9,417 research outputs found
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
Dominance Product and High-Dimensional Closest Pair under
Given a set of points in , the Closest Pair problem is
to find a pair of distinct points in at minimum distance. When is
constant, there are efficient algorithms that solve this problem, and fast
approximate solutions for general . However, obtaining an exact solution in
very high dimensions seems to be much less understood. We consider the
high-dimensional Closest Pair problem, where for some , and the underlying metric is .
We improve and simplify previous results for Closest Pair, showing
that it can be solved by a deterministic strongly-polynomial algorithm that
runs in time, and by a randomized algorithm that runs in
expected time, where is the time bound for computing the
{\em dominance product} for points in . That is a matrix ,
such that ; this is the
number of coordinates at which dominates . For integer coordinates
from some interval , we obtain an algorithm that runs in
time, where
is the exponent of multiplying an matrix by an
matrix.
We also give slightly better bounds for , by using more recent
rectangular matrix multiplication bounds. Computing the dominance product
itself is an important task, since it is applied in many algorithms as a major
black-box ingredient, such as algorithms for APBP (all pairs bottleneck paths),
and variants of APSP (all pairs shortest paths)
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