Analysis of approximate nearest neighbor searching with clustered point sets

We present an empirical analysis of data structures for approximate nearest neighbor searching. We compare the well-known optimized kd-tree splitting method against two alternative splitting methods. The first, called the sliding-midpoint method, which attempts to balance the goals of producing subdivision cells of bounded aspect ratio, while not producing any empty cells. The second, called the minimum-ambiguity method is a query-based approach. In addition to the data points, it is also given a training set of query points for preprocessing. It employs a simple greedy algorithm to select the splitting plane that minimizes the average amount of ambiguity in the choice of the nearest neighbor for the training points. We provide an empirical analysis comparing these two methods against the optimized kd-tree construction for a number of synthetically generated data and query sets. We demonstrate that for clustered data and query sets, these algorithms can provide significant improvements over the standard kd-tree construction for approximate nearest neighbor searching.Comment: 20 pages, 8 figures. Presented at ALENEX '99, Baltimore, MD, Jan 15-16, 199

Robust Proximity Search for Balls using Sublinear Space

Given a set of n disjoint balls b1, . . ., bn in IRd, we provide a data structure, of near linear size, that can answer (1 \pm \epsilon)-approximate kth-nearest neighbor queries in O(log n + 1/\epsilon^d) time, where k and \epsilon are provided at query time. If k and \epsilon are provided in advance, we provide a data structure to answer such queries, that requires (roughly) O(n/k) space; that is, the data structure has sublinear space requirement if k is sufficiently large

Faster tuple lattice sieving using spherical locality-sensitive filters

To overcome the large memory requirement of classical lattice sieving algorithms for solving hard lattice problems, Bai-Laarhoven-Stehl\'{e} [ANTS 2016] studied tuple lattice sieving, where tuples instead of pairs of lattice vectors are combined to form shorter vectors. Herold-Kirshanova [PKC 2017] recently improved upon their results for arbitrary tuple sizes, for example showing that a triple sieve can solve the shortest vector problem (SVP) in dimension $d$ in time $2^{0.3717d + o(d)}$, using a technique similar to locality-sensitive hashing for finding nearest neighbors. In this work, we generalize the spherical locality-sensitive filters of Becker-Ducas-Gama-Laarhoven [SODA 2016] to obtain space-time tradeoffs for near neighbor searching on dense data sets, and we apply these techniques to tuple lattice sieving to obtain even better time complexities. For instance, our triple sieve heuristically solves SVP in time $2^{0.3588d + o(d)}$. For practical sieves based on Micciancio-Voulgaris' GaussSieve [SODA 2010], this shows that a triple sieve uses less space and less time than the current best near-linear space double sieve.Comment: 12 pages + references, 2 figures. Subsumed/merged into Cryptology ePrint Archive 2017/228, available at https://ia.cr/2017/122