In this paper we propose algorithms for generation of frequent itemsets by successive construction of the nodes of a lexicographic tree of itemsets. We discuss di erent strategies in generation and traversal of the lexicographic tree such as breadth-rst search, depth-rst search or a combination of the two. These techniques provide di erent trade-o s in terms of the I O, memory and computational time requirements. We use the hierarchical structure of the lexicographic tree to successively project transactions at each node of the lexicographic tree, and use matrix counting on this reduced set of transactions for nding frequent itemsets. We tested our algorithm on both real and synthetic data. We provide an implementation of the tree projection method which is up to one order of magnitude faster than other recent techniques in the literature. The algorithm has a well structured data access pattern which provides data locality and reuse of data for multiple levels of the cache. We also discuss methods for parallelization of the TreeProjection algorithm.

A new atomic-parameters least-squares refinement method is presented which makes use of the fast Fourier transform algorithm at all stages of the computation. For large structures, the amount of computation is almost proportional to the size of the structure making it very attractive for large biological structures such as proteins. In addition the method has a radius of convergence of approximately 0.75/~, making it applicable at a very early stage of the structure-determination process. The method has been tested on hypothetical as well as real structures. The method has been used to refine the structure of insulin at 1.5/~ resolution, barium beauvuricin complex at 1.2/~ resolution, and myoglobin at 2 A resolution. Details of the method and brief summaries of its applications are presented in the paper.

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A three-dimensional (3D) matrix multiplication algorithm for massively parallel processing systems is presented. The P processors are con gured as a \virtual" processing cube with dimensions p 1 , p 2 , and p 3 proportional to the matrices' dimensions|M, N, and K. Each processor performs a single local matrix multiplication of size M=p 1 N=p 2 K=p 3. Before the local computation can be carried out, each subcube must receive a single submatrix of A and B. After the single matrix multiplication has completed, K=p 3 submatrices of this product must be sent to their respective destination processors and then summed together with the resulting matrix C. The 3D parallel matrix multiplication approach has a factor P 1=6 less communication than the 2D parallel algorithms. This algorithm has been implemented on IBM POWERparallel T M SP2 T M systems (up to 216 nodes) and has yielded close to the peak performance of the machine. The algorithm has been combined with Winograd's variant of Strassen's algorithm to achieve performance which exceeds the theoretical peak of the system. (We assume the MFLOPS rate of matrix multiplication to be 2MNK.

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