Minimum projective linearizations of trees in linear time

The Minimum Linear Arrangement problem (MLA) consists of finding a mapping $\pi$ from vertices graph to distinct integers that minimizes $\sum_{\{u,v\}\in E}|\pi(u) - \pi(v)|$. In setting, are often assumed lie on horizontal line and edges drawn as semicircles above said line. For trees, various algorithms available solve the in polynomial time $n=|V|$. There exist variants MLA which arrangements constrained. Iordanskii, later Hochberg Stallmann (HS), put forward $O(n)$-time when constrained be planar (also known one-page book embeddings). We also consider linear rooted trees projective (planar embeddings where root is not covered by any edge). Gildea Temperley (GT) sketched an algorithm for they claimed runs $O(n)$ but did provide justification its cost. contrast, Park Levy GT's $O(n \log d_{max})$ $d_{max}$ maximum degree sufficient detail. Here we correct error HS's case, show relationship with derive simple cases run without doubt time.