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Paper submission to PAAM’99.

In search for general equilibrium in multi-commodity markets, price-oriented schemes are normally used. That is, a set of prices (one price for each commodity) is updated until supply meets demand for each commodity. In some cases such an approach is very inef cient, and a resource-oriented scheme can be highly competitive. In a resource-oriented scheme the allocations are updated until the market equilibrium is found. It is well known that in a two-commodity market resource-oriented schemes are possible. In this paper we show that resource-oriented algorithms can be used for the general multi-commodity case as well, and present and analyze a speci c algorithm. The algorithm has been implemented and some performance properties, for a speci c example, are presented. 1

We present a parallel method for matrix multiplication on distributedmemory MIMD architectures based on Strassen's method. Our timing tests, performed on an Intel Paragon, demonstrate that our method realizes the potential of the Strassen's method with a complexity of 4:7M 2:807 at the system level rather than the node level at which several earlier works have been focused. The parallel efficiency is nearly perfect when the processor number is divisible by 7. The parallelized Strassen's method is always faster than the traditional matrix multiplication methods whose complexity is 2M 3 coupled with the BMR method and the Ring method at the system level. The speed gain depends on matrix order M : 20% for M ß 1000 and more than 100% for M ß 5000. Key words: matrix multiplication, parallel computation, Strassen's method AMS (MOS) Subject Classification: 65F30, 65Y05, 68Q25 Submitted to SIAM Journal on Scientific Computing y To whom correspondence should be sent. His email address...

The URLs given were last checked and found valid in November 2001. To Alla, for all her help and care To Lady Mathematics, for all the fun and moments of enlightenment iv Acknowledgments First and foremost I want to thank my advisor and friend, Olof Widlund, for proposing the thesis subject, and for all his support and help in the last six years, four of them as his student. I also want to thank Yu Chen and Jonathan Goodman for their willingness to serve as readers on short notice. I thank all the faculty, staff, and students of the Courant Institute who contributed to create a warm and motivating atmosphere. I thank all the professors who transmitted some of their excitement about mathematics to me, I especially want to mention John Rinzel and Bud Mishra. I also want to thank all who fed my neverending interest in all things mathematical and who were as curious as me about mathematics, physics and all that. Thank you, Sávio and Franz. I have had the privilege of becoming very good friends with four fantastic people during my several

In computational markets utilizing algorithms that establish a general equilibrium, competitive behavior is usually assumed: each agent makes its demand (supply) decisions so as to maximize its utility (profit) assuming that it has no impact on market prices. However, there is a potential gain from strategic behavior via speculating about others because an agent does affect the market prices, which affect the supply/demand decisions of others, which again affect the market prices that the agent faces. Determining the optimal strategy when the speculator has perfect knowledge about the other agents is a well known problem which has been studied in oligopoly theory in economics. We describe the computation of such a strategy, and focus on an issue that has received little attention in economics, but which is of fundamental importance in computational markets: the revelation of demand strategies that drive the market to the desired equilibrium. The more

Problems related to computing optimal paths have been abundant in computer science since its emergence as a field. Yet for a large number of such problems we still do not know whether the state-of-the-art algorithms are the best possible. A notable example of this phenomenon is the all pairs shortest paths problem in a directed graph with real edge weights. The best algorithm (modulo small polylogarithmic improvements) for this problem runs in cubic time, a running time known since the 1960s (by Floyd and Warshall). Our grasp of many such fundamental algorithmic questions is far from optimal, and the major goal of this thesis is to bring some new insights into efficiently solving path problems in graphs. We focus on several path problems optimizing different measures: shortest paths, maximum bottleneck paths, minimum nondecreasing paths, and various extensions. For the all-pairs versions of these path problems we use an algebraic approach. We obtain improved algorithms using reductions

Modern architectures have complex memory hierarchies and increasing parallelism (e.g., multicores). These features make achieving and maintaining good performance across rapidly changing architectures increasingly difficult. Performance has become a complex trade-off, not just a simple matter of counting cost of simple CPU operations. We present a novel, hybrid, and adaptive recursive Strassen-Winograd’s matrix multiplication (MM) that uses automatically tuned linear algebra software (ATLAS) or GotoBLAS. Our algorithm applies to any size and shape matrices stored in either row or column major layout (in double-precision in this work) and thus is efficiently applicable to both C and FORTRAN implementations. In addition, our algorithm divides the computation into equivalent in-complexity sub-MMs and does not require any extra computation to combine the intermediary sub-MM results. We achieve up to 22 % execution-time reduction versus GotoBLAS/ATLAS alone for a single core system and up to 19 % for a 2 dual-core processor system. Most importantly, even for small matrices such as 1500×1500, our approach attains already 10 % execution-time reduction and, for MM of matrices larger than 3000×3000, it delivers performance that would correspond, for a classic O(n3) algorithm, to faster-than-processor peak performance (i.e., our algorithm delivers the equivalent of 5 GFLOPS performance on a system with 4.4 GFLOPS peak performance and where GotoBLAS achieves only 4 GFLOPS). This is a result of the savings in operations (and thus FLOPS). Therefore, our algorithm is faster than any classic MM algorithms could ever be for matrices of this size. Furthermore, we present experimental evidence based on established methodologies found in the literature that our algorithm is, for a family of matrices, as accurate as the classic 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 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 non-trivial 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 non-zero coefficients. Sparsity is measured in geometric terms, namely, by the Newton polytope

The aim of this survey is to show that Number Theoretical Transforms (NTTs) can provide real benefits in the form of error-free computation and reduced computational complexity. Several practical applications that can benefit from the use of NTTs are identified and some pitfalls to watch out for are pointed out. Keywords--- Finite rings, finite fields, NTTs. I. Introduction NTTs are discrete Fourier transforms, defined over finite fields or rings. The problems experienced by early investigators lead to NTTs gaining a reputation for being difficult to design and apply. This is unfortunate, as the benefits gained from their error-free properties, as well as the potentially lower complexity, can be of great benefit to a number of applications. In the sequel several examples of such applications are surveyed. The space available prevents an exhaustive treatment, though. A generic forward and inverse NTT par is defined by X(k) = h N \Gamma1 X n=0 x(n)ff nk i M ; k = 0; 1; : : : ; ...