In SoCG 2022, Conroy and T\'oth presented several constructions of sparse, low-hop spanners in geometric intersection graphs, including an $O(n\log n)$-size 3-hop spanner for $n$ disks (or fat convex objects) in the plane, and an $O(n\log^2 n)$-size 3-hop spanner for $n$ axis-aligned rectangles in the plane. Their work left open two major questions: (i) can the size be made closer to linear by allowing larger constant stretch? and (ii) can near-linear size be achieved for more general classes of intersection graphs? We address both questions simultaneously, by presenting new constructions of constant-hop spanners that have almost linear size and that hold for a much larger class of intersection graphs. More precisely, we prove the existence of an $O(1)$-hop spanner for arbitrary string graphs with $O(n\alpha_k(n))$ size for any constant $k$, where $\alpha_k(n)$ denotes the $k$-th function in the inverse Ackermann hierarchy. We similarly prove the existence of an $O(1)$-hop spanner for intersection graphs of $d$-dimensional fat objects with $O(n\alpha_k(n))$ size for any constant $k$ and $d$. We also improve on some of Conroy and T\'oth's specific previous results, in either the number of hops or the size: we describe an $O(n\log n)$-size 2-hop spanner for disks (or more generally objects with linear union complexity) in the plane, and an $O(n\log n)$-size 3-hop spanner for axis-aligned rectangles in the plane. Our proofs are all simple, using separator theorems, recursion, shifted quadtrees, and shallow cuttings.

翻译：在2022年计算几何研讨会（SoCG 2022）上，Conroy与Tóth针对几何交图提出了若干稀疏、低跳数生成子的构造方法，包括针对平面上$n$个圆盘（或胖凸体）的$O(n\log n)$规模3跳生成子，以及针对平面上$n$个轴对齐矩形的$O(n\log^2 n)$规模3跳生成子。他们的工作遗留了两个关键问题：（i）能否通过允许更大的常数伸缩因子使规模更接近线性？（ii）对于更一般的交图类别，是否可实现近线性规模？我们通过提出具有近线性规模且适用于更广交图类别的恒定跳数生成子新构造，同时解决了这两个问题。更具体地，我们证明了对于任意字符串图，存在$O(1)$跳生成子，其规模为$O(n\alpha_k(n))$（$k$为任意常数），其中$\alpha_k(n)$表示逆阿克曼层级中的第$k$个函数。类似地，对于$d$维胖体交图，我们证明了存在$O(1)$跳生成子，其规模为$O(n\alpha_k(n))$（$k$和$d$为任意常数）。我们还改进了Conroy与Tóth的部分先前结果（体现在跳数或规模上）：针对平面上圆盘（或具有线性并复杂性的更一般物体），给出了$O(n\log n)$规模2跳生成子；针对平面上轴对齐矩形，给出了$O(n\log n)$规模3跳生成子。所有证明均简洁明了，利用了分割定理、递归、平移四叉树及浅层切割技术。