51,797 research outputs found
Enumeration Algorithms for Conjunctive Queries with Projection
We investigate the enumeration of query results for an important subset of CQs with projections, namely star and path queries. The task is to design data structures and algorithms that allow for efficient enumeration with delay guarantees after a preprocessing phase. Our main contribution is a series of results based on the idea of interleaving precomputed output with further join processing to maintain delay guarantees, which maybe of independent interest. In particular, we design combinatorial algorithms that provide instance-specific delay guarantees in linear preprocessing time. These algorithms improve upon the currently best known results. Further, we show how existing results can be improved upon by using fast matrix multiplication. We also present {new} results involving tradeoff between preprocessing time and delay guarantees for enumeration of path queries that contain projections. CQs with projection where the join attribute is projected away is equivalent to boolean matrix multiplication. Our results can therefore be also interpreted as sparse, output-sensitive matrix multiplication with delay guarantees
On Nondeterministic Derandomization of {F}reivalds' Algorithm: {C}onsequences, Avenues and Algorithmic Progress
Motivated by studying the power of randomness, certifying algorithms and barriers for fine-grained reductions, we investigate the question whether the multiplication of two matrices can be performed in near-optimal nondeterministic time . Since a classic algorithm due to Freivalds verifies correctness of matrix products probabilistically in time , our question is a relaxation of the open problem of derandomizing Freivalds' algorithm. We discuss consequences of a positive or negative resolution of this problem and provide potential avenues towards resolving it. Particularly, we show that sufficiently fast deterministic verifiers for 3SUM or univariate polynomial identity testing yield faster deterministic verifiers for matrix multiplication. Furthermore, we present the partial algorithmic progress that distinguishing whether an integer matrix product is correct or contains between 1 and erroneous entries can be performed in time -- interestingly, the difficult case of deterministic matrix product verification is not a problem of "finding a needle in the haystack", but rather cancellation effects in the presence of many errors. Our main technical contribution is a deterministic algorithm that corrects an integer matrix product containing at most errors in time . To obtain this result, we show how to compute an integer matrix product with at most nonzeroes in the same running time. This improves upon known deterministic output-sensitive integer matrix multiplication algorithms for nonzeroes, which is of independent interest
On Nondeterministic Derandomization of Freivalds\u27 Algorithm: Consequences, Avenues and Algorithmic Progress
Motivated by studying the power of randomness, certifying algorithms and barriers for fine-grained reductions, we investigate the question whether the multiplication of two n x n matrices can be performed in near-optimal nondeterministic time O~(n^2). Since a classic algorithm due to Freivalds verifies correctness of matrix products probabilistically in time O(n^2), our question is a relaxation of the open problem of derandomizing Freivalds\u27 algorithm.
We discuss consequences of a positive or negative resolution of this problem and provide potential avenues towards resolving it. Particularly, we show that sufficiently fast deterministic verifiers for 3SUM or univariate polynomial identity testing yield faster deterministic verifiers for matrix multiplication. Furthermore, we present the partial algorithmic progress that distinguishing whether an integer matrix product is correct or contains between 1 and n erroneous entries can be performed in time O~(n^2) - interestingly, the difficult case of deterministic matrix product verification is not a problem of "finding a needle in the haystack", but rather cancellation effects in the presence of many errors.
Our main technical contribution is a deterministic algorithm that corrects an integer matrix product containing at most t errors in time O~(sqrt{t} n^2 + t^2). To obtain this result, we show how to compute an integer matrix product with at most t nonzeroes in the same running time. This improves upon known deterministic output-sensitive integer matrix multiplication algorithms for t = Omega(n^{2/3}) nonzeroes, which is of independent interest
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
Computing the Boolean product of two n\times n Boolean matrices using O(n^2) mechanical operation
We study the problem of determining the Boolean product of two n\times n
Boolean matrices in an unconventional computational model allowing for
mechanical operations. We show that O(n^2) operations are sufficient to compute
the product in this model.Comment: 11 pages, 7 figure
Improving Quantum Query Complexity of Boolean Matrix Multiplication Using Graph Collision
The quantum query complexity of Boolean matrix multiplication is typically
studied as a function of the matrix dimension, n, as well as the number of 1s
in the output, \ell. We prove an upper bound of O (n\sqrt{\ell}) for all values
of \ell. This is an improvement over previous algorithms for all values of
\ell. On the other hand, we show that for any \eps < 1 and any \ell <= \eps
n^2, there is an \Omega(n\sqrt{\ell}) lower bound for this problem, showing
that our algorithm is essentially tight.
We first reduce Boolean matrix multiplication to several instances of graph
collision. We then provide an algorithm that takes advantage of the fact that
the underlying graph in all of our instances is very dense to find all graph
collisions efficiently
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