27,774 research outputs found
Evaluating parametric holonomic sequences using rectangular splitting
We adapt the rectangular splitting technique of Paterson and Stockmeyer to
the problem of evaluating terms in holonomic sequences that depend on a
parameter. This approach allows computing the -th term in a recurrent
sequence of suitable type using "expensive" operations at the cost
of an increased number of "cheap" operations.
Rectangular splitting has little overhead and can perform better than either
naive evaluation or asymptotically faster algorithms for ranges of
encountered in applications. As an example, fast numerical evaluation of the
gamma function is investigated. Our work generalizes two previous algorithms of
Smith.Comment: 8 pages, 2 figure
Faster Algorithms for Rectangular Matrix Multiplication
Let {\alpha} be the maximal value such that the product of an n x n^{\alpha}
matrix by an n^{\alpha} x n matrix can be computed with n^{2+o(1)} arithmetic
operations. In this paper we show that \alpha>0.30298, which improves the
previous record \alpha>0.29462 by Coppersmith (Journal of Complexity, 1997).
More generally, we construct a new algorithm for multiplying an n x n^k matrix
by an n^k x n matrix, for any value k\neq 1. The complexity of this algorithm
is better than all known algorithms for rectangular matrix multiplication. In
the case of square matrix multiplication (i.e., for k=1), we recover exactly
the complexity of the algorithm by Coppersmith and Winograd (Journal of
Symbolic Computation, 1990).
These new upper bounds can be used to improve the time complexity of several
known algorithms that rely on rectangular matrix multiplication. For example,
we directly obtain a O(n^{2.5302})-time algorithm for the all-pairs shortest
paths problem over directed graphs with small integer weights, improving over
the O(n^{2.575})-time algorithm by Zwick (JACM 2002), and also improve the time
complexity of sparse square matrix multiplication.Comment: 37 pages; v2: some additions in the acknowledgment
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
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