Basic building blocks for the switch-preserving transformation of shuffle interconnections

Abstract

The transformations of a two-dimensional shuffle on a 4 × 4 array into a one-dimensional shuffle on a vector of length 16 (and vice versa) are recognized as basic building blocks for the switch-preserving transformation of shuffle patterns of any size. [The switch-preserving transformation means the transformation of one-dimensional shuffles into two-dimensional and d-dimensional shuffles (d ≥ 3) and vice versa without the subdivision of switches.] The switch-preserving transformation of shuffle patterns on large arrays is defined recursively by means of the presented basic building blocks. Thi concept of the transformation is restricted to the two-dimensional symmetric generalized perfect shuffle on arrays with equal sides (squares) being an even multiple of four. (Generalized means arbitrary decomposition of each coordinate, and thus an arbitrary shuffle may be defined; perfect is the counterpart to imperfect and refers to the regularity or absence of failures; symmetric means the same decomposition of the data length for both coordinates.) The relationship of the results to multistage interconnection networks is clarified.