L_1 Geodesic Farthest Neighbors in a Simple Polygon and Related Problems

We investigate the L_1 geodesic farthest neighbors in a simple polygon P , and address several fundamental problems related to farthest neighbors. Given a subset S ⊆ P , an L_1 geodesic farthest neighbor of p ∈ P from S is one that maximizes the length of L_1 shortest path from p in P . Our list of problems include: computing the diameter, radius, center, farthest-neighbor Voronoi diagram, and two-center of S under the L_1 geodesic distance. We show that all these problems can be solved in linear or near-linear time based on our new observations on farthest neighbors and extreme points. Among them, the key observation shows that there are at most four extreme points of any compact subset S ⊆ P with respect to the L_1 geodesic distance after removing redundancy.