  Given a directed acyclic graph $G,$ and a set of values $y$ on the vertices,
the Isotonic Regression of $y$ is a vector $x$ that respects the partial order
described by $G,$ and minimizes $||x-y||,$ for a specified norm. This paper
gives improved algorithms for computing the Isotonic Regression for all
weighted $\ell_{p}$-norms with rigorous performance guarantees. Our algorithms
are quite practical, and their variants can be implemented to run fast in
practice.
