A limit field for orthogonal range searches in two-dimensional random point search trees

We consider the cost of general orthogonal range queries in random quadtrees. The cost of a given query is encoded into a (random) function of four variables which characterize the coordinates of two opposite corners of the query rectangle. We prove that, when suitably shifted and rescaled, the random cost function converges uniformly in probability towards a random field that is characterized as the unique solution to a distributional fixed-point equation. We also state similar results for $2$-d trees. Our results imply for instance that the worst case query satisfies the same asymptotic estimates as a typical query, and thereby resolve an old question of Chanzy, Devroye and Zamora-Cura [\emph{Acta Inf.}, 37:355--383, 2000]Comment: 24 pages, 8 figure

Design, Analysis and Implementation of New Variants of Kd-trees

The representation of multidimensional data is a central issue in database design, as well as in many other elds, including computer graphics, com- putational geometry, pattern recognition, geographic information systems and others. Indeed, multidimensional points can represent locations, as well as more general records that arise in database management systems. For instance, consider an employee record that has attributes corresponding to the employee's name, address, sex, age, height and weight. Although the di erent dimensions have di erent data types (name and address are strings of characters; sex is a binary eld; and age, height and weight are numbers), these records can be treated as points in a six-dimensional space. We may see a database as a collection of records. Each record has several attributes, some of which are keys. The associative retrieval problem consists of answering queries with respect to a le of multidimensional records. Such an associative query requires the retrieval of those records in the le whose key attributes satisfy a certain condition. Examples of associative queries are intersection queries and nearest neighbor queries. In order to facilitate the retrieval of records based on some conditions on its key attributes, it is usually helpful to assumed the existence of an ordering for its values. In the case of numeric keys, such an ordering is quite obvious. In the case of alphanumeric keys, the ordering is usually based on the alphabetic sequence of the characters making up the attribute value. Furthermore, certain queries, like nearest neighbor searches, require the existence of a distance function

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

Search for the Higgs boson in lepton, tau and jets final states

We present a search for the standard model Higgs boson in final states with an electron or muon and a hadronically decaying tau lepton in association with two or more jets using 9.7 fb^{-1} of Run II Fermilab Tevatron Collider data collected with the D0 detector. The analysis is sensitive to Higgs boson production via gluon fusion, associated vector boson production, and vector boson fusion, followed by the Higgs boson decay to tau lepton pairs or to W boson pairs. The ratios of 95% C.L. upper limits on the cross section times branching ratio to those predicted by the standard model are obtained for orthogonal subsamples that are enriched in either H -> tau tau decays or H -> WW decays, and for the combination of these subsample limits. The observed and expected limit ratios for the combined subsamples at a Higgs boson mass of 125 GeV are 11.3 and 9.0 respectively

Efficient intra- and inter-night linking of asteroid detections using kd-trees

The Panoramic Survey Telescope And Rapid Response System (Pan-STARRS) under development at the University of Hawaii's Institute for Astronomy is creating the first fully automated end-to-end Moving Object Processing System (MOPS) in the world. It will be capable of identifying detections of moving objects in our solar system and linking those detections within and between nights, attributing those detections to known objects, calculating initial and differentially-corrected orbits for linked detections, precovering detections when they exist, and orbit identification. Here we describe new kd-tree and variable-tree algorithms that allow fast, efficient, scalable linking of intra and inter-night detections. Using a pseudo-realistic simulation of the Pan-STARRS survey strategy incorporating weather, astrometric accuracy and false detections we have achieved nearly 100% efficiency and accuracy for intra-night linking and nearly 100% efficiency for inter-night linking within a lunation. At realistic sky-plane densities for both real and false detections the intra-night linking of detections into `tracks' currently has an accuracy of 0.3%. Successful tests of the MOPS on real source detections from the Spacewatch asteroid survey indicate that the MOPS is capable of identifying asteroids in real data.Comment: Accepted to Icaru