On Optimal Merging Networks

Kazuyuki Amano? and Akira Maruoka Graduate S hool of Information S ien es, Tohoku University Aoba 05, Aramaki, Sendai 980-8579 JAPAN fama|maruokag e ei.tohoku.a .jp Tel:+81-22-217-7149, Fax:+81-22-263-9414 Abstra t. We prove that Bat her's odd-even (m;n)-merging networks are exa tly optimal for (m;n) = (3; 4k + 2) and (4; 4k + 2) for k 0 in terms of the number of omparators used. For other ases where m 4, the optimality of Bat her's (m;n)-merging networks has been proved. So we an on lude that Bat her's odd-even merge yields optimal (m;n)-merging networks for every m 4 and for every n. The ru ial part of the proof is hara terizing the stru ture of optimal (2; n)-merging networks. 1 Introdu tion A omparator network, whi h onsists of omparators, has been widely investigated as a model of oblivious omparison-based algorithm. A omparator is a 2-input 2-output gate, where one output omputes the minimum of the two inputs and the other output omputes the maximum (see Fig. 1). An (m;n)-merging network is a omparator netx