Fast Hierarchical Clustering and Other Applications of Dynamic Closest Pairs

We develop data structures for dynamic closest pair problems with arbitrary distance functions, that do not necessarily come from any geometric structure on the objects. Based on a technique previously used by the author for Euclidean closest pairs, we show how to insert and delete objects from an n-object set, maintaining the closest pair, in O(n log^2 n) time per update and O(n) space. With quadratic space, we can instead use a quadtree-like structure to achieve an optimal time bound, O(n) per update. We apply these data structures to hierarchical clustering, greedy matching, and TSP heuristics, and discuss other potential applications in machine learning, Groebner bases, and local improvement algorithms for partition and placement problems. Experiments show our new methods to be faster in practice than previously used heuristics.Comment: 20 pages, 9 figures. A preliminary version of this paper appeared at the 9th ACM-SIAM Symp. on Discrete Algorithms, San Francisco, 1998, pp. 619-628. For source code and experimental results, see http://www.ics.uci.edu/~eppstein/projects/pairs

Enumerating the k closest pairs mechanically

Let $S$ be a set of $n$ points in $D$-dimensional space, where $D$ is a constant, and let $k$ be an integer between $1$ and $n \choose 2$. An algorithm is given that computes the $k$ closest pairs in the set $S$ in $O(n \log n + k)$ time, using $O(n+k)$ space. The algorithm fits in the algebraic decision tree model and is, therefore, optimal

One of the closest exoplanet pairs to the 3:2 Mean Motion Resonance: K2-19b \& c

The K2 mission has recently begun to discover new and diverse planetary systems. In December 2014 Campaign 1 data from the mission was released, providing high-precision photometry for ~22000 objects over an 80 day timespan. We searched these data with the aim of detecting further important new objects. Our search through two separate pipelines led to the independent discovery of K2-19b \& c, a two-planet system of Neptune sized objects (4.2 and 7.2 $R_\oplus$), orbiting a K dwarf extremely close to the 3:2 mean motion resonance. The two planets each show transits, sometimes simultaneously due to their proximity to resonance and alignment of conjunctions. We obtain further ground based photometry of the larger planet with the NITES telescope, demonstrating the presence of large transit timing variations (TTVs), and use the observed TTVs to place mass constraints on the transiting objects under the hypothesis that the objects are near but not in resonance. We then statistically validate the planets through the \texttt{PASTIS} tool, independently of the TTV analysis.Comment: 18 pages, 10 figures, accepted to A&A, updated to match published versio

A new projection method for finding the closest point in the intersection of convex sets

In this paper we present a new iterative projection method for finding the closest point in the intersection of convex sets to any arbitrary point in a Hilbert space. This method, termed AAMR for averaged alternating modified reflections, can be viewed as an adequate modification of the Douglas--Rachford method that yields a solution to the best approximation problem. Under a constraint qualification at the point of interest, we show strong convergence of the method. In fact, the so-called strong CHIP fully characterizes the convergence of the AAMR method for every point in the space. We report some promising numerical experiments where we compare the performance of AAMR against other projection methods for finding the closest point in the intersection of pairs of finite dimensional subspaces