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<p><b>Abstract</b>—Sorting is a fundamental problem with applications in all areas of computer science and engineering. In this work, we address the problem of sorting on mesh connected computers enhanced by endowing each row and each column with its own dedicated high-speed bus. This architecture, commonly referred to as a <it>mesh with multiple broadcasting</it>, is commercially available and has been adopted by the DAP family of multiprocessors. Somewhat surprisingly, the problem of sorting <it>m</it>, (<it>m</it>≤<it>n</it>), elements on a mesh with multiple broadcasting of size <tmath>$\sqrt n\times \sqrt n$</tmath> has been studied, thus far, only in the sparse case, where <tmath>$m\in \Theta \left( {\sqrt n} \right)$</tmath> and in the <it>dense</it> case, where <it>m</it>∈Θ(<it>n</it>). Yet, many applications require using an existing platform of size <tmath>$\sqrt n\times \sqrt n$</tmath> for sorting <it>m</it> elements, with <tmath>$\sqrt n<m\le n.$</tmath> Our main contribution is to present the first known <it>adaptive</it> time- and VLSI-optimal sorting algorithm for meshes with multiple broadcasting. Specifically, we show that, for every choice of a constant <tmath>$0<\epsilon\le {{1 \over 2}},$</tmath> it is possible to sort <it>m</it> elements, <tmath>$n^{{{1 \over 2}}+\epsilon}\le m\le n,$</tmath> stored in the leftmost <tmath>$\left\lceil {{{m \over {\sqrt n}}}} \right\rceil$</tmath> columns of a mesh with multiple broadcasting of size <tmath>$\sqrt n\times \sqrt n$</tmath> in <tmath>$\Theta \left( {{{m \over {\sqrt n}}}} \right)$</tmath> time.</p>
Parallel algorithms, sorting, lower bounds, VLSI-optimality, time-optimality, meshes with multiple broadcasting.

S. Olariu, L. Wilson, J. Zhang, J. L. Schwing, H. Gurla and D. Bhagavathi, "Time- and VLSI-Optimal Sorting on Enhanced Meshes," in IEEE Transactions on Parallel & Distributed Systems, vol. 9, no. , pp. 929-937, 1998.
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