Foundations of Multidimensional and Metric Data Structures

Transcription

1 Foundations of Multidimensional and Metric Data Structures Hanan Samet University of Maryland, College Park ELSEVIER AMSTERDAM BOSTON HEIDELBERG LONDON NEW YORK OXFORD PARIS SAN DIEGO SAN FRANCISCO SINGAPORE SYDNEY TOKYO Morgan Kaufmann Publishers is an imprint of Elsevier MORGAN KAUFMANN PUBLISHERS

2 Contents Foreword Preface xv xvii 1 Multidimensional Point Data Introduction Range Trees Priority Search Trees Quadtrees Point Quadtrees Insertion Deletion Search Trie-Based Quadtrees MX Quadtree PR Quadtrees Comparison of Point and Trie-Based Quadtrees K-d Trees Point K-d Trees Insertion Deletion Search Point K-d Tree Variants Trie-Based K-d Trees PR K-d Tree, 71, Sliding-Midpoint K-d Tree Bucket PR K-d Tree and PMR K-d Tree Path-Compressed PR K-d Tree Path-Level Compressed PR K-d Tree BD-Trees Balanced Box-Decomposition Tree (BBD-Tree) , Conjugation Tree j One-Dimensional Orderings * Bucket Methods atree Directory Methods K-d-B-Tree Hybrid Tree 101 /i LSD Tree hb-tree K-d-B-Tries BV-Tree Definition _ Building a BV-Tree 120 IX

3 Contents Searching a BV-Tree ' Performance Summary Static Methods Grid Directory Methods Grid File EXCELL Linear Hashing Adapting Linear Hashing to Multidimensional Data Multidimensional Extendible Hashing (MDEH) , Dynamic Z Hashing Linear Hashing with Partial Expansions (LHPE) Alternative Implementations of Linear Hashing Quantile Hashing Piecewise Linear Order Preserving (PLOP) Hashing Spiral Hashing Summary Comparison Storage Utilization PK-Trees Motivation Overview Definition Comparison with Bucket Methods : Operations Representation and Searching Insertion Deletion Discussion Conclusion : Object-Based and Image-Based Image Representations Interior-Based Representations Unit-Size Cells Cell Shapes and Tilings Ordering Space Array Access Structures Tree Access Structures Blocks Decomposition into Arbitrarily Sized Rectangular Blocks Medial Axis Transformation (MAT) Irregular Grid Region Quadtree and Region Octree ATree 220 ^ Bintree Adaptation of the Point Quadtree for Regions Adaptation of the K-d Tree for Regions 223 A f X-Y Tree, Treemap, and Puzzletree Hexagonal Blocks Triangular Blocks 231» Nonorthogonal Blocks BSPTree 233 n K-Structure Separating Chain Overview 242 / Layered Dag Arbitrary Objects Coverage-Based Splitting Density-Based Splitting Hierarchical Interior-Based Representations 264

4 Image-Based Hierarchical Interior-Based Representations Contents (Pyramids), Object-Based Hierarchical Interior-Based Representations (R-trees, Bounding Box Hierarchies) Overview Ordering-Based Aggregation Techniques Extent-Based Aggregation Techniques R*-Tree : Bulk Insertion Bulk Loading Shortcomings and Solutions Disjoint Object-Based Hierarchical Interior-Based Representations (k-d-b-tree,.r+-tree, and Cell Tree) Boundary-Based Representations The Boundary Model (BRep) Overview Edge-Based Methods Using Winged Edges Vertex-Based and Face-Based Methods Using Laths Lath Data Structures for Manifold Objects Lath Data Structures for Meshes with Boundaries Summary Voronoi Diagrams, Delaunay Graphs, and Delaunay Triangulations Constrained and Conforming Delaunay Triangulations Delaunay Tetrahedralizations Applications of the Winged-Edge Data Structure (Triangle Table, Corner Table) Image-Based Boundary Representations Line Quadtree MX Quadtree and MX Octree Edge Quadtree Face Octree Adaptively Sampled Distance Fields PM Quadtrees PM Octree ; Bucket PM Quadtree and PMR Quadtree Sector Tree Cone Tree Object-Based Boundary Representations Strip Tree, Arc Tree, and BSPR Prism Tree, HAL Tree Simplification Methods and Data Structures Surface-Based Boundary Representations Difference-Based Compaction Methods ,1 Runlength Encoding Chain Code Vertex Representation Historical Overview Intervals and Small Rectangles Plane-Sweep Methods and the Rectangle Intersection Problem 428 3Tl.1 Segment Tree Interval Tree Priority Search Tree L4 Alternative Solutions and Related Problems Plane-Sweep Methods and the Measure Problem Point-Based Methods Representative Points Collections of Representative Points Corner Stitching _. 459 XI

