We prove that computing the product of two n × n matrices over the binary field requires at least 2.5 n 2 - o ( n 2 ) multiplications. Key Words : matrix multiplication, arithmetic complexity, lower bounds, linear codes. 1. INTRODUCTION Let x = ( x 1 , . . . , x n ) T and y = ( y 1 , . . . , y m ) T be column vectors of indeterminates. A straight-line algorithm for computing a set of bilinear forms in x and y is called quadratic ( respectively bilinear ), if all its non-scalar multiplication are of the shape l ( x , y ) . l ( x , y ) , (respectively l ( x ) . l ( y ) ) where l and l are linear forms of the indeterminates. 1 In this paper we establish the new 2.5 n 2 - o ( n 2 ) lower bound on the multiplicative complexity of quadratic algorithms for multiplying n × n matrices over the binary field Z 2 . Let M F ( n , m , k ) and M ## F ( n , m , k ) denote the number of multiplications required to compute the product of n ×m and m ×k matrices by means of quadratic ...

AbstraCt The paper considers the complexity of bilinear forms in a noncommutative ring. The dual of a computation is defined and applied to matrix multiplication and other bilinear forms. It is shown that the dual of an optimal computation gives an optimal computation for a dual problem. An nxm by mxp matrix product is shown to be the dual of an nxp by pxm or an mxn by nxp matrix product implying that each of the matrix products requires the same number of multiplications to compute. Finally an algorithm for computing a single bilinear form over a noncommutative ring with a minimum number of multiplications is derived by considering a dual problem.

Abstract. We survey results in algebraic complexity theory, focusing on matrix multiplication. Our goals are (i) to show how open questions in algebraic complexity theory are naturally posed as questions in geometry and representation theory, (ii) to motivate researchers to work on these questions, and (iii) to point out relations with more general problems in geometry. The key geometric objects for our study are the secant varieties of Segre varieties. We explain how these varieties are also useful for algebraic statistics, the study of phylogenetic invariants, and quantum computing.

The model of bulk-synchronous parallel (BSP) computation is an emerging paradigm of general-purpose parallel computing. This thesis presents a systematic approach to the design and analysis of BSP algorithms. We introduce an extension of the BSP model, called BSPRAM, which reconciles shared-memory style programming with efficient exploitation of data locality. The BSPRAM model can be optimally simulated by a BSP computer for a broad range of algorithms possessing certain characteristic properties: obliviousness, slackness, granularity. We use BSPRAM to design BSP algorithms for problems from three large, partially overlapping domains: combinatorial computation, dense matrix computation, graph computation. Some of the presented algorithms are adapted from known BSP algorithms (butterfly dag computation, cube dag computation, matrix multiplication). Other algorithms are obtained by application of established non-BSP techniques (sorting, randomised list contraction, Gaussian elimination without pivoting and with column pivoting, algebraic path computation), or use original techniques specific to the BSP model (deterministic list contraction, Gaussian elimination with nested block pivoting, communication-efficient multiplication of Boolean matrices, synchronisation-efficient shortest paths computation). The asymptotic BSP cost of each algorithm is established, along with its BSPRAM characteristics. We conclude by outlining some directions for future research.

ABSTRACT: Limiting consideration to algorithms satisfying various numerical stability requirements may change lower bounds for computational complexity and/or make lower bounds easier to prove. We will show that, under a sufficiently strong restriction upon numerical stability, any algorithm for multiplying two n x n matrices using only +,- and x requires at least n 3 multiplications. We conclude with a survey of results concerning the numerical stability of several algorithms which have been considered by complexity theorists. I.

this paper we propose a generic symmetry-breaking schema in a DLL procedure [5], inspired from the approach of Brown, Finkelstein and Purdom. Like their approach, we exploit symmetries in every node of a search tree so that only one object in every equivalence class is searched by the DLL procedure. Unlike their approach, we are in propositional reasoning case and don't use results of computational group theory. We present our approach using two This work is supported by French CNRS under grant number SUB/2001/0111/DR16 149 hard search problems containing numerous symmetries. The experimental results show that even simple approaches to symmetry breaking can save a lot of time

The communication cost of algorithms (also known as I/Ocomplexity) is shown to be closely related to the expansion properties of the corresponding computation graphs. We demonstrate this on Strassen’s and other fast matrix multiplication algorithms, and obtain the first lower bounds on their communication costs. For sequential algorithms these bounds are attainable and so optimal. 1.

One of the leading problems of algebraic complexity theory is matrix multiplication. The naïve multiplication of two n × n matrices uses n 3 multiplications. In 1969, Strassen [20] presented an explicit algorithm for multiplying 2 × 2 matrices using seven multiplications. In the opposite direction, Hopcroft and Kerr [12] and

L’évaluation numérique à grande précision de constantes comme π, e, γ, ln 2, etc., de fonctions élémentaires comme exp et arctan, puis de fonctions spéciales comme Γ ou ζ est un problème classique. D’un point de vue informatique, « grande précision » s’oppose à précision fixe, mais sous-entend aussi que l’on cherche des algorithmes asymptotiquement efficaces quand le nombre de chiffres demandés grandit. Le développement d’algorithmes de complexité quasilinéaire en le nombre de chiffres du résultat remonte aux années 1970, avec par exemple les travaux de Richard Brent, Eugene Salamin ou R. William Gosper. Les fonctions holonomes sont les solutions d’équations différentielles linéaires à coefficients polynomiaux. Leurs propriétés élémentaires sont bien connues depuis le dix-neuvième siècle, mais elles ont pris une place importante en combinatoire (comme séries génératrices) et en calcul formel (en tant que classe de fonctions bénéficiant de propriétés algorithmiques agréables, tant du point de vue de la calculabilité que de celui de la complexité) depuis les années 1980. Parmi les responsables de ce regain d’intérêt, on peut citer Richard Stanley, Leonard Lipshitz et Doron Zeilberger. Mon travail de stage s’incrit dans une démarche générale du projet Algo de développer pour toute la classe des fonctions holonomes une algorithmique efficace utilisant

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