高效导航随机Delaunay三角剖分
Efficiently navigating a random Delaunay triangulation
摘要：
可平面图导航是一个重要的问题，它对几何数据结构中的点定位和网络中的路由都有着重要的意义。然而，虽然已经提出了一些算法和存在性证明，很少的分析是可用于所生成的路径的属性和所需的计算资源，以生成它们的输入下的广义过程假设。在本文中，我们分析了一个新的确定性的平面导航算法，具有恒定的竞争力，以下的顶点邻接的Delaunay三角剖分。我们称之为圆锥行走策略。我们证明了给定$n$在单位面积的光滑凸区域上的一致点，并且对于任意的起始点$z$和查询点$q$，对$z$和$q$应用锥行走，当$n$趋于无穷大时，将访问最多的具有复杂度$O(|zq|\sqrt{n} \log \log n + \log^7 n)$且概率趋于1的$O(|zq|\sqrt{n} +\log^7 n)$站点.我们还表明，在这个模型中，锥行走是$(\log ^{3+ξ} n)$ -无记忆的高概率域中的任何一对开始和查询点，任何积极的$ξ$。我们在整个过程中特别小心，以确保我们的边界是有效的，即使查询点任意接近边界。
英文摘要：
Planar graph navigation is an important problem with significant implications to both point location in geometric data structures and routing in networks. However, whilst a number of algorithms and existence proofs have been proposed, very little analysis is available for the properties of the paths generated and the computational resources required to generate them under a random distribution hypothesis for the input. In this paper we analyse a new deterministic planar navigation algorithm with constant competitiveness which follows vertex adjacencies in the Delaunay triangulation. We call this strategy cone walk. We prove that given $n$ uniform points in a smooth convex domain of unit area, and for any start point $z$ and query point $q$; cone walk applied to $z$ and $q$ will access at most $O(|zq|\sqrt{n} +\log^7 n)$ sites with complexity $O(|zq|\sqrt{n} \log \log n + \log^7 n)$ with probability tending to 1 as $n$ goes to infinity. We additionally show that in this model, cone walk is $(\log ^{3+ξ} n)$-memoryless with high probability for any pair of start and query point in the domain, for any positive $ξ$. We take special care throughout to ensure our bounds are valid even when the query points are arbitrarily close to the border.
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