Minimum projective linearizations of trees in linear time

The minimum linear arrangement problem (MLA) consists of finding a mapping $\pi$ from vertices of a graph to integers that minimizes $\sum_{uv\in E}|\pi(u) - \pi(v)|$. For trees, various algorithms are available to solve the problem in polynomial time; the best known runs in subquadratic time in $n=|V|$. There exist variants of the MLA in which the arrangements are constrained to certain classes of projectivity. Iordanskii, and later Hochberg and Stallmann (HS), put forward $O(n)$-time algorithms that solve the problem when arrangements are constrained to be planar. We also consider linear arrangements of rooted trees that are constrained to be projective. Gildea and Temperley (GT) sketched an algorithm for the projectivity constraint which, as they claimed, runs in $O(n)$ but did not provide any justification of its cost. In contrast, Park and Levy claimed that GT's algorithm runs in $O(n \log d_{max})$ where $d_{max}$ is the maximum degree but did not provide sufficient detail. Here we correct an error in HS's algorithm for the planar case, show its relationship with the projective case, and derive an algorithm for the projective case that runs undoubtlessly in $O(n)$-time.