24,454 research outputs found
The I/O Complexity of Hybrid Algorithms for Square Matrix Multiplication
Asymptotically tight lower bounds are derived for the I/O complexity of a general class of hybrid algorithms computing the product of n x n square matrices combining "Strassen-like" fast matrix multiplication approach with computational complexity Theta(n^{log_2 7}), and "standard" matrix multiplication algorithms with computational complexity Omega (n^3). We present a novel and tight Omega ((n/max{sqrt M, n_0})^{log_2 7}(max{1,(n_0)/M})^3M) lower bound for the I/O complexity of a class of "uniform, non-stationary" hybrid algorithms when executed in a two-level storage hierarchy with M words of fast memory, where n_0 denotes the threshold size of sub-problems which are computed using standard algorithms with algebraic complexity Omega (n^3).
The lower bound is actually derived for the more general class of "non-uniform, non-stationary" hybrid algorithms which allow recursive calls to have a different structure, even when they refer to the multiplication of matrices of the same size and in the same recursive level, although the quantitative expressions become more involved. Our results are the first I/O lower bounds for these classes of hybrid algorithms. All presented lower bounds apply even if the recomputation of partial results is allowed and are asymptotically tight.
The proof technique combines the analysis of the Grigoriev\u27s flow of the matrix multiplication function, combinatorial properties of the encoding functions used by fast Strassen-like algorithms, and an application of the Loomis-Whitney geometric theorem for the analysis of standard matrix multiplication algorithms. Extensions of the lower bounds for a parallel model with P processors are also discussed
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{}
LINVIEW: Incremental View Maintenance for Complex Analytical Queries
Many analytics tasks and machine learning problems can be naturally expressed
by iterative linear algebra programs. In this paper, we study the incremental
view maintenance problem for such complex analytical queries. We develop a
framework, called LINVIEW, for capturing deltas of linear algebra programs and
understanding their computational cost. Linear algebra operations tend to cause
an avalanche effect where even very local changes to the input matrices spread
out and infect all of the intermediate results and the final view, causing
incremental view maintenance to lose its performance benefit over
re-evaluation. We develop techniques based on matrix factorizations to contain
such epidemics of change. As a consequence, our techniques make incremental
view maintenance of linear algebra practical and usually substantially cheaper
than re-evaluation. We show, both analytically and experimentally, the
usefulness of these techniques when applied to standard analytics tasks. Our
evaluation demonstrates the efficiency of LINVIEW in generating parallel
incremental programs that outperform re-evaluation techniques by more than an
order of magnitude.Comment: 14 pages, SIGMO
Computational linear algebra over finite fields
We present here algorithms for efficient computation of linear algebra
problems over finite fields
Communication-optimal Parallel and Sequential Cholesky Decomposition
Numerical algorithms have two kinds of costs: arithmetic and communication,
by which we mean either moving data between levels of a memory hierarchy (in
the sequential case) or over a network connecting processors (in the parallel
case). Communication costs often dominate arithmetic costs, so it is of
interest to design algorithms minimizing communication. In this paper we first
extend known lower bounds on the communication cost (both for bandwidth and for
latency) of conventional (O(n^3)) matrix multiplication to Cholesky
factorization, which is used for solving dense symmetric positive definite
linear systems. Second, we compare the costs of various Cholesky decomposition
implementations to these lower bounds and identify the algorithms and data
structures that attain them. In the sequential case, we consider both the
two-level and hierarchical memory models. Combined with prior results in [13,
14, 15], this gives a set of communication-optimal algorithms for O(n^3)
implementations of the three basic factorizations of dense linear algebra: LU
with pivoting, QR and Cholesky. But it goes beyond this prior work on
sequential LU by optimizing communication for any number of levels of memory
hierarchy.Comment: 29 pages, 2 tables, 6 figure
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