VLSI layout of Benes networks

Abstract

The Benes network consists of back-to-back butterflies. There exist a number of topological representations that are used to describe butterfly -like architectures. We identify a new topological representation of Benes network. The significance of this representation is demonstrated by solving two problems, one related to VLSI layout and the other related to robotics. An important VLSI layout network problem is to produce the smallest possible grid area for realizing a given network. We propose an elegant VLSI layout of r-dimensional Benes networks using this representation. The area of this layout is O(2 2r ) whereas the lower bound for the area of the VLSI layout of Benes networks is O(2 2r ). This lower bound is estimated by applying Thompson result.

Generalization of edge general position problem

“…It is known that diam(BN(r)) = diam(BF(r)) = 2r, cf. [18,21,24], and that the edge set of BF(r) can be partitioned with respect to 2 r diametral paths [21]. It follows that the edge set of BN(r) can be partitioned with respect to 2 r+1 diametral paths of BN(r).…”

“…An alternative way to represent Benes networks is that BN(r) consists of two backto-back butterflies BF(r), cf. [18,21,24], that is, of two copies of BF(r) sharing level r vertices. It is known that diam(BN(r)) = diam(BF(r)) = 2r, cf.…”

The Edge General Position Problem

“…It is known that diam(BN(r)) = diam(BF(r)) = 2r, cf. [12,15,17], and that the edge set of BF(r) can be partitioned with respect to 2 r diametral paths [15]. It follows that the edge set of BN(r) can be partitioned with respect to 2 r+1 diametral paths of BN(r).…”

“…An alternative way to represent Benes networks is that BN(r) consists of two back-to-back butterflies BF(r), cf. [12,15,17], that is, of two copies of BF(r) sharing level r vertices. It is known that diam(BN(r)) = diam(BF(r)) = 2r, cf.…”