An algorithm is presented within the context of the calculation of the time-relaxation behavior of the hydration shells around atomic sites in biomolecules. We report a calculation of the time-relaxation behavior of the first and second hydration shells of polar, hydrophobic, and charged groups in a protein, crambin. The water mean residence times around protein groups are obtained from averages over configurations sampled during a 325-ps molecular dynamics simulation of crambin in solution. A convolution arising in the calculation of the mean relaxation time is implemented using a parallel prefix operator. A new characterization is given of the parallel prefii operator as a linear transformation, and this formulation enables us to derive efficient factorization of the convolution as a product of two parallel prefix operations. The parallel prefix operations are implemented in logarithmic time.
We reproduce the author's abstract: "This thesis describes techniques for the design of parallel programs that solve well-structured problems with inherent symmetry.Part I demonstrates the reduction of such problems to generalized matrix multiplication by a group-equivariant matrix. Fast techniques for this multiplication are described, including factorization, orbit decomposition, and Fourier transforms over finite groups. Our algorithms entail interaction between two symmetry groups: one arising at the software level from the problem's symmetry and the other arising at the hardware level from the processors' communication network.Part II illustrates the applicability of our symmetry-exploitation techniques by presenting a series of case studies of the design and implementation of parallel programs.First, a parallel program that solves chess endgames by factorization of an associated dihedral group-equivariant matrix is described. This code runs faster than previous serial programs, and discovered a number of new results.Second, parallel algorithms for Fourier transforms for finite groups are developed, and preliminary parallel implementations for group transforms of dihedral and of symmetric groups are described. Applications in learning, vision, pattern recognition, and statistics arc proposed.Third, parallel implementations solving several computational-science problems are described, including the direct n-body problem, convolutions arising from molecular biology, and some communication primitives such as broadcast and reduce. Some of our implementations ran orders of magnitude faster than previous techniques, and were used in the investigation of various physical phenomena."The thesis is available by anonymous ftp from ftp.cs.jhu.edu:pub/stiller/thesis/thesis-600dpi.ps, with a stated restriction that it will only print out properly on 600 dpi printers, for which double-sided printing is recommended. The directory named also contains a file thesis-300dpi.ps, suitable for previewing only.
Efficient space and time exploitation of symmetry in domains on highly parallel, distributed-memory architecture is, in certain cases, equivalent to routing along a labeled group action graph, with computation associated with each group element label, where the group of symmetries acts on the processors. The algebraic structure of the group can sometimes be analyzed to determine, a priori, space and time efficient routing schedules on the hardware network (which, in practice, is often another group action graph). The algorithms we develop were implemented on a 64K-processor CM-2 and used to solve certain natural classes of chess endgames, part of whose search space is invariant under a noncommutative crystallographic group. This program runs 400 times faster than any previous implementation, and discovered many interesting new results in the area; some of these results are not solvable in practice with current serial techniques because the time and space requirements are too large. It seems interesting that it was possible, albeit with difficulty, to implement efficiently certain irregular chess rules on the CM-2, which is optimized for regular data sets.
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