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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. [ABSTRACT FROM AUTHOR]

Peer-to-peer (P2P) systems have provided a scalable and cost effective way for file sharing and multimedia streaming in the past decade. The implementation of P2P systems usually involves two transmission schemes: routing (store-and-forward), and network coding. In this paper, we make a theoretical investigation of the rate that can be achieved by routing vs. network coding. We model P2P networks as node-capacitated networks with constraints on both node upload capacity and node download capacity. We compare routing and network coding for unicast, multicast and broadcast transmissions with both single-source scenario and multi-source scenario. Our results present not only a unification of existing results, but also extensions to new scenarios for node-capacitated networks. [ABSTRACT FROM PUBLISHER]