随机排布中边交叉数方差的快速计算
Fast calculation of the variance of edge crossings in random arrangements
AI总结：
针对随机嵌入图的边交叉数方差计算需求，本文基于新型算术表达式，提出了适用于多种随机布局的高效算法，分别将任意图和森林的计算复杂度优化至$O(nm)$与$O(n)$，为相关应用提供支撑。
摘要：
图$G$的交叉数$\text{cr}(G)$，是指在特定曲面上绘制图时产生的最少边交叉数。确定$\text{cr}(G)$是图论中极为重要的问题。其最大化变体，即最大交叉数$\text{max-cr}(G)$，正受到越来越多的关注。本文并未研究交叉数的优化问题，而是考虑任意图的顶点在某空间中均匀随机嵌入时，边交叉数的方差。在其开创性研究中，Moon推导了完全一分图和完全二分图在随机线性排布下的该方差。鉴于需要高效算法来支撑这类研究，同时鉴于空间网络（即顶点嵌入在某空间中的网络）中边交叉数的研究热度日益提升，本文推导了一种在$o(nm^2)$时间内计算任意图方差的算法，通过复用计算量，我们将其转化为运行时间为$O(nm)$的算法。我们还推导了一种适用于森林的算法，运行时间为$O(n)$。这些算法适用于广泛的随机布局（不仅限于Moon的布局），且基于我们从先前理论工作出发推导出的用于方差计算的新型算术表达式。这为诸多依赖快速且精确的方差计算的应用铺平了道路。
英文摘要：
The crossing number of a graph $G$, $\mathrm{cr}(G)$, is the minimum number of edge crossings arising when drawing a graph on a certain surface. Determining $\mathrm{cr}(G)$ is a problem of great importance in Graph Theory. Its maximum variant, i.e. the maximum crossing number, $\mathrm{max-cr}(G)$, is receiving growing attention. Instead of an optimization problem on the number of crossings, here we consider the variance of the number of edge crossings, when embedding the vertices of an arbitrary graph uniformly at random in some space. In his pioneering research, Moon derived this variance on random linear arrangements of complete unipartite and bipartite graphs. Given the need of efficient algorithms to support this sort of research and given also the growing interest of the number of edge crossings in spatial networks, networks where vertices are embedded in some space, here we derive an algorithm to calculate the variance in arbitrary graphs in time $o(nm^2)$ that we transform into one that runs in time $O(nm)$ by reusing computations. We also derive one for forests that runs in time $O(n)$. These algorithms work on a wide range of random layouts (not only on Moon's) and are based on novel arithmetic expressions for the calculation of the variance that we develop from previous theoretical work. This paves the way for many applications that rely on a fast but exact calculation of the variance.
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