We study SINGLE-SOURCE SHORTEST PATH (SSSP) on unweighted intersection graphs whose node set corresponds to a set of $n$ constant-complexity objects in the plane. We prove SSSP can be solved in $O(U(n)\ \mathrm{polylog}\,n)$ expected time for any class of objects whose union complexity is $U(n)$. In particular, we obtain an $O(n 2^{α(n)}\log^2 n)$ algorithm for arbitrary pseudodisks, and an $O(λ_{s+2}(n)2^{O(\log^* n)} \log^2 n)$ algorithm for locally fat objects. This significantly extends the class of objects for which SSSP can be solved in $O(n\ \mathrm{polylog}\, n)$ time: so far, $O(n\ \mathrm{polylog}\, n)$ SSSP algorithms were not even known for pseudodisks that are fat and convex and similarly-sized. Our second result concerns the DIAMETER problem, which asks for the maximum distance between any two nodes in a graph. Even for intersection graphs, near-quadratic algorithms are difficult to obtain, and the $O(n^2\ \mathrm{polylog}\, n)$ running time that follows from our SSSP algorithm is the first near-quadratic running time for such general classes of intersection graphs. Obtaining subquadratic running time is even more challenging. We prove that the diameter of a set of arbitrary pseudodisks can be computed almost exactly, namely up to an additive error of 2, in $\tilde{O}(n^{2-1/14})$ expected time. This generalizes and speeds up a recent algorithm by Chang, Gao, and Le~(SoCG 2024) that works for similarly-sized disks (or similarly-sized pseudodisks that are fat and satisfy a strong monotonicity assumption) and runs in $\tilde{O}(n^{2-1/18})$ time. To this end, we develop a so-called star-based $r$-clustering for intersection graphs of pseudodisks, which is interesting in its own right. Our star-based $r$-clustering can also be used to obtain an almost exact distance oracle for pseudodisks that uses $O(n^{2-1/13})$ storage and has $O(1)$ query time.

翻译：我们研究平面中一组$n$个常复杂度对象对应的无权交集图中的单源最短路径（SSSP）问题。对于任何并复杂度为$U(n)$的对象类，我们证明可以在期望时间$O(U(n)\ \mathrm{polylog}\,n)$内求解SSSP。特别地，对于任意伪圆盘，我们得到$O(n 2^{α(n)}\log^2 n)$算法；对于局部胖物体，得到$O(λ_{s+2}(n)2^{O(\log^* n)} \log^2 n)$算法。这显著扩展了可在$O(n\ \mathrm{polylog}\, n)$时间内求解SSSP的对象类：此前，即使对于胖凸且尺寸相似的伪圆盘，也未知$O(n\ \mathrm{polylog}\, n)$的SSSP算法。第二个结果涉及直径问题，即求图中任意两点间的最大距离。即便对于交集图，近二次算法也难以获得，而我们的SSSP算法导出的$O(n^2\ \mathrm{polylog}\, n)$运行时间是针对如此一般交集图类的首个近二次算法。获得次二次运行时间更具挑战性。我们证明任意伪圆盘集的直径可在$\tilde{O}(n^{2-1/14})$期望时间内几乎精确计算（即加法误差不超过2）。这推广并加速了Chang、Gao和Le（SoCG 2024）针对尺寸相似圆盘（或满足强单调性假设的胖伪圆盘）的算法，其运行时间为$\tilde{O}(n^{2-1/18})$。为此，我们发展了伪圆盘交集图的一种基于星形的$r$-聚类方法，该方法本身具有独立意义。该星形$r$-聚类还可用于构建伪圆盘的几乎精确距离预言机，该预言机使用$O(n^{2-1/13})$存储空间并支持$O(1)$查询时间。