Appears in the Proceedings of the CCCG 2018. Compared to v2: the enumeration of Propositions, Lemmas and Theorems is changed to follow the CCCG 2018 format, Figure 2(b) was changed, and some further small additions and changes were made
A geometric graph $G$ is $xy-$monotone if each pair of vertices of $G$ is connected by a $xy-$monotone path. We study the problem of producing the $xy-$monotone spanning geometric graph of a point set $P$ that (i) has the minimum cost, where the cost of a geometric graph is the sum of the Euclidean lengths of its edges, and (ii) has the least number of edges, in the cases that the Cartesian System $xy$ is specified or freely selected. Building upon previous results, we easily obtain that the two solutions coincide when the Cartesian System is specified and are both equal to the rectangle of influence graph of $P$. The rectangle of influence graph of $P$ is the geometric graph with vertex set $P$ such that two points $p,q \in P$ are adjacent if and only if the rectangle with corners $p$ and $q$ does not include any other point of $P$. When the Cartesian System can be freely chosen, we note that the two solutions do not necessarily coincide, however we show that they can both be obtained in $O(|P|^3)$ time. We also give a simple $2-$approximation algorithm for the problem of computing the spanning geometric graph of a $k-$rooted point set $P$, in which each root is connected to all the other points (including the other roots) of $P$ by $y-$monotone paths, that has the minimum cost.
On the minimum spanning tree problem in imprecise set-up
在这篇文章中，我们研究了欧氏最小生成树问题的不精确设置。该问题在文献中称为\emph{Minimum Spanning Tree Problem with Neighborhoods}。我们研究的问题表示为非交叉线段的邻域。给定$I\!\!R^2$中$n$不相交线段的集合${\cal S}$，目标是找到一个最小生成树（MST），该最小生成树包含$\cal S$中每个线段的一个端点，并且该最小生成树的代价在$2^n$可能的最小生成树中最小.我们发现，找到这样的MST是NP-困难的一般情况下，并提出了一个$2α$因子近似算法相同，其中$α$是最有名的近似算法的近似因子，以计算最小成本Steiner树在无向图与非负边权重。作为我们约简的一个含义，我们可以证明问题的无限制版本（即，必须从每个段中选择一个点，使得MST的成本尽可能最小）也是NP难的。我们还提出了一个参数化算法的问题的基础上的“可分性”参数定义的部分。
A Surprisingly Simple Method for Distributed Euclidean-Minimum Spanning Tree / Single Linkage Dendrogram Construction from High Dimensional Embeddings via Distance Decomposition
