946 research outputs found
Fast algorithms for computing with integer matrices: normal forms and applications
The focus of this thesis is on fundamental computational problems in exact integer linear algebra. Specifically, for a nonsingular integer input matrix A of dimension n, we consider problems such as linear system solving and computing integer matrix normal forms.
Our goal is to design algorithms that have complexity about the same as the cost of multiplying together two integer matrices of the same dimension and size of entries as the input matrix A. If 2 ≤ ω ≤ 3 is a valid exponent for matrix multiplication, that is, if two n × n matrices can be multiplied in O(n^ω) basic operations from the domain of entries, then our target complexity is O(n^ω log ||A||) bit operations, up to some missing log n and loglog ||A|| factors. Here ||A|| denotes the largest entry in A in absolute value.
The first contribution is solving the problem of computing the Smith normal form S of a nonsingular matrix A along with computing unimodular matrices U, V such that AV = US within our target cost. The algorithm we give is a Las Vegas probabilistic algorithm which means that we are able to verify the correctness of its output.
The second contribution of the thesis is with respect to linear system solving. We present a deterministic reduction to matrix multiplication for the problem of linear system solving: given as input a nonsingular A and a vector b, solve the system Ax = b. The system solution x is computed within our target complexity
Fast, deterministic computation of the Hermite normal form and determinant of a polynomial matrix
Given a nonsingular matrix of univariate polynomials over a
field , we give fast and deterministic algorithms to compute its
determinant and its Hermite normal form. Our algorithms use
operations in ,
where is bounded from above by both the average of the degrees of the rows
and that of the columns of the matrix and is the exponent of matrix
multiplication. The soft- notation indicates that logarithmic factors in the
big- are omitted while the ceiling function indicates that the cost is
when . Our algorithms are based
on a fast and deterministic triangularization method for computing the diagonal
entries of the Hermite form of a nonsingular matrix.Comment: 34 pages, 3 algorithm
Deterministic Polynomial Time Algorithms for Matrix Completion Problems
We present new deterministic algorithms for several cases of the maximum rank
matrix completion problem (for short matrix completion), i.e. the problem of
assigning values to the variables in a given symbolic matrix as to maximize the
resulting matrix rank. Matrix completion belongs to the fundamental problems in
computational complexity with numerous important algorithmic applications,
among others, in computing dynamic transitive closures or multicast network
codings (Harvey et al SODA 2005, Harvey et al SODA 2006).
We design efficient deterministic algorithms for common generalizations of
the results of Lovasz and Geelen on this problem by allowing linear functions
in the entries of the input matrix such that the submatrices corresponding to
each variable have rank one. We present also a deterministic polynomial time
algorithm for finding the minimal number of generators of a given module
structure given by matrices. We establish further several hardness results
related to matrix algebras and modules. As a result we connect the classical
problem of polynomial identity testing with checking surjectivity (or
injectivity) between two given modules. One of the elements of our algorithm is
a construction of a greedy algorithm for finding a maximum rank element in the
more general setting of the problem. The proof methods used in this paper could
be also of independent interest.Comment: 14 pages, preliminar
Algorithms in algebraic number theory
In this paper we discuss the basic problems of algorithmic algebraic number
theory. The emphasis is on aspects that are of interest from a purely
mathematical point of view, and practical issues are largely disregarded. We
describe what has been done and, more importantly, what remains to be done in
the area. We hope to show that the study of algorithms not only increases our
understanding of algebraic number fields but also stimulates our curiosity
about them. The discussion is concentrated of three topics: the determination
of Galois groups, the determination of the ring of integers of an algebraic
number field, and the computation of the group of units and the class group of
that ring of integers.Comment: 34 page
Tropical polyhedra are equivalent to mean payoff games
We show that several decision problems originating from max-plus or tropical
convexity are equivalent to zero-sum two player game problems. In particular,
we set up an equivalence between the external representation of tropical convex
sets and zero-sum stochastic games, in which tropical polyhedra correspond to
deterministic games with finite action spaces. Then, we show that the winning
initial positions can be determined from the associated tropical polyhedron. We
obtain as a corollary a game theoretical proof of the fact that the tropical
rank of a matrix, defined as the maximal size of a submatrix for which the
optimal assignment problem has a unique solution, coincides with the maximal
number of rows (or columns) of the matrix which are linearly independent in the
tropical sense. Our proofs rely on techniques from non-linear Perron-Frobenius
theory.Comment: 28 pages, 5 figures; v2: updated references, added background
materials and illustrations; v3: minor improvements, references update
Fast Quantum Algorithm for Solving Multivariate Quadratic Equations
In August 2015 the cryptographic world was shaken by a sudden and surprising
announcement by the US National Security Agency NSA concerning plans to
transition to post-quantum algorithms. Since this announcement post-quantum
cryptography has become a topic of primary interest for several standardization
bodies. The transition from the currently deployed public-key algorithms to
post-quantum algorithms has been found to be challenging in many aspects. In
particular the problem of evaluating the quantum-bit security of such
post-quantum cryptosystems remains vastly open. Of course this question is of
primarily concern in the process of standardizing the post-quantum
cryptosystems. In this paper we consider the quantum security of the problem of
solving a system of {\it Boolean multivariate quadratic equations in
variables} (\MQb); a central problem in post-quantum cryptography. When ,
under a natural algebraic assumption, we present a Las-Vegas quantum algorithm
solving \MQb{} that requires the evaluation of, on average,
quantum gates. To our knowledge this is the fastest algorithm for solving
\MQb{}
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