15,109 research outputs found
Partial words and the critical factorization theorem revisited
In this paper, we consider one of the most fundamental results on the periodicity of words, namely the critical factorization theorem. Given a word w and nonempty words u, v satisfying w = uv, the minimal local period associated with the factorization (u, v) is the length of the shortest square at position |u| - 1. The critical factorization theorem shows that for any word, there is always a factorization whose minimal local period is equal to the minimal period (or global period) of the word.
Crochemore and Perrin presented a linear time algorithm (in the length of the word) that finds a critical factorization from the computation of the maximal suffixes of the word with respect to two total orderings on words: the lexicographic ordering related to a fixed total ordering on the alphabet, and the lexicographic ordering obtained by reversing the order of letters in the alphabet. Here, by refining Crochemore and Perrin’s algorithm, we give a version of the critical factorization theorem for partial words (such sequences may contain ?do not know? symbols or ?holes?). Our proof provides an efficient algorithm which computes a critical factorization when one exists. Our results extend those of Blanchet-Sadri and Duncan for partial words with one hole. A World Wide Web server interface at http://www.uncg.edu/mat/research/cft2/ has been established for automated use of the program
Hybrid static/dynamic scheduling for already optimized dense matrix factorization
We present the use of a hybrid static/dynamic scheduling strategy of the task
dependency graph for direct methods used in dense numerical linear algebra.
This strategy provides a balance of data locality, load balance, and low
dequeue overhead. We show that the usage of this scheduling in communication
avoiding dense factorization leads to significant performance gains. On a 48
core AMD Opteron NUMA machine, our experiments show that we can achieve up to
64% improvement over a version of CALU that uses fully dynamic scheduling, and
up to 30% improvement over the version of CALU that uses fully static
scheduling. On a 16-core Intel Xeon machine, our hybrid static/dynamic
scheduling approach is up to 8% faster than the version of CALU that uses a
fully static scheduling or fully dynamic scheduling. Our algorithm leads to
speedups over the corresponding routines for computing LU factorization in well
known libraries. On the 48 core AMD NUMA machine, our best implementation is up
to 110% faster than MKL, while on the 16 core Intel Xeon machine, it is up to
82% faster than MKL. Our approach also shows significant speedups compared with
PLASMA on both of these systems
The Existential Theory of Equations with Rational Constraints in Free Groups is PSPACE-Complete
It is known that the existential theory of equations in free groups is
decidable. This is a famous result of Makanin. On the other hand it has been
shown that the scheme of his algorithm is not primitive recursive. In this
paper we present an algorithm that works in polynomial space, even in the more
general setting where each variable has a rational constraint, that is, the
solution has to respect a specification given by a regular word language. Our
main result states that the existential theory of equations in free groups with
rational constraints is PSPACE-complete. We obtain this result as a corollary
of the corresponding statement about free monoids with involution.Comment: 45 pages. LaTeX sourc
Order Preservation in Limit Algebras
The matrix units of a digraph algebra, A, induce a relation, known as the
diagonal order, on the projections in a masa in the algebra. Normalizing
partial isometries in A act on these projections by conjugation; they are said
to be order preserving when they respect the diagonal order. Order preserving
embeddings, in turn, are those embeddings which carry order preserving
normalizers to order preserving normalizers. This paper studies operator
algebras which are direct limits of finite dimensional algebras with order
preserving embeddings. We give a complete classification of direct limits of
full triangular matrix algebras with order preserving embeddings. We also
investigate the problem of characterizing algebras with order preserving
embeddings.Comment: 43 pages, AMS-TEX v2.
Threefold Flops via Matrix Factorization
The explicit McKay correspondence, as formulated by Gonzalez-Sprinberg and
Verdier, associates to each exceptional divisor in the minimal resolution of a
rational double point a matrix factorization of the equation of the rational
double point. We study deformations of these matrix factorizations, and show
that they exist over an appropriate "partially resolved" deformation space for
rational double points of types A and D. As a consequence, all simple flops of
lengths 1 and 2 can be described in terms of blowups defined from matrix
factorizations. We also formulate conjectures which would extend these results
to rational double points of type E and simple flops of length greater than 2.Comment: v2: minor change
On braid monodromy factorizations
We introduce and develop a language of semigroups over the braid groups for a
study of braid monodromy factorizations (bmf's) of plane algebraic curves and
other related objects. As an application we give a new proof of Orevkov's
theorem on realization of a bmf over a disc by algebraic curves and show that
the complexity of such a realization can not be bounded in terms of the types
of the factors of the bmf. Besides, we prove that the type of a bmf is
distinguishing Hurwitz curves with singularities of inseparable types up to
-isotopy and -holomorphic cuspidal curves in \C P^2 up to symplectic
isotopy.Comment: 52 pages, AMS-Te
Finding Structure with Randomness: Probabilistic Algorithms for Constructing Approximate Matrix Decompositions
Low-rank matrix approximations, such as the truncated singular value decomposition and the rank-revealing QR decomposition, play a central role in data analysis and scientific computing. This work surveys and extends recent research which demonstrates that randomization offers a powerful tool for performing low-rank matrix approximation. These techniques exploit modern computational architectures more fully than classical methods and open the possibility of dealing with truly massive data sets. This paper presents a modular framework for constructing randomized algorithms that compute partial matrix decompositions. These methods use random sampling to identify a subspace that captures most of the action of a matrix. The input matrix is then compressed—either explicitly or
implicitly—to this subspace, and the reduced matrix is manipulated deterministically to obtain the desired low-rank factorization. In many cases, this approach beats its classical competitors in terms of accuracy, robustness, and/or speed. These claims are supported by extensive numerical experiments and a detailed error analysis. The specific benefits of randomized techniques depend on the computational environment. Consider the model problem of finding the k dominant components of the singular value decomposition of an m × n matrix. (i) For a dense input matrix, randomized algorithms require O(mn log(k))
floating-point operations (flops) in contrast to O(mnk) for classical algorithms. (ii) For a sparse input matrix, the flop count matches classical Krylov subspace methods, but the randomized approach is more robust and can easily be reorganized to exploit multiprocessor architectures. (iii) For a matrix that is too large to fit in fast memory, the randomized techniques require only a constant number of passes over the data, as opposed to O(k) passes for classical algorithms. In fact, it is sometimes possible to perform matrix approximation with a single pass over the data
- …