Implicit Additively Weighted Geodesic Voronoi Diagrams and Applications

Let $P$ be a simple polygon with $m$ vertices. After $O(m)$-time preprocessing of $P$, for any set $S$ of $n$ additively weighted point sites, we compute an implicit additively weighted geodesic Voronoi diagram in $O(n\log n+n\log^2 m)$ time, supporting nearest-neighbor queries in $O(\log(n+m))$ time. In contrast, explicitly constructing the diagram requires $Ω(m)$ time. Our algorithm introduces a new sweep-line paradigm that avoids explicitly constructing the diagram, together with a novel technique for point location in implicit subdivisions. As applications, we obtain new dynamic data structures for additively weighted geodesic nearest-neighbor searching and the first efficient algorithms for shortest paths in geodesic disk graphs. After the same $O(m)$-time preprocessing, single-source shortest paths in a geodesic disk graph of $n$ sites can be computed in $O(n\log n+n\log^2 m)$ time in the unweighted case and $O(n^{3/2}(\log n+\log^2 m))$ time in the weighted case. We also extend our techniques to implicit farthest geodesic Voronoi diagrams for unweighted sites, obtaining analogous bounds and dynamic farthest-neighbor data structures.