24,611 research outputs found
Boolean Inner product Spaces and Boolean Matrices
This article discusses the concept of Boolean spaces endowed with a Boolean
valued inner product and their matrices. A natural inner product structure for
the space of Boolean n-tuples is introduced. Stochastic boolean vectors and
stochastic and unitary Boolean matrices are studied. A dimension theorem for
orthonormal bases of a Boolean space is proven. We characterize the invariant
stochastic Boolean vectors for a Boolean stochastic matrix and show that they
can be used to reduce a unitary matrix. Finally, we obtain a result on powers
of stochastic and unitary matrices.Comment: 36 page
Quantum and approximation algorithms for maximum witnesses of Boolean matrix products
The problem of finding maximum (or minimum) witnesses of the Boolean product
of two Boolean matrices (MW for short) has a number of important applications,
in particular the all-pairs lowest common ancestor (LCA) problem in directed
acyclic graphs (dags). The best known upper time-bound on the MW problem for
n\times n Boolean matrices of the form O(n^{2.575}) has not been substantially
improved since 2006. In order to obtain faster algorithms for this problem, we
study quantum algorithms for MW and approximation algorithms for MW (in the
standard computational model). Some of our quantum algorithms are input or
output sensitive. Our fastest quantum algorithm for the MW problem, and
consequently for the related problems, runs in time
\tilde{O}(n^{2+\lambda/2})=\tilde{O}(n^{2.434}), where \lambda satisfies the
equation \omega(1, \lambda, 1) = 1 + 1.5 \, \lambda and \omega(1, \lambda, 1)
is the exponent of the multiplication of an n \times n^{\lambda}$ matrix by an
n^{\lambda} \times n matrix. Next, we consider a relaxed version of the MW
problem (in the standard model) asking for reporting a witness of bounded rank
(the maximum witness has rank 1) for each non-zero entry of the matrix product.
First, by adapting the fastest known algorithm for maximum witnesses, we obtain
an algorithm for the relaxed problem that reports for each non-zero entry of
the product matrix a witness of rank at most \ell in time
\tilde{O}((n/\ell)n^{\omega(1,\log_n \ell,1)}). Then, by reducing the relaxed
problem to the so called k-witness problem, we provide an algorithm that
reports for each non-zero entry C[i,j] of the product matrix C a witness of
rank O(\lceil W_C(i,j)/k\rceil ), where W_C(i,j) is the number of witnesses for
C[i,j], with high probability. The algorithm runs in
\tilde{O}(n^{\omega}k^{0.4653} +n^2k) time, where \omega=\omega(1,1,1).Comment: 14 pages, 3 figure
A bound on the scrambling index of a primitive matrix using Boolean rank
The scrambling index of an primitive matrix is the smallest
positive integer such that , where denotes the
transpose of and denotes the all ones matrix. For an
Boolean matrix , its {\it Boolean rank} is the smallest
positive integer such that for some Boolean matrix
and Boolean matrix . In this paper, we give an upper bound on
the scrambling index of an primitive matrix in terms of its
Boolean rank . Furthermore we characterize all primitive matrices that
achieve the upper bound.Comment: 13 pages, 3 table
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