description of the Burrows–Wheeler transform and a brief survey of recent work on processing its output, the paper examines the coding of the zero-runs from the MTF recoding stage, an aspect with little prior treatment. It is concluded that the original scheme proposed by Wheeler is extremely efficient and unlikely to be much improved.The paper then proposes some new interpretations and uses of the Burrows–Wheeler transform, with new insights and approaches to lossless compression, perhaps including techniques from error correction

Parallel sorting and Star-P data movement and tree flattening

Thesis (M. Eng.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2005.Includes bibliographical references (p. 81-84).This thesis studies three problems in the field of parallel computing. The first result provides a deterministic parallel sorting algorithm that empirically shows an improvement over two sample sort algorithms. When using a comparison sort, this algorithm is 1-optimal in both computation and communication. The second study develops some extensions to the Star-P system [7, 6] that allows it to solve more real problems. The timings provided indicate the scalability of the implementations on some systems. The third problem concerns automatic parallelization. By representing a computation as a binary tree, which we assume is given, it can be shown that the height corresponds to the parallel execution time, given enough processors. The main result of the chapter is an algorithm that uses tree rotations to reduce the height of an arbitrary binary tree to become logarithmic in the number of its inputs. This method can solve more general problems as the definition of tree rotation is slightly altered; examples are given that derive the parallel prefix algorithm, and give a speedup in the dynamic programming approach to the computation of Fibonacci numbers.by David R. Cheng.M.Eng

Efficient Algorithms with Asymmetric Read and Write Costs

In several emerging technologies for computer memory (main memory), the cost of reading is significantly cheaper than the cost of writing. Such asymmetry in memory costs poses a fundamentally different model from the RAM for algorithm design. In this paper we study lower and upper bounds for various problems under such asymmetric read and write costs. We consider both the case in which all but O(1) memory has asymmetric cost, and the case of a small cache of symmetric memory. We model both cases using the (M,omega)-ARAM, in which there is a small (symmetric) memory of size M and a large unbounded (asymmetric) memory, both random access, and where reading from the large memory has unit cost, but writing has cost omega >> 1. For FFT and sorting networks we show a lower bound cost of Omega(omega*n*log_{omega*M}(n)), which indicates that it is not possible to achieve asymptotic improvements with cheaper reads when omega is bounded by a polynomial in M. Moreover, there is an asymptotic gap (of min(omega,log(n)/log(omega*M)) between the cost of sorting networks and comparison sorting in the model. This contrasts with the RAM, and most other models, in which the asymptotic costs are the same. We also show a lower bound for computations on an n*n diamond DAG of Omega(omega*n^2/M) cost, which indicates no asymptotic improvement is achievable with fast reads. However, we show that for the minimum edit distance problem (and related problems), which would seem to be a diamond DAG, we can beat this lower bound with an algorithm with only O(omega*n^2/(M*min(omega^{1/3},M^{1/2}))) cost. To achieve this we make use of a "path sketch" technique that is forbidden in a strict DAG computation. Finally, we show several interesting upper bounds for shortest path problems, minimum spanning trees, and other problems. A common theme in many of the upper bounds is that they require redundant computation and a tradeoff between reads and writes

On the Impact of Morphisms on BWT-Runs

Morphisms are widely studied combinatorial objects that can be used for generating infinite families of words. In the context of Information theory, injective morphisms are called (variable length) codes. In Data compression, the morphisms, combined with parsing techniques, have been recently used to define new mechanisms to generate repetitive words. Here, we show that the repetitiveness induced by applying a morphism to a word can be captured by a compression scheme based on the Burrows-Wheeler Transform (BWT). In fact, we prove that, differently from other compression-based repetitiveness measures, the measure r_bwt (which counts the number of equal-letter runs produced by applying BWT to a word) strongly depends on the applied morphism. More in detail, we characterize the binary morphisms that preserve the value of r_bwt(w), when applied to any binary word w containing both letters. They are precisely the Sturmian morphisms, which are well-known objects in Combinatorics on words. Moreover, we prove that it is always possible to find a binary morphism that, when applied to any binary word containing both letters, increases the number of BWT-equal letter runs by a given (even) number. In addition, we derive a method for constructing arbitrarily large families of binary words on which BWT produces a given (even) number of new equal-letter runs. Such results are obtained by using a new class of morphisms that we call Thue-Morse-like. Finally, we show that there exist binary morphisms ? for which it is possible to find words w such that the difference r_bwt(?(w))-r_bwt(w) is arbitrarily large