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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 27 (19 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.
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 13 (5 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.
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 5 (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:
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 4 (2 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.
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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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.
Cryptography from tensor problems
, 2012
"... We describe a new proposal for a trapdoor oneway function. The new proposal belongs to the “multivariate quadratic” family but the trapdoor is different from existing methods, and is simpler. Known quantum algorithms do not appear to help an adversary attack this trapdoor. (Beyond the asymptotic ..."
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We describe a new proposal for a trapdoor oneway function. The new proposal belongs to the “multivariate quadratic” family but the trapdoor is different from existing methods, and is simpler. Known quantum algorithms do not appear to help an adversary attack this trapdoor. (Beyond the asymptotic squarerootspeedup which applies to all oracle search problems.)
Algebraic Algorithms for Matching
, 2011
"... Given a set of nodes and edges between them, what’s the maximum of number of disjoint edges? This problem is known as the graph matching problem, and its study has had an enormous impact on the develpoment of algorithms, combinatorics, optimization theory, and even complexity theory. Mathematicians ..."
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Given a set of nodes and edges between them, what’s the maximum of number of disjoint edges? This problem is known as the graph matching problem, and its study has had an enormous impact on the develpoment of algorithms, combinatorics, optimization theory, and even complexity theory. Mathematicians have been interested in the matching problem since the 19th century, leading to celebrated theorems in graph theory due to Tutte, Menger, König, and
unknown title
, 2012
"... These notes are based on a lecture given at the Toronto Student Seminar on February 9, 2012. The material is taken mostly from the classic paper by Coppersmith and Winograd [CW]. Other sources are §15.7 of Algebraic Complexity Theory [ACT], Stothers’s thesis [Sto] and V. Williams’s recent paper [Wil ..."
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These notes are based on a lecture given at the Toronto Student Seminar on February 9, 2012. The material is taken mostly from the classic paper by Coppersmith and Winograd [CW]. Other sources are §15.7 of Algebraic Complexity Theory [ACT], Stothers’s thesis [Sto] and V. Williams’s recent paper [Wil]. Starred sections are the ones we didn’t have time to cover. We present three different algorithms, all taken from [CW], in rapid succession. All these algorithms are based on Strassen’s groundbreaking laser method. Strassen’s original ideas are described in the appendix.