222 research outputs found
Minimal Suffix and Rotation of a Substring in Optimal Time
For a text given in advance, the substring minimal suffix queries ask to
determine the lexicographically minimal non-empty suffix of a substring
specified by the location of its occurrence in the text. We develop a data
structure answering such queries optimally: in constant time after linear-time
preprocessing. This improves upon the results of Babenko et al. (CPM 2014),
whose trade-off solution is characterized by product of these
time complexities. Next, we extend our queries to support concatenations of
substrings, for which the construction and query time is preserved. We
apply these generalized queries to compute lexicographically minimal and
maximal rotations of a given substring in constant time after linear-time
preprocessing.
Our data structures mainly rely on properties of Lyndon words and Lyndon
factorizations. We combine them with further algorithmic and combinatorial
tools, such as fusion trees and the notion of order isomorphism of strings
Almost Linear Time Computation of Maximal Repetitions in Run Length Encoded Strings
We consider the problem of computing all maximal repetitions contained in a string that is given in run-length encoding.
Given a run-length encoding of a string, we show that the maximum number of maximal repetitions contained in the string is at most m+k-1, where m is the size of the run-length encoding, and k is the number of run-length factors whose exponent is at least 2.
We also show an algorithm for computing all maximal repetitions in O(m alpha(m)) time and O(m) space, where alpha denotes the inverse Ackermann function
Zeroing the Output of a Nonlinear System Without Relative Degree
The goal of this paper is to establish some facts concerning the problem of
zeroing the output of an input-output system that does not have relative
degree. The approach taken is to work with systems that have a Chen-Fliess
series representation. The main result is that a class of generating series
called primely nullable series provides the building blocks for solving this
problem using the shuffle algebra. It is shown that the shuffle algebra on the
set of generating polynomials is a unique factorization domain so that any
polynomial can be uniquely factored modulo a permutation into its irreducible
elements for the purpose of identifying nullable factors. This is achieved
using the fact that this shuffle algebra is isomorphic to the symmetric algebra
over the vector space spanned by Lyndon words. A specific algorithm for
factoring generating polynomials into its irreducible factors is presented
based on the Chen-Fox-Lyndon factorization of words
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