Orders with low crossing number, introduced by Welzl, are a fundamental tool in range searching and computational geometry. Recently, they have found important applications in structural graph theory: set systems with linear shatter functions correspond to graph classes with linear neighborhood complexity. For such systems, Welzl's theorem guarantees the existence of orders with only $\mathcal{O}(\log^2 n)$ crossings. A series of works has progressively improved the runtime for computing such orders, from Chazelle and Welzl's original $\mathcal{O}(|U|^3 |\mathcal{F}|)$ bound, through Har-Peled's $\mathcal{O}(|U|^2|\mathcal{F}|)$, to the recent sampling-based methods of Csikós and Mustafa. We present a randomized algorithm that computes Welzl orders for set systems with linear primal and dual shatter functions in time $\mathcal{O}(\|S\| \log \|S\|)$, where $\|S\| = |U| + \sum_{X \in \mathcal{F}} |X|$ is the size of the canonical input representation. As an application, we compute compact neighborhood covers in graph classes with (near-)linear neighborhood complexity in time \(\mathcal{O}(n \log n)\) and improve the runtime of first-order model checking on monadically stable graph classes from $\mathcal{O}(n^{5+\varepsilon})$ to $\mathcal{O}(n^{3+\varepsilon})$.

翻译：由Welzl引入的低交叉数序是范围搜索和计算几何学中的基本工具。最近，它们在结构图论中发现了重要应用：具有线性粉碎函数的集合系统对应于具有线性邻域复杂度的图类。对于此类系统，Welzl定理保证了仅存在$\mathcal{O}(\log^2 n)$交叉的序。一系列研究逐步改进了计算此类序的运行时间，从Chazelle和Welzl最初的$\mathcal{O}(|U|^3 |\mathcal{F}|)$界，到Har-Peled的$\mathcal{O}(|U|^2|\mathcal{F}|)$，再到Csikós和Mustafa最近的基于抽样的方法。我们提出了一种随机算法，可在$\mathcal{O}(\|S\| \log \|S\|)$时间内为具有线性原对偶粉碎函数的集合系统计算Welzl序，其中$\|S\| = |U| + \sum_{X \in \mathcal{F}} |X|$是规范输入表示的大小。作为应用，我们在$\mathcal{O}(n \log n)$时间内计算具有（近）线性邻域复杂度的图类中的紧致邻域覆盖，并将一阶模型检测在单演稳定图类上的运行时间从$\mathcal{O}(n^{5+\varepsilon})$改进为$\mathcal{O}(n^{3+\varepsilon})$。