Planar point location using persistent search trees
Communications of the ACM
Computing the geodesic center of a simple polygon
Discrete & Computational Geometry
Optimal shortest path queries in a simple polygon
Journal of Computer and System Sciences
Computing geodesic furthest neighbors in simple polygons
Journal of Computer and System Sciences
Matrix searching with the shortest path metric
STOC '93 Proceedings of the twenty-fifth annual ACM symposium on Theory of computing
Davenport-Schinzel sequences and their geometric applications
Davenport-Schinzel sequences and their geometric applications
Handbook of discrete and computational geometry
An Optimal Algorithm for Euclidean Shortest Paths in the Plane
SIAM Journal on Computing
Farthest-polygon Voronoi diagrams
ESA'07 Proceedings of the 15th annual European conference on Algorithms
The geodesic diameter of polygonal domains
ESA'10 Proceedings of the 18th annual European conference on Algorithms: Part I
Farthest-polygon Voronoi diagrams
Computational Geometry: Theory and Applications
Farthest voronoi diagrams under travel time metrics
WALCOM'12 Proceedings of the 6th international conference on Algorithms and computation
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We investigate the farthest-site Voronoi diagram of k point sites with respect to the geodesic distance in a polygonal domain of n corners and h (≥ 0) holes. In the case of h=0, Aronov et al. [2] in 1993 proved that there are at most O(k) faces in the diagram and the complexity of the diagram is at most O(n+k). However, any nontrivial upper bound on the geodesic farthest-site Voronoi diagram in a polygonal domain when h 0 remains unknown afterwards. In this paper, we show that the diagram in a polygonal domain consists of Θ(hk) faces and its total combinatorial complexity is Θ(nk) in the worst case for any h ≥ 1. Interestingly, the worst-case complexity of the diagram is independent from the number h of holes if h ≥ 1 while the maximum possible number of faces is dependent on h rather than on the complexity n of the polygonal domain. Also, we present an O(nk log2(n+k) log k)-time algorithm that constructs the diagram explicitly.