Lower bounds for merging networks

• Authors: Shigeki Iwata
• Venue: Information and Computation
• Year: 2001

A lower bound theorem is established for the number of comparators in a merging network. Let M(m, n) be the least number of comparators required in the (m, n)-merging networks, and let C(m, n) be the number of comparators in Batcher's (m, n)-merging network, respectively. We prove for n 隆脻 1 that M(4, n) = C(4, n) for n 2 mod 4, and M(5, = C(5, n) for n 0, 1, 5 mod 8. Futhermore Batcher's (6, 8k +6)-, (7, 8k + 7)-, and (8, 8k + 8)- merging networks are optimal for k 隆脻 0. Our lower bound for (m, n)- merging networks, m 隆Ü n, has the same terms as C(m, n) has as far as n is concerned. Thus Batcher's (m, n)-merging network is optimal up to a constant number of comparators, where the constant depends only on m. An open problem posed by Yao and Yao (Lower bounds on merging network, J. Assoc. Comput. Mach. 23, 566-571) is solved: limn---∞M (m, n)/n=log m/2+m/2[log m] Copyright 2001 Academic Press.