5 \ Contents Balanced Four-Dimensional K-d Tree ' Grid File! LSD Tree Two-Dimensional Representations Summary Area-Based Methods ' MX-CIF Quadtree Definition Operations Comparison with Other Representations Alternatives to the MX-CIF Quadtree Multiple Quadtree Block Representations High-Dimensional Data Best-First Nearest Neighbor Finding Motivation Search Hierarchy Algorithm Algorithm with Duplicate Instances of Objects Algorithm Extensions (K-Nearest, AT-Farthest, Skylines) Nearest Neighbors in a Spatial Network Related Work The Depth-First /st-nearest Neighbor Algorithm Basic Algorithm Pruning Rules Effects of Clustering Methods on Pruning Ordering the Processing of the Elements of the Active List Improved Algorithm Incorporating MAXNEARESTDIST in a Best-First Algorithm Example Comparison Approximate Nearest Neighbor Finding Multidimensional Indexing Methods X-Tree Bounding Sphere Methods: Sphere Tree, SS-Tree, Balltree, and SR-Tree Increasing the Fanout: TV-Tree, Hybrid Tree, and A-Tree Methods Based on the Voronoi Diagram: OS-Tree Approximate Voronoi Diagram (AVD) Avoiding Overlapping All of the Leaf Blocks Pyramid Technique Methods Based on a Sequential Scan Distance-Based Indexing Methods Distance Metric and Search Pruning Ball Partitioning Methods Vp-Tree Structure Search 606 f Vp sb -Tree Mvp-Tree Other Methods Related to Ball Partitioning Generalized Hyperplane Partitioning Methods Gh-Tree 613 s GNAT Bisector Tree and Mb-Tree 617 f Other Methods Related to Generalized Hyperplane Partitioning / M-Tree Structure Search Sa-Tree Definition..." Search 633 Xll

6 4.5.6 knn Graph 637 Contents Distance Matrix Methods AESA LAESA Other Distance Matrix Methods SASH: Indexing without Using the Triangle Inequality Dimension Reduction Methods Searching in the Dimension Reduced Space Range Queries Nearest Neighbor Queries Using Only One Dimension Representative Point Methods Transformation into a Different and Smaller Feature Set SVD Discrete Fourier Transform (DFT) Summary Embedding Methods Introduction." Lipschitz Embeddings Definition Selecting Reference Sets Example of a Lipschitz Embedding SparseMap FastMap Mechanics of FastMap Choosing Pivot Objects Deriving the First Coordinate Value Projected Distance Subsequent Iterations Pruning Property Expansion Heuristic for Non-Euclidean Metrics Summary and Comparison with Other Embeddings Locality Sensitive Hashing 711 Appendix A: Overview of B-Trees 717 Appendix B: Linear Hashing 729 Appendix C: Spiral Hashing 735 Appendix D: Description of Pseudocode Language 743 Solutions to Exercises 747 References 877 Reference Keyword Index 947 Author and Credit Index 953 Subject Index 969 XU1