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<p>Abstract—We introduce a new class of parallel algorithms for the exact computation of systems with pairwise mutual interactions of <it>n</it> elements, so called <it>n</it><super>2</super>-problems. Hitherto, practical conventional parallelization strategies could achieve a complexity of <it>O</it>(<it>np</it>) with respect to the inter-processor communication, <it>p</it> being the number of processors. Our new approach can reduce the inter-processor communication complexity to a number <tmath>$O(np^{{{1 \over 2}}}).$</tmath> In the framework of Additive Number Theory, the determination of the optimal communication pattern can be formulated as <it>h</it>-range minimization problem that can be solved numerically. Based on a complexity model, the scaling behavior of the new algorithm is numerically tested on the connection machine CM5. As a real life example, we have implemented a fast code for globular cluster <it>n</it>-body simulations, a generic <it>n</it><super>2</super>-problem, on the CRAY T3D, with striking success. Our parallel method promises to be useful in various scientific and engineering fields like polymer chain computations, protein folding, signal processing, and, in particular, for parallel level-3 BLAS.</p>
Systolic algorithm, hyper-systolic algorithm, n-body computation, n2-loop computation, parallel computer, connection machine CM5 and Cray T3D, novel complexity class.

A. Bode, K. Schilling, A. Seyfried and T. Lippert, "Hyper-Systolic Parallel Computing," in IEEE Transactions on Parallel & Distributed Systems, vol. 9, no. , pp. 97-108, 1998.
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