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  1. (, Proceedings of the 32nd Canadian Conference on Computational Geometry)
    Keil, Mark; Mondal, Debajyoti (Ed.)
    We adapt and generalize a heuristic for k-center clustering to the permutation case, where every pre x of the ordering is a guaranteed approximate solution. The one-hop greedy permutations work by choosing at each step the farthest unchosen point and then looking in its local neighborhood for a point that covers the most points at a certain scale. This balances the competing demands of reducing the coverage radius and also covering as many points as possible. This idea first appeared in the work of Garcia-Diaz et al. and their algorithm required O(n2 log n) time for a fi xed k (i.e. not the whole permutation). We show how to use geometric data structures to approximate the entire permutation in O(n log D) time for metrics sets with spread D. Notably, this running time is asymptotically the same as the running time for computing the ordinary greedy permutation. 
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    Accepted Manuscript Full Text Available
  2. ; (, Proceedings of the 32nd Canadian Conference on Computational Geometry)
    Keil, Mark; Mondal, Debajyoti (Ed.)
    Finding the kth nearest neighbor to a query point is a ubiquitous operation in many types of metric computations, especially those in unsupervised machine learning. In many such cases, the distance to k sample points is used as an estimate of the local density of the sample. In this paper, we give an algorithm that takes a finite metric (P,d) and an integer k and produces a subset S ⊆ P with the property that for any q ∈ P, the distance to the second nearest point of S to q is a constant factor approximation to the distance to the kth nearest point of P to q. Thus, the sample S may be used in lieu of P. In addition to being much smaller than P, the distance queries on S only require finding the second nearest neighbor instead of the kth nearest neighbor. This is a significant improvement, especially because theoretical guarantees on kth nearest neighbor methods often require k to grow as a function of the input size n. 
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    Accepted Manuscript Full Text Available