# Computing Farthest Neighbors on a Convex Polytope

## Abstract

Let *N* be a set of *n* points in convex position in ∝^{3}. The farthest-point Voronoi diagram of *N* partitions ∝^{3} into *n* convex cells. We consider the intersection *G*(*N*) of the diagram with the boundary of the convex hull of *N*. We give an algorithm that computes an implicit representation of *G*(*N*) in expected *O*(*n* log^{2} *n*) time. More precisely, we compute the combinatorial structure of *G*(*N*), the coordinates of its vertices, and the equation of the plane de?ning each edge of *G*(*N*). The algorithm allows us to solve the all-pairs farthest neighbor problem for *N* in expected time *O*(*n* log^{2} *n*), and to perform farthest-neighbor queries on N in *O*(log^{2} *n*) time with high probability. This can be applied to find a Euclidean maximum spanning tree and a diameter 2-clustering of *N* in expected *O*(*n* log^{4} *n*) time.

## Keywords

Voronoi Diagram Voronoi Cell Implicit Representation Convex Polytope Connected Subgraph## Preview

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## References

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