Unified algebraic theory of sorting, routing, multicasting, and concentration networks

Multistage interconnection networks (MINs) are commonly deployed in sorting, switching, and other applications. Arithmetic over MINs for various applications is hereby unified in algebra. The key is to structure the signal alphabet as a distributive lattice instead of an ordered set. The theoretic unification enhances the mathematical understanding of properties of MINs and, in particular, demystifies various 0-1 principles. Conventional applications of MINs are all for point-to-point transmissions, while the unified theory applies to multicasting as well. The Multicast Concentrator Theorem is generalized into the Boolean Concentrator Theorem, which is useful in recursive construction of multicast switches with the feature of priority treatment. Meanwhile, the concomitant theory of cut-through coding is introduced for delay-free signal propagation through a MIN.