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In this paper, we consider minimum cost lossless source coding for multiple multicast sessions. Each session comprises a set of correlated sources whose information is demanded by a set of sink nodes. We propose a distributed end- to-end algorithm which operates over given multicast trees, and a back-pressure algorithm which optimizes routing and coding over the whole network. Unlike other existing algorithms, the source rates need not be centrally coordinated; the sinks control transmission rates across the sources. With random network coding, the proposed approach yields completely distributed and optimal algorithms for intra-session network coding. We prove the convergence of our proposed algorithms. Some practical considerations are also discussed. Experimental results are provided to complement our theoretical analysis.