Abstract
We investigate what computational tasks can be performed on a point set in \({\mathbb {R}}^d\), if we are only given black-box access to it via nearest-neighbor search. This is a reasonable assumption if the underlying point set is either provided implicitly, or it is stored in a data structure that can answer such queries. In particular, we show the following: (A) One can compute an approximate bi-criteria k-center clustering of the point set, and more generally compute a greedy permutation of the point set. (B) One can decide if a query point is (approximately) inside the convex-hull of the point set. We also investigate the problem of clustering the given point set, such that meaningful proximity queries can be carried out on the centers of the clusters, instead of the whole point set.
Highlights
Many problems in Computational Geometry involve sets of points in Rd
If the oracle answers (1 + ε)-approximate nearest-neighbor (ANN) queries only, for any k, the permutation generated is competitive with the optimal k-center clustering, considering the first O k log1/ε Φ points in this permutation, where Φ is the spread of the point set
We assume that the sole access to P is through “black-box” data structures Tnn and Tann, which given a query point q, return the nearest neighbor (NN) and ANN, respectively, to q in P
Summary
Many problems in Computational Geometry involve sets of points in Rd. Traditionally, such a point set is presented explicitly, say, as a list of coordinate vectors. Pck} is an O(1)-approximation to the optimal k center clustering radius, where c is a constant depending only on the dimension This result uses exact proximity queries, and only one query per sequence point generated. If the oracle answers (1 + ε)-ANN queries only, for any k, the permutation generated is competitive with the optimal k-center clustering, considering the first O k log1/ε Φ points in this permutation, where Φ is (roughly) the spread of the point set. Our main new contribution for the convex-hull membership problem is showing that the iterative algorithm can be applied to implicit point sets using nearest-neighbor queries. We assume that all the data is known and the goal is to come up with a useful clustering that can help in proximity search queries
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