MULTI-RESOLUTION DATA STRUCTURES FOR SPHERICALLY MAPPED POINT DATA by ROBERT WILLIAM YOUNGREN A DISSERTATION Submitted in partial fulfillment of the requirements for the Degree of Doctor of Philosophy in Computer Science to The School of Graduate Studies of The University of Alabama in Huntsville HUNTSVILLE, ALABAMA 2017 ABSTRACT The School of Graduate Studies The University of Alabama in Huntsville Degree Doctor of Philosophy College/Dept Science/Computer Science_ Name of Candidate Robert Youngren__________________________ Title Multi-Resolution Data Structures for Spherically Mapped Point Data_______ Data describing entities or objects whose locations may be treated as points on the surface of a sphere are said to be spherically mapped. A number of data structures specifically designed to store and access spherically mapped data have been developed. One of them, Hierarchical Equal Area iso-Latitude Pixelization (HEALPix), has been successfully used for numerous applications, notably including organizing and analyzing cosmic microwave background data. However, for applications involving relatively sparse spherically mapped point datasets, HEALPix has some drawbacks, including inefficient memory requirements due to fixed resolution, overwriting of data for closely proximate points, and return of spurious points in response to certain queries. A multi-resolution variant of the HEALPix data structure optimized for point data was developed to address these issues. The new data structure is multi-resolution in that different portions of the sphere may be subdivided at different levels of resolution in the same data structure, depending on the data to be stored. It combines the best aspects of iv HEALPix with the advantages of multi-resolution, including reduced memory requirements, improved query efficiency, and flexible handling of proximate points. The new Multi-resolution HEALPix (MRH) data structure uses multiple quadtrees and the Morton cell addressing scheme. An implementation of the MRH data structure was tested using four sets of spherically mapped point data from different scientific applications (warhead fragmentation trajectories, weather station locations, redshifted galaxy locations, and synthetic locations). A large set of randomly generated range queries of four different types (disc, polygon, latitude strip, and neighbor) was applied to each data structure for each dataset. MRH used from two to four orders of magnitude less memory than HEALPix to store the same data, and on average MRH queries executed 72% faster. Additional effort to develop a three dimensional variant of MRH was explored. The new data structure, Multi-MRH, adds an additional degree of freedom (temporal or spatial) into an entirely new data and applications. In Multi-MRH, multiple instances of MRH are utilized to store either temporal, same data point locations at different times, or spatial, data points with spherical coordinates including radius, spherical data. The Multi-MRH data structure consists of a sorted list of MRH maps. An implementation of the Multi-MRH data structure was tested using three sets of spherically mapped point data from different scientific applications (synthetic locations, warhead fragmentation trajectories, and NEXRAD wind velocity data). A large set of randomly generated range queries of four different types (cone, prism, band, and ray) was v ACKNOWLEDGMENTS I owe a huge debt of gratitude to my academic advisor Dr. Mikel D. Petty. His endless enthusiasm, patience, and persistence have helped me keep making steady forward progress towards the dissertation finish line. I would also like to thank Dr. Wesley N. Colley, who first brought our attention to the HEALPix data structure. Special thank you to my mother-in-law Mary Ann who spent many hours editing this dissertation for grammar and provided so many useful suggestions to improve the language and flow. I also must thank my wife Kathy and my dogs, Trails, Cairo, Tracks, and Sparta for putting up with me in this endless seeming endeavor and constantly motivating me to persist to the finish line. Lastly, I want to dedicate this dissertation and all the hard work that came with it and before it to my father, Ronald William Youngren, who sadly passed away just as I started down this long path; he would have really appreciated the effort, sympathized with my struggles and have been proud of the final result. vii Table of Contents Page LIST OF FIGURES ............................................................................................. xvi LIST OF TABLES ...............................................................................................xxx 1 Introduction .......................................................................................................1 2 Survey of Spherical Data Structures .................................................................7 2.1 Introduction ................................................................................................7 2.2 Quadtrees and Octrees ................................................................................8 2.3 Platonic Spherical Data Structures ...........................................................20 2.3.1 Hierarchical Triangular Mesh............................................................21 2.3.2 Quaternary Triangular Mesh .............................................................26 2.3.3 Hierarchical Spatial Data Structure ...................................................36 2.3.4 Semi-Quad Code ATM......................................................................38 2.3.5 Sphere Quadtree ................................................................................39 2.3.6 Truncated Icosahedron ......................................................................44 2.3.7 Quadrilateralized Spherical Cube ......................................................47 2.3.8 Hexagonal Sampling Grids with Icosahedral Symmetry ..................50 2.3.9 Linear Quadtree .................................................................................52 2.3.10 Icosahedral Snyder Equal Area Variants ...........................................56 viii 2.3.11 Diamond Partitioning Platonic Solids ...............................................58 2.3.12 Linear Quadtree QTM .......................................................................61 2.4 Igloo Spherical Data Structures ................................................................64 2.4.1 Igloo Type Constructions ..................................................................64 2.4.2 Gauss-Legendre Sky Pixelization ......................................................74 2.5 Hierarchical Equal Area iso-Latitude Pixelization ...................................77 2.5.1 HEALPix: Applications.....................................................................82 2.5.2 Recent Independent Developments Involving HEALPix..................85 2.5.3 HEALPix Anomalies .........................................................................94 2.5.4 Problems with HEALPix and Point Data ..........................................94 3 Research overview ..........................................................................................99 4 Multi-Resolution HEALPix Data Structure ..................................................101 4.1 Overview ................................................................................................101 4.2 MRH Map Description ...........................................................................103 4.3 Morton Cell Addressing Overview ........................................................104 4.4 Morton Datatype Definition ...................................................................108 4.5 Morton and HEALPix Address Relationship .........................................109 4.6 MLQ Definition ......................................................................................113 ix 4.7 Building the MRH Data Structure ..........................................................116 4.8 Searching the MRH Data Structure ........................................................121 4.8.1 Search MRH Data Structure by (Φ, θ) ............................................122 4.8.2 Search MRH Data Structure by HEALPix (index, level) Pair ........123 4.9 MRH Range Queries ..............................................................................126 4.9.1 Summary of Query Types and Algorithms .....................................127 4.9.2 Query Worst Case Computational Complexity Summary ..............130 4.9.3 Polygon Query Description .............................................................132 4.9.4 Polygon Query Worst Case Analysis ..............................................142 4.9.5 Disc Query Description ...................................................................147 4.9.6 Disc Query Worst Case Analysis ....................................................155 4.9.7 Latitude Strip Query Description ....................................................160 4.9.8 Latitude Strip Query Worst Case Analysis .....................................164 4.9.9 Neighbor Query Description ...........................................................165 4.9.10 Neighbor Query Worst Case Analysis ............................................172 4.10 Archiving the MRH Data Structure .....................................................174 5 Benchmarking Datasets and Methodology ...................................................177 5.1 Warhead Fragmentation Flyout Data .....................................................179