Single-Source Shortest Paths and Almost Exact Diameter in Pseudodisk Graphs
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arXiv
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| Natura: | Preprint |
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2026
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| author | de Berg, Mark Jansen, Bart M. P. Lamme, Jeroen S. K. |
| author_facet | de Berg, Mark Jansen, Bart M. P. Lamme, Jeroen S. K. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_23171 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Single-Source Shortest Paths and Almost Exact Diameter in Pseudodisk Graphs de Berg, Mark Jansen, Bart M. P. Lamme, Jeroen S. K. Computational Geometry Data Structures and Algorithms F.2.2 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. |
| title | Single-Source Shortest Paths and Almost Exact Diameter in Pseudodisk Graphs |
| topic | Computational Geometry Data Structures and Algorithms F.2.2 |
| url | https://arxiv.org/abs/2604.23171 |