Dialogue discourse parsing aims to uncover the internal structure of a multi-participant conversation by finding all the discourse~\emph{links} and corresponding~\emph{relations}. Previous work either treats this task as a series of independent multiple-choice problems, in which the link existence and relations are decoded separately, or the encoding is restricted to only local interaction, ignoring the holistic structural information. In contrast, we propose a principled method that improves upon previous work from two perspectives: encoding and decoding. From the encoding side, we perform structured encoding on the adjacency matrix followed by the matrix-tree learning algorithm, where all discourse links and relations in the dialogue are jointly optimized based on latent tree-level distribution. From the decoding side, we perform structured inference using the modified Chiu-Liu-Edmonds algorithm, which explicitly generates the labeled multi-root non-projective spanning tree that best captures the discourse structure. In addition, unlike in previous work, we do not rely on hand-crafted features; this improves the model's robustness. Experiments show that our method achieves new state-of-the-art, surpassing the previous model by 2.3 on STAC and 1.5 on Molweni (F1 scores). \footnote{Code released at~\url{https://github.com/chijames/structured_dialogue_discourse_parsing}.}

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