Randomized incremental construction (RIC) is one of the most important paradigms for building geometric data structures. Clarkson and Shor developed a general theory that led to numerous algorithms which are both simple and efficient in theory and in practice. Randomized incremental constructions are usually space-optimal and time-optimal in the worst case, as exemplified by the construction of convex hulls, Delaunay triangulations, and arrangements of line segments. However, the worst-case scenario occurs rarely in practice and we would like to understand how RIC behaves when the input is nice in the sense that the associated output is significantly smaller than in the worst case. For example, it is known that the Delaunay triangulation of nicely distributed points in $${\mathbb {E}}^d$$ or on polyhedral surfaces in $${\mathbb {E}}^3$$ has linear complexity, as opposed to a worst-case complexity of $$\Theta (n^{\lfloor d/2\rfloor })$$ in the first case and quadratic in the second. The standard analysis does not provide accurate bounds on the complexity of such cases and we aim at establishing such bounds in this paper. More precisely, we will show that, in the two cases above and variants of them, the complexity of the usual RIC is $$O(n\log n)$$ , which is optimal. In other words, without any modification, RIC nicely adapts to good cases of practical value. At the heart of our proof is a bound on the complexity of the Delaunay triangulation of random subsets of $${\varepsilon }$$ -nets. Along the way, we prove a probabilistic lemma for sampling without replacement, which may be of independent interest.

中文翻译：

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尼斯点集的 Delaunay 三角剖分的随机增量构造

随机增量构造 (RIC) 是构建几何数据结构的最重要范例之一。Clarkson 和 Shor 开发了一个通用理论，导致了许多算法，这些算法在理论和实践中既简单又有效。在最坏的情况下，随机增量构造通常是空间最优和时间最优的，例如凸包的构造、Delaunay 三角剖分和线段的排列。然而，最坏的情况在实践中很少发生，我们想了解当输入很好时 RIC 的行为，因为相关的输出明显小于最坏情况下的输出。例如，众所周知，$${\mathbb {E}}^d$$ 或 $${\mathbb {E}}^3$$ 中的多面体表面上分布良好的点的 Delaunay 三角剖分具有线性复杂度，而不是$$\Theta (n^{\lfloor d/2\rfloor })$$ 在第一种情况下的最坏情况复杂度，在第二种情况下是二次的。标准分析并未对此类案例的复杂性提供准确的界限，我们的目标是在本文中建立此类界限。更准确地说，我们将证明，在上述两种情况及其变体中，通常 RIC 的复杂性是 $$O(n\log n)$$ ，这是最优的。换句话说，无需任何修改，RIC 就很好地适应了具有实用价值的好案例。我们证明的核心是对 $${\varepsilon }$$ -nets 的随机子集的 Delaunay 三角剖分的复杂性的界限。一路上，