Lower Bounds on Merging Networks

Let <italic>M</italic>(<italic>m, n</italic>) be the minimum number or comparators needed in an (<italic>m, n</italic>)-merging network. It is shown that <italic>M</italic>(<italic>m, n</italic>) ≥ <italic>n</italic>(lg(<italic>m</italic> + 1))/2, which implies that Batcher's merging networks are optimal up to a factor of 2 + ε for almost all values of <italic>m</italic> and <italic>n</italic>. The limit <italic>r</italic><subscrpt>m</subscrpt> = lim<subscrpt>n→∞</subscrpt> <italic>M</italic>(<italic>m, n</italic>)/<italic>n</italic> is determined to within 1. It is also proved that <italic>M</italic>(2, <italic>n</italic>) = [3<italic>n</italic>/2].