WF Smyth

McMaster University, Computing and Software, Emeritus

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Papers

String Covering: A Survey

arXiv (Cornell University), Nov 21, 2022

On the Parikh-de-Bruijn grid

arXiv (Cornell University), Nov 16, 2017

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String Comparison in $V$-Order: New Lexicographic Properties & On-line Applications

arXiv (Cornell University), Jul 24, 2015

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Indeterminate Strings, Prefix Arrays & Undirected Graphs

Abstract. An integer array y = y[1..n] is said to be feasible if and only if y[1] = n and, for ev... more Abstract. An integer array y = y[1..n] is said to be feasible if and only if y[1] = n and, for every i ∈ 2..n, i ≤ i+y[i] ≤ n+1. A string is said to be indeterminate if and only if at least one of its elements is a subset of cardinality greater than one of a given alphabet Σ; otherwise it is said to be regular. A feasible array y is said to be regular if and only if it is the prefix array of some regular string. We show using a graph model that every feasible array of integers is a prefix array of some (indetermi-nate or regular) string, and for regular strings corresponding to y, we use the model to provide a lower bound on the alphabet size. We show fur-ther that there is a 1–1 correspondence between labelled simple graphs and indeterminate strings, and we show how to determine the minimum alphabet size σ of an indeterminate string x based on its associated graph Gx. Thus, in this sense, indeterminate strings are a more natural object of combinatorial interest than the strings on ...

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Author manuscript, published in "18th Data Compression Conference (DCC'08), United States (2008)" A simple algorithm for computing the Lempel–Ziv

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Enhanced Covers of Regular and Indeterminate Strings Using Prefix Tables

ArXiv, 2016

A \itbf{cover} of a string $x = x[1..n]$ is a proper substring $u$ of $x$ such that $x$ can be co... more A \itbf{cover} of a string $x = x[1..n]$ is a proper substring $u$ of $x$ such that $x$ can be constructed from possibly overlapping instances of $u$. A recent paper \cite{FIKPPST13} relaxes this definition --- an \itbf{enhanced cover} $u$ of $x$ is a border of $x$ (that is, a proper prefix that is also a suffix) that covers a {\it maximum} number of positions in $x$ (not necessarily all) --- and proposes efficient algorithms for the computation of enhanced covers. These algorithms depend on the prior computation of the \itbf{border array} $\beta[1..n]$, where $\beta[i]$ is the length of the longest border of $x[1..i]$, $1 \le i \le n$. In this paper, we first show how to compute enhanced covers using instead the \itbf{prefix table}: an array $\pi[1..n]$ such that $\pi[i]$ is the length of the longest substring of $x$ beginning at position $i$ that matches a prefix of $x$. Unlike the border array, the prefix table is robust: its properties hold also for \itbf{indeterminate strings} ...

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Lyndon array construction during Burrows–Wheeler inversion

Journal of Discrete Algorithms, 2018

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Frequency Covers for Strings

Fundamenta Informaticae, 2018

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An approach to phrase selection for offline data compression

ACSC '02 Proceedings of the twenty-fifth Australasian conference on Computer science - Volume 4, 2002

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An Improved Method for Reducing the Bandwidth of Sparse Symmetric Matrices

World Computer Congress, 1971

Generic Algorithms for Factoring Strings

Lecture Notes in Computer Science, 2013

ABSTRACT In this paper we describe algorithms for factoring words over sets of strings known as c... more ABSTRACT In this paper we describe algorithms for factoring words over sets of strings known as circ-UMFFs, generalizations of the well-known Lyndon words based on lexorder, whose properties were first studied in 1958 by Chen, Fox and Lyndon. In 1983 Duval designed an elegant linear-time sequential (RAM) Lyndon factorization algorithm; a corresponding parallel (PRAM) algorithm was described in 1994 by Daykin, Iliopoulos and Smyth. In 2003 Daykin and Daykin introduced various circ-UMFFs, including one based on V-words and V-ordering; in 2011 linear string comparison and sequential factorization algorithms based on V-order were given by Daykin, Daykin and Smyth. Here we first describe generic RAM and PRAM algorithms for factoring a word over any circ-UMFF; then we show how to customize these generic algorithms to yield optimal parallel Lyndon-like V-word factorization.

