Generalized algorithms for systematic synthesis of Branch-and-Combine clock networks for meshes, tori, and hypercubes

Branch-and-Combine (BaC) clock distribution has recently been introduced. The most interesting aspect of the new scheme is its ability to bound skew by a constant irrespective of network size. In this paper, we introduce algorithms for systematic synthesis of BaC networks for clocking meshes, tori, and hypercubes of different dimensionalities. For meshes our approach relies on filing techniques. We start with the identification of basic proper tiles satisfying certain criteria. We define a set of valid transformations on tiles. By appropriately applying a sequence of transformations on a basic proper tile, one could synthesize a valid BaC network. We formally introduce methods and procedures for applying the above steps to systematically construct different valid BaC network designs for 2D and 3D meshes. To construct BaC networks for clocking hypercubes of any dimensionality we describe a formal methodology. In this case, we utilize an approach called replication which is based on constructing larger hypercube clocking networks from smaller ones. We combine the techniques for 2D, 3D meshes with replication techniques to formulate a methodology applicable to meshes and tori of dimensionality greater than three. We provide proofs of correctness for the algorithms we introduce. Besides, we formally define an optimality criterion based on link costs which is utilized to check the optimality of the synthesized network designs. In the case of meshes, we show that the majority of synthesized networks are optimal with respect to our defined criterion. For those suboptimal networks, we describe a procedure for identifying and removing unnecessary (redundant) links. The procedure is guaranteed to optimize the network without changing its behavioral parameters