We consider the problem of establishing minimumcost multicast connections over coded packet networks, i.e. packet networks where the contents of outgoing packets are arbitrary, causal functions of the contents of received packets. We consider both wireline and wireless packet networks as well as both static multicast (where membership of the multicast group remains constant for the duration of the connection) and dynamic multicast (where membership of the multicast group changes in time, with nodes joining and leaving the group).For static multicast, we reduce the problem to a polynomialtime solvable optimization problem, and we present decentralized algorithms for solving it. These algorithms, when coupled with existing decentralized schemes for constructing network codes, yield a fully decentralized approach for achieving minimumcost multicast. By contrast, establishing minimum-cost static multicast connections over routed packet networks is a very difficult problem even using centralized computation, except in the special cases of unicast and broadcast connections.For dynamic multicast, we reduce the problem to a dynamic programming problem and apply the theory of dynamic programming to suggest how it may be solved.

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We give an information flow interpretation for multicasting using network coding. This generalizes the fluid model used to represent flows to a single receiver. Using the generalized model, we present a decentralized algorithm to minimize the number of packets that undergo network coding. We also propose a decentralized algorithm to construct capacity achieving multicast codes when the processing at some nodes is restricted to routing. The proposed algorithms can be coupled with existing decentralized schemes to achieve minimum cost muticast.

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