Showing papers by "William F. Smyth published in 2014"

Fast and Simple Computations Using Prefix Tables Under Hamming and Edit Distance

Abstract: In this article, we introduce a new and simple data structure, the prefix table under Hamming distance, and present two algorithms to compute it efficiently: one asymptotically fast; the other very fast on average and in practice. Because the latter approach avoids the computation of global data structures, such as the suffix array and the longest common prefix array, it yields algorithms much faster in practice than existing methods. We show how this data structure can be used to solve two string problems of interest: (a) approximate string matching under Hamming distance; and (b) longest approximate overlap under Hamming distance. Analogously, we introduce the prefix table under edit distance, and present an efficient algorithm for its computation. In the process, we also define the border array under both distance measures, and provide an algorithm for conversion between prefix tables and border arrays.

Simple Linear Comparison of Strings in V -Order (Extended Abstract)

Abstract: In this paper we focus on a total (but non-lexicographic) or- dering of strings called V-order. We devise a new linear-time algorithm for computing the V -comparison of two finite strings. In comparison with the previous algorithm in the literature, our algorithm is both conceptu- ally simpler, based on recording letter positions in increasing order, and more straightforward to implement, requiring only linked lists.

Two squares canonical factorization

Abstract: We present a new combinatorial structure in a string: a canonical factor- ization for any two squares that occur at the same position and satisfy some size restrictions. We believe that this canonical factorization will have application to re- lated problems such as the New Periodicity Lemma, Crochemore-Rytter Three Squares Lemma, and ultimately the maximum-number-of-runs conjecture.

Large-scale detection of repetitions

Abstract: 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...

Simple Linear Comparison of Strings in V -Order

Abstract: In this paper we focus on a total (but non-lexicographic) ordering of strings called V-order. We devise a new linear-time algorithm for computing the V-comparison of two finite strings. In comparison with the previous algorithm in the literature, our algorithm is both conceptually simpler, based on recording letter positions in increasing order, and more straightforward to implement, requiring only linked lists.

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 every i \in 2..n, i \le i+y[i] \le 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 Sigma; 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 (indeterminate 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 further that there is a 1-1 correspondence between labelled simple graphs and indeterminate strings, and we show how to determine the minimum alphabet size |Sigma| 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 elements of Sigma that have traditionally been studied.

Inferring an indeterminate string from a prefix graph

Abstract: An indeterminate string (or, more simply, just a string ) x=x[1..n] on an alphabet σ is a sequence of nonempty subsets of σ. We say that x[i1] and x[i2] match (written x[i1]≈x[i2]) if and only if x[i1]∩x[i2]≠θ. A feasible array is an array y=y[1..n] of integers such that y[1]=n and for every i∈2..n, y[i]∈0..n-i+1. A prefix table of a string x is an array π=π[1..n] of integers such that, for every i∈1..n, π[i]=j if and only if x[i..i+j-1] is the longest substring at position i of x that matches a prefix of x It is known from [6] that every feasible array is a prefix table of some indeterminate string. A prefix graph P=Py is a labelled simple graph whose structure is determined by a feasible array y In this paper we show, given a feasible array y , how to use Py to construct a lexicographically least indeterminate string on a minimum alphabet whose prefix table π=y.

Computing Covers Using Prefix Tables

Abstract: An \emph{indeterminate string} $x = x[1..n]$ on an alphabet $\Sigma$ is a sequence of nonempty subsets of $\Sigma$; $x$ is said to be \emph{regular} if every subset is of size one. A proper substring $u$ of regular $x$ is said to be a \emph{cover} of $x$ iff for every $i \in 1..n$, an occurrence of $u$ in $x$ includes $x[i]$. The \emph{cover array} $\gamma = \gamma[1..n]$ of $x$ is an integer array such that $\gamma[i]$ is the longest cover of $x[1..i]$. Fifteen years ago a complex, though nevertheless linear-time, algorithm was proposed to compute the cover array of regular $x$ based on prior computation of the border array of $x$. In this paper we first describe a linear-time algorithm to compute the cover array of regular string $x$ based on the prefix table of $x$. We then extend this result to indeterminate strings.