2,094 research outputs found
Subsequence Automata with Default Transitions
Let be a string of length with characters from an alphabet of size
. The \emph{subsequence automaton} of (often called the
\emph{directed acyclic subsequence graph}) is the minimal deterministic finite
automaton accepting all subsequences of . A straightforward construction
shows that the size (number of states and transitions) of the subsequence
automaton is and that this bound is asymptotically optimal.
In this paper, we consider subsequence automata with \emph{default
transitions}, that is, special transitions to be taken only if none of the
regular transitions match the current character, and which do not consume the
current character. We show that with default transitions, much smaller
subsequence automata are possible, and provide a full trade-off between the
size of the automaton and the \emph{delay}, i.e., the maximum number of
consecutive default transitions followed before consuming a character.
Specifically, given any integer parameter , , we
present a subsequence automaton with default transitions of size
and delay . Hence, with we
obtain an automaton of size and delay . On
the other extreme, with , we obtain an automaton of size and delay , thus matching the bound for the standard subsequence
automaton construction. Finally, we generalize the result to multiple strings.
The key component of our result is a novel hierarchical automata construction
of independent interest.Comment: Corrected typo
k-Universality of Regular Languages
A subsequence of a word w is a word u such that u = w[i1]w[i2] . . . w[ik], for some set of indices 1 ≤ i1 < i2 < · · · < ik ≤ |w|. A word w is k-subsequence universal over an alphabet Σ if every word in Σk appears in w as a subsequence. In this paper, we study the intersection between the set of k-subsequence universal words over some alphabet Σ and regular languages over Σ. We call a regular language L k-∃-subsequence universal if there exists a k-subsequence universal word in L, and k-∀-subsequence universal if every word of L is k-subsequence universal. We give algorithms solving the problems of deciding if a given regular language, represented by a finite automaton recognising it, is k-∃-subsequence universal and, respectively, if it is k-∀-subsequence universal, for a given k. The algorithms are FPT w.r.t. the size of the input alphabet, and their run-time does not depend on k; they run in polynomial time in the number n of states of the input automaton when the size of the input alphabet is O(log n). Moreover, we show that the problem of deciding if a given regular language is k-∃-subsequence universal is NP-complete, when the language is over a large alphabet. Further, we provide algorithms for counting the number of k-subsequence universal words (paths) accepted by a given deterministic (respectively, nondeterministic) finite automaton, and ranking an input word (path) within the set of k-subsequence universal words accepted by a given finite automaton
Compact Recognizers of Episode Sequences
Abstract Mikhail J. Atallah t Purdue University Given two strings T = at ... an and P = hI .. .h m over an alphabet E, the problem of testing whether P occurs as a subsequence of T is trivially solved in linear time. It is also known that a simple D(nlog lEI) time preprocessing ofT makes it easy to decide subsequently for any P and in at most IPJIog lEI character comparisons, whether P is a subsequence of T. These problems become more complicated if onc asks instead whether P occurs as a subsequence of some substring Y of T of bounded length. This paper presents an automaton built on the textstring T and capable of identifying all distinct minimal substrings Y of X having P as a subsequence. By a substring Y being minimal with respect to P, it is meant that P is not a subsequence of any proper substring of Y. For every minimal substring Y, the automaton recognizes the occurrence of P having lexicographically smallest sequence of symbol positions in Y. It is not difficult to realize such an automaton in time and space 0(n 2 ) for a text of n characters. One result of this paper consists of bringing those bounds down to linear or O(nlogn), respectively, depending on whether the alphabet is bounded or of arbitrary size, thereby matching the respective complexities of off-line exact string searching. Having built the automaton, the search for all lexicographically earliest occurrences of P in X is carried out in time O(n + k l rocc, . i . log n . log I~I), where rocc, is the number of distinct minimal substrings of T having b 1 ... b; as a subsequence. All log factors appearing in the above bounds can be further reduced to log log by resort to known integer-handling data structures. Index Terms -Algorithms, pattern matching, subsequence and episode searching, DAWG, suffix automaton, compact subsequence automaton, skip-edge DAWG, forward failure function, skip-link
Fast and Compact Regular Expression Matching
We study 4 problems in string matching, namely, regular expression matching,
approximate regular expression matching, string edit distance, and subsequence
indexing, on a standard word RAM model of computation that allows
logarithmic-sized words to be manipulated in constant time. We show how to
improve the space and/or remove a dependency on the alphabet size for each
problem using either an improved tabulation technique of an existing algorithm
or by combining known algorithms in a new way
The separation problem for regular languages by piecewise testable languages
Separation is a classical problem in mathematics and computer science. It
asks whether, given two sets belonging to some class, it is possible to
separate them by another set of a smaller class. We present and discuss the
separation problem for regular languages. We then give a direct polynomial time
algorithm to check whether two given regular languages are separable by a
piecewise testable language, that is, whether a sentence can
witness that the languages are indeed disjoint. The proof is a reformulation
and a refinement of an algebraic argument already given by Almeida and the
second author
Completeness Results for Parameterized Space Classes
The parameterized complexity of a problem is considered "settled" once it has
been shown to lie in FPT or to be complete for a class in the W-hierarchy or a
similar parameterized hierarchy. Several natural parameterized problems have,
however, resisted such a classification. At least in some cases, the reason is
that upper and lower bounds for their parameterized space complexity have
recently been obtained that rule out completeness results for parameterized
time classes. In this paper, we make progress in this direction by proving that
the associative generability problem and the longest common subsequence problem
are complete for parameterized space classes. These classes are defined in
terms of different forms of bounded nondeterminism and in terms of simultaneous
time--space bounds. As a technical tool we introduce a "union operation" that
translates between problems complete for classical complexity classes and for
W-classes.Comment: IPEC 201
Order preserving pattern matching on trees and DAGs
The order preserving pattern matching (OPPM) problem is, given a pattern
string and a text string , find all substrings of which have the
same relative orders as . In this paper, we consider two variants of the
OPPM problem where a set of text strings is given as a tree or a DAG. We show
that the OPPM problem for a single pattern of length and a text tree
of size can be solved in time if the characters of are
drawn from an integer alphabet of polynomial size. The time complexity becomes
if the pattern is over a general ordered alphabet. We
then show that the OPPM problem for a single pattern and a text DAG is
NP-complete
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