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33
Communicationoptimal parallel algorithm for Strassen’s matrix multiplication
 In Proceedings of the 24th ACM Symposium on Parallelism in Algorithms and Architectures, SPAA ’12
, 2012
"... Parallel matrix multiplication is one of the most studied fundamental problems in distributed and high performance computing. We obtain a new parallel algorithm that is based on Strassen’s fast matrix multiplication and minimizes communication. The algorithm outperforms all known parallel matrix mul ..."
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Cited by 28 (17 self)
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Parallel matrix multiplication is one of the most studied fundamental problems in distributed and high performance computing. We obtain a new parallel algorithm that is based on Strassen’s fast matrix multiplication and minimizes communication. The algorithm outperforms all known parallel matrix multiplication algorithms, classical and Strassenbased, both asymptotically and in practice. A critical bottleneck in parallelizing Strassen’s algorithm is the communication between the processors. Ballard, Demmel, Holtz, and Schwartz (SPAA’11) prove lower bounds on these communication costs, using expansion properties of the underlying computation graph. Our algorithm matches these lower bounds, and so is communicationoptimal. It exhibits perfect strong scaling within the maximum possible range.
New lower bounds for the border rank of matrix multiplication
"... Abstract. The border rank of the matrix multiplication operator for n × n matrices is a standard measure of its complexity. Using techniques from algebraic geometry and representation theory, we show the border rank is at least 2n 2 − n. Our bounds are better than the previous lower bound (due to Li ..."
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Cited by 15 (7 self)
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Abstract. The border rank of the matrix multiplication operator for n × n matrices is a standard measure of its complexity. Using techniques from algebraic geometry and representation theory, we show the border rank is at least 2n 2 − n. Our bounds are better than the previous lower bound (due to Lickteig in 1985) of 3 2 n2 + n 2 − 1 for all n ≥ 3. The bounds are obtained by finding new equations that bilinear maps of small border rank must satisfy, i.e., new equations for secant varieties of triple Segre products, that matrix multiplication fails to satisfy.
Destination Prediction by SubTrajectory Synthesis and Privacy Protection Against Such Prediction
"... Abstract — Destination prediction is an essential task for many emerging location based applications such as recommending sightseeing places and targeted advertising based on destination. A common approach to destination prediction is to derive the probability of a location being the destination bas ..."
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Cited by 15 (6 self)
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Abstract — Destination prediction is an essential task for many emerging location based applications such as recommending sightseeing places and targeted advertising based on destination. A common approach to destination prediction is to derive the probability of a location being the destination based on historical trajectories. However, existing techniques using this approach suffer from the “data sparsity problem”, i.e., the available historical trajectories is far from being able to cover all possible trajectories. This problem considerably limits the number of query trajectories that can obtain predicted destinations. We propose a novel method named SubTrajectory Synthesis (SubSyn) algorithm to address the data sparsity problem. SubSyn algorithm first decomposes historical trajectories into subtrajectories comprising two neighbouring locations, and then connects the subtrajectories into “synthesised ” trajectories. The number of query trajectories that can have predicted destinations is exponentially increased by this means. Experiments based on real datasets show that SubSyn algorithm can predict destinations for up to ten times more query trajectories than a baseline algorithm while the SubSyn prediction algorithm runs over two orders of magnitude faster than the baseline algorithm. In this paper, we also consider the privacy protection issue in case an adversary uses SubSyn algorithm to derive sensitive location information of users. We propose an efficient algorithm to select a minimum number of locations a user has to hide on her trajectory in order to avoid privacy leak. Experiments also validate the high efficiency of the privacy protection algorithm. I.
On the Complexity of Solving quadratic boolean systems
, 2012
"... A fundamental problem in computer science is to find all the common zeroes of m quadratic polynomials in n unknowns over F2. The cryptanalysis of several modern ciphers reduces to this problem. Up to now, the best complexity bound was reached by an exhaustive search in 4log 2 n2 n operations. We giv ..."
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Cited by 13 (3 self)
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A fundamental problem in computer science is to find all the common zeroes of m quadratic polynomials in n unknowns over F2. The cryptanalysis of several modern ciphers reduces to this problem. Up to now, the best complexity bound was reached by an exhaustive search in 4log 2 n2 n operations. We give an algorithm that reduces the problem to a combination of exhaustive search and sparse linear algebra. This algorithm has several variants depending on the method used for the linear algebra step. Under precise algebraic assumptions on the input system, we show that the deterministic variant of our algorithm has complexity bounded by O(2 0.841n) when m = n, while a probabilistic variant of the Las Vegas type has expected complexity O(2 0.792n). Experiments on random systems show that the algebraic assumptions are satisfied with probability very close to 1. We also give a rough estimate for the actual threshold between our method and exhaustive search, which is as low as 200, and thus very relevant for cryptographic applications.
A fast solver for a class of linear systems
, 2008
"... The solution of linear systems is a problem of fundamental theoretical importance but also one with a myriad of applications in numerical mathematics, engineering and science. Linear systems that are generated by realworld applications frequently fall into special classes. Recent research led to a ..."
