The Maximum Linear Arrangement problem (MaxLA) consists of finding a mapping $\pi$ from the $n$ vertices of a graph $G$ to distinct consecutive integers that maximizes $D_{\pi}(G)=\sum_{uv\in E(G)}|\pi(u) - \pi(v)|$. In this setting, vertices are considered to lie on a horizontal line and edges are drawn as semicircles above the line. There exist variants of MaxLA in which the arrangements are constrained. In the planar variant edge crossings are forbidden. In the projective variant for rooted trees arrangements are planar and the root cannot be covered by any edge. Here we present $O(n)$-time and $O(n)$-space algorithms that solve Planar and Projective MaxLA for trees. We also prove several properties of maximum projective and planar arrangements.
翻译:最大线性安排问题( MaxLA) 包括从图形的美元顶端找到一个绘图 $\ pion $G$到不同的连续整数,使 $D ⁇ pi} (G) ⁇ sum ⁇ uv\ in E(G) ⁇ pi(u) -\\ pi(v)\\\ $) 。 在此设置中, 脊椎被视为位于水平线上, 边缘被画成线上方的半二次曲线 。 有 MaxLA 的变量, 其安排受到限制 。 在平面变式边缘过境点中, 禁止 。 根树安排的投影变量是平面的, 根不能被任何边缘覆盖 。 在此, 我们提出用于解决树的平面和投影 MaxLA 。 我们还证明了最大投影和平面安排的若干特性 。