Complexity in Prefix-Free Regular Languages

We examine deterministic and nondeterministic state complexities of regular operations on prefix-free languages. We strengthen several results by providing witness languages over smaller alphabets, usually as small as possible. We next provide the tight bounds on state complexity of symmetric difference, and deterministic and nondeterministic state complexity of difference and cyclic shift of prefix-free languages.Comment: In Proceedings DCFS 2010, arXiv:1008.127

Reversal Distances for Strings with Few Blocks or Small Alphabets

International audienceWe study the String Reversal Distance problem, an extension of the well-known Sorting by Reversals problem. String Reversal Distance takes two strings S and T as input, and asks for a minimum number of reversals to obtain T from S. We consider four variants: String Reversal Distance, String Prefix Reversal Distance (in which any reversal must include the first letter of the string), and the signed variants of these problems, namely Signed String Reversal Distance and Signed String Prefix Reversal Distance. We study algorithmic properties of these four problems, in connection with two parameters of the input strings: the number of blocks they contain (a block being maximal substring such that all letters in the substring are equal), and the alphabet size Σ. For instance, we show that Signed String Reversal Distance and Signed String Prefix Reversal Distance are NP-hard even if the input strings have only one letter

On the aperiodic avoidability of binary patterns with variables and reversals

In this work we present a characterisation of the avoidability of all unary and binary patterns, that do not only contain variables but also reversals of their instances, with respect to aperiodic infinite words. These types of patterns were studied recently in either more general or particular cases

Harmonious Hilbert curves and other extradimensional space-filling curves

This paper introduces a new way of generalizing Hilbert's two-dimensional space-filling curve to arbitrary dimensions. The new curves, called harmonious Hilbert curves, have the unique property that for any d' < d, the d-dimensional curve is compatible with the d'-dimensional curve with respect to the order in which the curves visit the points of any d'-dimensional axis-parallel space that contains the origin. Similar generalizations to arbitrary dimensions are described for several variants of Peano's curve (the original Peano curve, the coil curve, the half-coil curve, and the Meurthe curve). The d-dimensional harmonious Hilbert curves and the Meurthe curves have neutral orientation: as compared to the curve as a whole, arbitrary pieces of the curve have each of d! possible rotations with equal probability. Thus one could say these curves are `statistically invariant' under rotation---unlike the Peano curves, the coil curves, the half-coil curves, and the familiar generalization of Hilbert curves by Butz and Moore. In addition, prompted by an application in the construction of R-trees, this paper shows how to construct a 2d-dimensional generalized Hilbert or Peano curve that traverses the points of a certain d-dimensional diagonally placed subspace in the order of a given d-dimensional generalized Hilbert or Peano curve. Pseudocode is provided for comparison operators based on the curves presented in this paper.Comment: 40 pages, 10 figures, pseudocode include

Penalty-Based Aggregation of Strings

International Summer School on Aggregation Operators (2019. Olomouc, Czech Republic

Avoidability index for binary patterns with reversal

For every pattern $p$ over the alphabet $\{x,y,x^R,y^R\}$, we specify the least $k$ such that $p$ is $k$-avoidable.Comment: 15 pages, 1 figur

siEDM: an efficient string index and search algorithm for edit distance with moves

Although several self-indexes for highly repetitive text collections exist, developing an index and search algorithm with editing operations remains a challenge. Edit distance with moves (EDM) is a string-to-string distance measure that includes substring moves in addition to ordinal editing operations to turn one string into another. Although the problem of computing EDM is intractable, it has a wide range of potential applications, especially in approximate string retrieval. Despite the importance of computing EDM, there has been no efficient method for indexing and searching large text collections based on the EDM measure. We propose the first algorithm, named string index for edit distance with moves (siEDM), for indexing and searching strings with EDM. The siEDM algorithm builds an index structure by leveraging the idea behind the edit sensitive parsing (ESP), an efficient algorithm enabling approximately computing EDM with guarantees of upper and lower bounds for the exact EDM. siEDM efficiently prunes the space for searching query strings by the proposed method, which enables fast query searches with the same guarantee as ESP. We experimentally tested the ability of siEDM to index and search strings on benchmark datasets, and we showed siEDM's efficiency.Comment: 23 page