Geodesic farthest-point Voronoi diagram in linear time
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Let P be a simple polygon with n vertices. For any two points in P, the geodesic distance between them is the length of the shortest path that connects them among all paths contained in P. Given a set S of m sites being a subset of the vertices of P, we present a randomized algorithm to compute the geodesic farthest-point Voronoi diagram of S in P running in expected O(n + m) time. That is, a partition of P into cells, at most one cell per site, such that every point in a cell has the same farthest site with respect to the geodesic distance. In particular, this algorithm can be extended to run in expected O(n + m m) time when S is an arbitrary set of m sites contained in P, thereby solving the open problem posed by Mitchell in Chapter 27 of the Handbook of Computational Geometry.
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