Space Exploration via Proximity Search

We investigate what computational tasks can be performed on a point set in $\Re^d$, 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

Geometry-Oblivious FMM for Compressing Dense SPD Matrices

We present GOFMM (geometry-oblivious FMM), a novel method that creates a hierarchical low-rank approximation, "compression," of an arbitrary dense symmetric positive definite (SPD) matrix. For many applications, GOFMM enables an approximate matrix-vector multiplication in $N \log N$ or even $N$ time, where $N$ is the matrix size. Compression requires $N \log N$ storage and work. In general, our scheme belongs to the family of hierarchical matrix approximation methods. In particular, it generalizes the fast multipole method (FMM) to a purely algebraic setting by only requiring the ability to sample matrix entries. Neither geometric information (i.e., point coordinates) nor knowledge of how the matrix entries have been generated is required, thus the term "geometry-oblivious." Also, we introduce a shared-memory parallel scheme for hierarchical matrix computations that reduces synchronization barriers. We present results on the Intel Knights Landing and Haswell architectures, and on the NVIDIA Pascal architecture for a variety of matrices.Comment: 13 pages, accepted by SC'1

Net and Prune: A Linear Time Algorithm for Euclidean Distance Problems

We provide a general framework for getting expected linear time constant factor approximations (and in many cases FPTAS's) to several well known problems in Computational Geometry, such as $k$-center clustering and farthest nearest neighbor. The new approach is robust to variations in the input problem, and yet it is simple, elegant and practical. In particular, many of these well studied problems which fit easily into our framework, either previously had no linear time approximation algorithm, or required rather involved algorithms and analysis. A short list of the problems we consider include farthest nearest neighbor, $k$-center clustering, smallest disk enclosing $k$ points, $k$th largest distance, $k$th smallest $m$-nearest neighbor distance, $k$th heaviest edge in the MST and other spanning forest type problems, problems involving upward closed set systems, and more. Finally, we show how to extend our framework such that the linear running time bound holds with high probability