几何交叉最小化--一种可扩展的随机方法
Geometric Crossing-Minimization -- A Scalable Randomized Approach
摘要：
我们考虑图$G=(V, E)$的几何绘图中边交叉的最小化，即，在每个边缘被描绘为线段的附图中。相应的决策问题是NP难的[Bienstock，'91]。与理论和拓扑设置相比，几何设置在实践中没有得到很多关注。先前的工作[Radermacher等人，ALENEX'18]仅限于具有少于$200$边的几何图中的交叉最小化。所描述的算法基于将单个顶点$v$移动到其交叉最小位置的原语操作，即，在$\mathbb{R}^2$中使$v$的边缘交叉数最小的位置。在本文中，我们介绍了一种技术，以加快计算的一个因素的$20$。这是必要的，但不足以应付几千边的图。为了处理更大的图，我们放弃了每个顶点$v$必须移动到其交叉最小位置的条件，并计算一个仅对边缘的一个小的随机子集最优的位置。在我们的理论贡献中，我们考虑包含每个边缘$uv \in E$和每个位置$p \in \mathbb{R}^2$的$v$ $o(|E|)$交叉的图纸。在这种情况下，我们证明，对于大小为$Θ(k \log k)$的边的随机子集，度$k$顶点$v$的共交叉数，即，不交叉的边缘对$uv \in E, e \in E$的数量可以用任意但固定的因子$δ$以高概率来近似。在我们的实验评估中，我们表明，随机化的方法减少了交叉的数量在图中与$13\,000$边缘相当。评估表明，根据度分布不同的策略，结果在最少的交叉点。
英文摘要：
We consider the minimization of edge-crossings in geometric drawings of graphs $G=(V, E)$, i.e., in drawings where each edge is depicted as a line segment. The respective decision problem is NP-hard [Bienstock, '91]. In contrast to theory and the topological setting, the geometric setting did not receive a lot of attention in practice. Prior work [Radermacher et al., ALENEX'18] is limited to the crossing-minimization in geometric graphs with less than $200$ edges. The described heuristics base on the primitive operation of moving a single vertex $v$ to its crossing-minimal position, i.e., the position in $\mathbb{R}^2$ that minimizes the number of crossings on edges incident to $v$. In this paper, we introduce a technique to speed-up the computation by a factor of $20$. This is necessary but not sufficient to cope with graphs with a few thousand edges. In order to handle larger graphs, we drop the condition that each vertex $v$ has to be moved to its crossing-minimal position and compute a position that is only optimal with respect to a small random subset of the edges. In our theoretical contribution, we consider drawings that contain for each edge $uv \in E$ and each position $p \in \mathbb{R}^2$ for $v$ $o(|E|)$ crossings. In this case, we prove that with a random subset of the edges of size $Θ(k \log k)$ the co-crossing number of a degree-$k$ vertex $v$, i.e., the number of edge pairs $uv \in E, e \in E$ that do not cross, can be approximated by an arbitrary but fixed factor $δ$ with high probability. In our experimental evaluation, we show that the randomized approach reduces the number of crossings in graphs with up to $13\,000$ edges considerably. The evaluation suggests that depending on the degree-distribution different strategies result in the fewest number of crossings.
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