动力学稳定的Delaunay图
Kinetic Stable Delaunay Graphs
摘要：
我们考虑的问题，保持欧几里德Delaunay三角剖分$\DT$的一组$P$的$n$移动点在平面上，沿代数轨迹的恒定描述的复杂性。由于在完整的$\DT$拓扑变化的数量上的最佳上限是接近立方的，我们试图保持它的一个合适的部分，这是不太易失性，但保留了许多有用的属性。我们引入了稳定Delaunay图的概念，它是Delaunay三角剖分的动态子图。稳定的Delaunay图（a）易于定义，（b）仅经历近似二次的离散变化，（c）在范数的小变化下是鲁棒的，以及（d）具有某些有用的性质。稳定Delaunay图（简称$\SDG$）是根据参数$α>0$定义的，并且由Delaunay边$pq$组成，其中$p$和$q$看到它们的Voronoi边$e_{pq}$的角度至少为$α$。我们证明了：（i）$\SDG$总是包含至少大约三分之一的Delaunay边;（ii）它包含$P$的$β$骨架，对于$β=1+Ω(α^2)$ ;（iii）它是稳定的，在这个意义上，它的边存活很长一段时间，只要连接$P$的（附近）点的片段的方向不改变太多;和（iv）稳定的Delaunay边保持稳定（与一个适当的重新定义的稳定性），如果我们用任何足够接近的范数取代欧几里德范数。特别地，我们可以通过多边形范数（即，具有$k=Θ(1/α)$的正则$k$ -边形）来近似欧几里德范数，并且通过在多边形范数下保持$P$的完整Delaunay三角剖分来跟踪欧几里德$\SDG$。我们描述了两种用于维护$\SDG$的动力学数据结构。两种结构都使用$O^*(n)$存储，并在运动期间处理$O^*(n^2)$事件，每个事件都在$O^*(1)$时间内。
英文摘要：
We consider the problem of maintaining the Euclidean Delaunay triangulation $\DT$ of a set $P$ of $n$ moving points in the plane, along algebraic trajectories of constant description complexity. Since the best known upper bound on the number of topological changes in the full $\DT$ is nearly cubic, we seek to maintain a suitable portion of it that is less volatile yet retains many useful properties. We introduce the notion of a stable Delaunay graph, which is a dynamic subgraph of the Delaunay triangulation. The stable Delaunay graph (a) is easy to define, (b) experiences only a nearly quadratic number of discrete changes, (c) is robust under small changes of the norm, and (d) possesses certain useful properties. The stable Delaunay graph ($\SDG$ in short) is defined in terms of a parameter $α>0$, and consists of Delaunay edges $pq$ for which the angles at which $p$ and $q$ see their Voronoi edge $e_{pq}$ are at least $α$. We show that (i) $\SDG$ always contains at least roughly one third of the Delaunay edges; (ii) it contains the $β$-skeleton of $P$, for $β=1+Ω(α^2)$; (iii) it is stable, in the sense that its edges survive for long periods of time, as long as the orientations of the segments connecting (nearby) points of $P$ do not change by much; and (iv) stable Delaunay edges remain stable (with an appropriate redefinition of stability) if we replace the Euclidean norm by any sufficiently close norm. In particular, we can approximate the Euclidean norm by a polygonal norm (namely, a regular $k$-gon, with $k=Θ(1/α)$), and keep track of a Euclidean $\SDG$ by maintaining the full Delaunay triangulation of $P$ under the polygonal norm. We describe two kinetic data structures for maintaining $\SDG$. Both structures use $O^*(n)$ storage and process $O^*(n^2)$ events during the motion, each in $O^*(1)$ time.
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