EXPOSED LAYOUTS OF THE BUTTERFLY NETWORK

Abstract

This work studies layouts of the Butterfly network under the restriction that its input and output verticles are exposed, that is, are placed on the boundary of the grid. Avior et al. have shown that the minimal area of a layout of the n-input Butterfly, without the above restriction, is (1 + o(1))n2). The effect of this restriction on the area was left as an open question. This paper reveals that exposing the input and output vertices is essentially free. That is, it presents an exposed layout of the Butterfly having the same area as above. In this layout, the fractions of the input and output vertices assigned to each side of the grid are as follows when scanning the sides in a circular fashion: half of the inputs, half of the outputs, half of the inputs, and half of the outputs. We refer to such a layout as a [Formula: see text]-layout. The main technique employed in this layout is the reduction of the layout problem of the Butterfly to a certain layout problem of a complete bipartite graph. We use the same technique to produce a (I, 0, O, 0)-layout (inputs on one side and outputs on the opposite side) in area 2 + o(1))n2. Finally, we show that the area of a [Formula: see text]-layout is greater than [Formula: see text]. Hence, this input-output configuration is more area-demanding than [Formula: see text].