有根均匀单调最小生成树
Rooted Uniform Monotone Minimum Spanning Trees
摘要：
研究了给定根点集$P$的最小生成几何图的构造问题，其中$P$的每个点都通过一条满足给定性质的路连接到根点集.我们专注于两个属性，即单调性w.r.t.一个单一的方向（$y$ -单调性）和单调性w.r.t.单对正交方向（$xy$ -单调性）。本文提出了在$O(|P|\log^2 |P|)$中计算$P$的根$y$ -单调（$xy$ -单调）最小生成树的算法（分别为.当方向（resp $O(|P|\log^3 |P|)$）在$O(|P|^2\log|P|)$时间内，当最优方向（分别为一对正交方向）具有单调性时，给出了最优方向（分别为一对正交方向）的单调性.一对正交方向）。我们还给出了简单的算法，给定一个有根的连接的几何图，决定如果根连接到每个其他顶点的路径都是单调的w.r.t.同一方向（一对正交方向）。
英文摘要：
We study the construction of the minimum cost spanning geometric graph of a given rooted point set $P$ where each point of $P$ is connected to the root by a path that satisfies a given property. We focus on two properties, namely the monotonicity w.r.t. a single direction ($y$-monotonicity) and the monotonicity w.r.t. a single pair of orthogonal directions ($xy$-monotonicity). We propose algorithms that compute the rooted $y$-monotone ($xy$-monotone) minimum spanning tree of $P$ in $O(|P|\log^2 |P|)$ (resp. $O(|P|\log^3 |P|)$) time when the direction (resp. pair of orthogonal directions) of monotonicity is given, and in $O(|P|^2\log|P|)$ time when the optimum direction (resp. pair of orthogonal directions) has to be determined. We also give simple algorithms which, given a rooted connected geometric graph, decide if the root is connected to every other vertex by paths that are all monotone w.r.t. the same direction (pair of orthogonal directions).
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
