Daniel Prigan | Searcharxiv

arXiv2026

Distances in Planar Graphs are Almost for Free!

We prove that, up to subpolynomial or polylogarithmic factors, there is no tradeoff between preprocessing time, query time, and size of exact distance oracles for planar graphs. Namely, we show how given an $n$-vertex weighted directed planar graph $G$, one can compute in $n^{1+o(1)}$ time and space a representation of $G$ from which one can extract the exact distance between any two vertices of $G$ in $\log^{2+o(1)}(n)$ time. Previously, it was only known how to construct oracles with these space and query time in $n^{3/2+o(1)}$ time [STOC 2019, SODA 2021, JACM 2023]. We show how to construct these oracles in $n^{1+o(1)}$ time.

cs.DS↗

arXiv2025

Faster Construction of a Planar Distance Oracle with \~{O}(1) Query Time

We show how to preprocess a weighted undirected $n$-vertex planar graph in $\tilde O(n^{4/3})$ time, such that the distance between any pair of vertices can then be reported in $\tilde O(1)$ time. This improves the previous $\tilde O(n^{3/2})$ preprocessing time [JACM'23]. Our main technical contribution is a near optimal construction of \emph{additively weighted Voronoi diagrams} in undirected planar graphs. Namely, given a planar graph $G$ and a face $f$, we show that one can preprocess $G$ in $\tilde O(n)$ time such that given any weight assignment to the vertices of $f$ one can construct the additively weighted Voronoi diagram of $f$ in near optimal $\tilde O(|f|)$ time. This improves the $\tilde O(\sqrt{n |f|})$ construction time of [JACM'23].

cs.DS↗