String Comparison and Lyndon-Like Factorization Using V-Order in Linear Time

Lecture Notes in Computer Science, 2011

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Inverted Files Versus Suffix Arrays for Locating Patterns in Primary Memory

Lecture Notes in Computer Science, 2006

Recent advances in the asymptotic resource costs of pattern matching with compressed suffix array... more Recent advances in the asymptotic resource costs of pattern matching with compressed suffix arrays are attractive, but a key rival structure, the compressed inverted file, has been dismissed or ignored in papers presenting the new structures. In this paper we examine the resource requirements of compressed suffix array algorithms against compressed inverted file data structures for general pattern matching in

Computing regularities in strings

2009 2nd IEEE International Conference on Computer Science and Information Technology, 2009

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An Algorithm for Finding the Diameter of a Graph

No abstract availabl

Large-scale detection of repetitions

Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2014

Combinatorics on words began more than a century ago with a demonstration that an infinitely long... more Combinatorics on words began more than a century ago with a demonstration that an infinitely long string with no repetitions could be constructed on an alphabet of only three letters. Computing all the repetitions (such as ⋯ TTT ⋯ or ⋯ CGACGA ⋯ ) in a given string x of length n is one of the oldest and most important problems of computational stringology, requiring time in the worst case. About a dozen years ago, it was discovered that repetitions can be computed as a by-product of the Θ ( n )-time computation of all the maximal periodicities or runs in x . However, even though the computation is linear, it is also brute force: global data structures, such as the suffix array , the longest common prefix array and the Lempel–Ziv factorization , need to be computed in a preprocessing phase. Furthermore, all of this effort is required despite the fact that the expected number of runs in a string is generally a small fraction of the string length. In this paper, I explore the possibilit...

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Sorting Suffixes of Two-Pattern Strings

International Journal of Foundations of Computer Science, 2005

Recently, several authors presented linear recursive algorithms for sorting suffixes of a string.... more Recently, several authors presented linear recursive algorithms for sorting suffixes of a string. All these algorithms employ a similar three-step approach, based on an initial division of the suffixes of x into two sets: Step 1 sorts the first set using recursive reduction of the problem, Step 2 determines the order of the suffixes in the second set based on the order of the suffixes in the first set, and Step 3 merges the two sets together. To optimize such an algorithm either for space or time, it may not be sufficient to optimize one of the three steps, since in doing so, one might increase the resources required for the others to an unacceptable extent. Franek, Lu, and Smyth introduced two-pattern strings as a generalization of Sturmian strings. Like Sturmian strings, two-pattern strings are generated by iterated morphisms, but they exhibit a much richer structure. In this paper we show that the suffixes of two-pattern strings can be sorted in linear time using a variant of the...

A fast average case algorithm for lyndon decomposition

International Journal of Computer Mathematics, 1995

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Computing patterns in strings

A string is just a sequence of letters. But strings can be massive. Plant and animal genomes are ... more A string is just a sequence of letters. But strings can be massive. Plant and animal genomes are strings billions of letters long on the simple alphabet Internet traffic among billions of websites is a collection of strings that amount to quadrillions of computer bits every day. Such strings are regularly searched, probably millions of times a day, for patterns of all kinds - genomic codes for genes and chromosomes, indicators of terrorist activity, and many others. The search for patterns is fundamental to many fields molecular biology, cryptography, data compression, computer vision, speech recognition, computational geometry. This book provides a basic general introduction to the algorithms (methods) that efficiently compute patterns in strings. It focuses on results that can be explained with reasonable economy and simplicity, but its 250 references also enable the reader to access current state-of-the-art methodology.