  We propose a new method for projective dependency parsing based on headed
spans. In a projective dependency tree, the largest subtree rooted at each word
covers a contiguous sequence (i.e., a span) in the surface order. We call such
a span marked by a root word \textit{headed span}.
  A projective dependency tree can be represented as a collection of headed
spans. We decompose the score of a dependency tree into the scores of the
headed spans and design a novel $O(n^3)$ dynamic programming algorithm to
enable global training and exact inference. Our model achieves state-of-the-art
or competitive results on PTB, CTB, and UD. Our code is publicly available at
\url{https://github.com/sustcsonglin/span-based-dependency-parsing}.
