边约束邻近图的本质约束
Essential Constraints of Edge-Constrained Proximity Graphs
摘要：
给定一个$|V| = n$点的平面森林$F = (V, E)$，我们找到最小边集$S \subseteq E$，使得顶点集$V$和约束集$S$上的边约束最小生成树包含$F$。我们提出了一个$O(n \log n )$时间算法，解决了这个问题。我们推广到其他邻近图的约束设置，如相对邻域图，加布里埃尔图，$β$骨架和Delaunay三角剖分。我们提出了一个算法，确定一个给定的平面图$I=(V,E)$的边的最小集$S\subseteq E$，使$I \subseteq CG_β(V, S)$为$1 \leq β\leq 2$，其中$CG_β(V, S)$是约束$β$ -骨架上的顶点集$V$和约束集$S$。只要给出$I$的约束Delaunay三角剖分，算法的运行时间为$O(n)$.
英文摘要：
Given a plane forest $F = (V, E)$ of $|V| = n$ points, we find the minimum set $S \subseteq E$ of edges such that the edge-constrained minimum spanning tree over the set $V$ of vertices and the set $S$ of constraints contains $F$. We present an $O(n \log n )$-time algorithm that solves this problem. We generalize this to other proximity graphs in the constraint setting, such as the relative neighbourhood graph, Gabriel graph, $β$-skeleton and Delaunay triangulation. We present an algorithm that identifies the minimum set $S\subseteq E$ of edges of a given plane graph $I=(V,E)$ such that $I \subseteq CG_β(V, S)$ for $1 \leq β\leq 2$, where $CG_β(V, S)$ is the constraint $β$-skeleton over the set $V$ of vertices and the set $S$ of constraints. The running time of our algorithm is $O(n)$, provided that the constrained Delaunay triangulation of $I$ is given.
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