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Cited by 7 (3 self)
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The solution of linear systems is a problem of fundamental theoretical importance but also one with a myriad of applications in numerical mathematics, engineering and science. Linear systems that are generated by realworld applications frequently fall into special classes. Recent research led to a fast algorithm for solving symmetric diagonally dominant (SDD) linear systems. We give an overview of this solver and survey the underlying notions and tools from algebra, probability and graph algorithms. We also discuss some of the many and diverse applications of SDD solvers.
Fast computation of common left multiples of linear ordinary differential operators
 in Proceedings of ISSAC’12
, 2012
"... We study tight bounds and fast algorithms for LCLMs of several linear differential operators with polynomial coefficients. We analyse the arithmetic complexity of existing algorithms for LCLMs, as well as the size of their outputs. We propose a new algorithm that recasts the LCLM computation in a li ..."
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Cited by 6 (1 self)
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We study tight bounds and fast algorithms for LCLMs of several linear differential operators with polynomial coefficients. We analyse the arithmetic complexity of existing algorithms for LCLMs, as well as the size of their outputs. We propose a new algorithm that recasts the LCLM computation in a linear algebra problem on a polynomial matrix. This algorithm yields sharp bounds on the coefficient degrees of the LCLM, improving by one order of magnitude the best bounds obtained using previous algorithms. The complexity of the new algorithm is almost optimal, in the sense that it nearly matches the arithmetic size of the output. Categories and Subject Descriptors:
Algebraic algorithms
"... This article, along with [Elkadi and Mourrain 1996], explain the correlation between residue theory and the Dixon matrix, which yields an alternative method for studying and approximating all common solutions. In 1916, Macaulay [1916] constructed a matrix whose determinant is a multiple of the class ..."
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Cited by 4 (0 self)
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This article, along with [Elkadi and Mourrain 1996], explain the correlation between residue theory and the Dixon matrix, which yields an alternative method for studying and approximating all common solutions. In 1916, Macaulay [1916] constructed a matrix whose determinant is a multiple of the classical resultant for n homogeneous polynomials in n variables. The Macaulay matrix si16 multaneously generalizes the Sylvester matrix and the coefficient matrix of a system of linear equations [Kapur and Lakshman Y. N. 1992]. As the Dixon formulation, the Macaulay determinant is a multiple of the resultant. Macaulay, however, proved that a certain minor of his matrix divides the matrix determinant so as to yield the exact resultant in the case of generic homogeneous polynomials. Canny [1990] has invented a general method that perturbs any polynomial system and extracts a nontrivial projection operator. Using recent results pertaining to sparse polynomial systems [Gelfand et al. 1994, Sturmfels 1991], a matrix formula for computing the sparse resultant of n + 1 polynomials in n variables was given by Canny and Emiris [1993] and consequently improved in [Canny and Pedersen 1993, Emiris and Canny 1995]. The determinant of the sparse resultant matrix, like the Macaulay and Dixon matrices, only yields a projection operation, not the exact resultant. Here, sparsity means that only certain monomials in each of the n + 1 polynomials have nonzero coefficients. Sparsity is measured in geometric terms, namely, by the Newton polytope
Algorithms for the universal decomposition algebra
"... Let k be a field and let f ∈ k [T] be a polynomial of degree n. The universal decomposition algebra A is the quotient of k [X1,..., Xn] by the ideal of symmetric relations (those polynomials that vanish on all permutations of the roots of f). We show how to obtain efficient algorithms to compute in ..."
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Let k be a field and let f ∈ k [T] be a polynomial of degree n. The universal decomposition algebra A is the quotient of k [X1,..., Xn] by the ideal of symmetric relations (those polynomials that vanish on all permutations of the roots of f). We show how to obtain efficient algorithms to compute in A. We use a univariate representation of A, i.e. an isomorphism of the form A ≃ k[T]/Q(T), since in this representation, arithmetic operations in A are known to be quasioptimal. We give details for two related algorithms, to find the isomorphism above, and to compute the characteristic polynomial of any element of A.
A FAST ALGORITHM FOR REVERSION OF POWER SERIES
, 1108
"... Abstract. We give an algorithm for reversion of formal power series, based on an efficient way to evaluate the Lagrange inversion formula. Our algorithm requires O(n 1/2 (M(n)+MM(n 1/2))) operations where M(n) and MM(n) are the costs of polynomial and matrix multiplication respectively. This matches ..."
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Abstract. We give an algorithm for reversion of formal power series, based on an efficient way to evaluate the Lagrange inversion formula. Our algorithm requires O(n 1/2 (M(n)+MM(n 1/2))) operations where M(n) and MM(n) are the costs of polynomial and matrix multiplication respectively. This matches an algorithm of Brent and Kung, but we achieve a constant factor speedup whose magnitude depends on the polynomial and matrix multiplication algorithms used. Benchmarks confirm that the algorithm performs well in practice. 1.