Towards Gossiping in 2 1/4 k on Butterflies

``regular'' butterflies (which have no wrap-around connections).
In the telephone communication model, for a butterfly of order $k$,
the best previous algorithms require about $2 1/2 * k$
and $3 * k$ communication rounds, respectively. By new asymptotic
methods we break through these bounds. We show that gossiping on a
class of ``column-based'' networks, which also contains CCCs, can be
reduced to the simpler problem of ``row-gossiping''. Row-gossiping
in turn is reduced to a special kind of broadcasting. This latter
problem is solved for butterflies with upto $15 \cdot 2^15$ nodes
with the help of a sophisticated computer program. In this way we
prove that on butterflies and regular butterflies of order $k$
gossiping can be performed in $2 1/4 * k + o(k)$
and $2 1/2 * k + o(k)$ rounds, respectively.