论文标题
Delaunay三角构造的有限算法
A Finite Algorithm for the Realizabilty of a Delaunay Triangulation
论文作者
论文摘要
点集的\ emph {delaunay graph} $ p \ subseteq \ mathbb {r}^2 $是带有顶点$ p $的平面图，以及包含$ \ {p，p，p，p'\} $的边缘 - 如果存在与$ p $ p $ pugy ats $ p $ ats $ p $ $ \ \ \ p，p，p，p，p，p，p，p，p，p，p，p，p'因此，如果存在一些$ p \ subseteq \ subseteq \ mathbb {r}^2 $的Delaunay图的三角剖分，则称为A \ emph {delaunay triangulation} $ p $ $ g $ g $ g $ g $。 \ textsc {delaunay实现}的目的是计算一个点集$ p \ subseteq \ mathbb {r}^2 $，它实现了给定的图形$ g $（如果存在这样的$ p $）。已知的算法不能求解\ textsc {delaunay实现}，因为它们是非构造性的。 Hiroshima等人提到了\ textsc {delaunay实现}的建设性算法{delaunay实现}，这是一个开放的问题。我们设计了$ n^{\ Mathcal {o}（n）} $ - \ textsc {delaunay obalization}的时间建设性算法。实际上，我们的算法以{\ em Integer}坐标输出点集。
The \emph{Delaunay graph} of a point set $P \subseteq \mathbb{R}^2$ is the plane graph with the vertex-set $P$ and the edge-set that contains $\{p,p'\}$ if there exists a disc whose intersection with $P$ is exactly $\{p,p'\}$. Accordingly, a triangulated graph $G$ is \emph{Delaunay realizable} if there exists a triangulation of the Delaunay graph of some $P \subseteq \mathbb{R}^2$, called a \emph{Delaunay triangulation} of $P$, that is isomorphic to $G$. The objective of \textsc{Delaunay Realization} is to compute a point set $P \subseteq \mathbb{R}^2$ that realizes a given graph $G$ (if such a $P$ exists). Known algorithms do not solve \textsc{Delaunay Realization} as they are non-constructive. Obtaining a constructive algorithm for \textsc{Delaunay Realization} was mentioned as an open problem by Hiroshima et al.~\cite{hiroshima2000}. We design an $n^{\mathcal{O}(n)}$-time constructive algorithm for \textsc{Delaunay Realization}. In fact, our algorithm outputs sets of points with {\em integer} coordinates.
