Foundations of Multidimensional and Metric Data Structures

Chapter 1: Multidimensional Point Data

1.1 Introduction

1.2 Range Trees

1.3 Priority Search Trees

1.4 Quadtrees

1.4.1 Point Quadtrees

1.4.2 Trie-based Quadtree

1.4.3 Comparison of Point and Trie-based Quadtrees

1.5 K-d Trees

1.5.1 Point K-d Trees

1.5.2 Trie-based K-d Trees

1.5.3 Conjugation Tree

1.6 One-dimensional Orderings

1.7 Bucket Methods

1.7.1 Tree Directory Methods (K-d-B-tree, Hybrid Tree, LSD Tree, hB-tree, K-d-B-trie, BV-tree)

1.7.2 Grid Directory Methods (Grid File, EXCELL, Linear Hashing, Spiral Hashing)

1.7.3 Storage Utilization

1.8 PK-tree

1.9 Conclusion

Chapter 2: Object-based and Image-based Image Representations

2.1 Interior-based Representations

2.1.1 Unit-size Cells

2.1.2 Blocks (Medial Axis Transform, Region Quadtree and Octree, Bintree, X-Y Tree)

2.1.3 Nonorthogonal Blocks (BSP Tree, Layered DAG)

2.1.4 Arbitrary Objects (Loose Octree, Field Tree, PMR Quadtree)

2.1.5 Hierarchical Interior-based Representations (Pyramid, R-tree, Hilbert R-tree, R*-tree, R+-tree, Packed R-tree, R-tree, Cell Tree, Bulk Loading)

2.2 Boundary-based Representations

2.2.1 The Boundary Model (CSG,BREP, Winged-edge, Voronoi Diagram, Delaunay Triangulation, Tetrahedra)

2.2.2 Image-based Boundary Representations (PM Quadtree and Octree, Adaptively Sampled Distance Field)

2.2.3 Object-based Boundary Representation (LOD, Strip Tree, Simplification Methods)

2.2.4 Surface-based Boundary Representations (TIN)

2.3 Difference-based Compaction Methods

2.3.1 Runlength Encoding

2.3.2 Chain Code

2.3.3 Vertex Representation

2.4 Historical Overview

Chapter 3: Intervals and Small Rectangles

3.1 Plane-sweep Methods and the Rectangle Intersection Problem

3.1.1 Segment Tree

3.1.2 Interval Tree

3.1.3 Priority Search Tree

3.1.4 Alternative Solutions and Related Problems

3.2 Plane-sweep Methods and the Measure Problem

3.3 Point-based Methods

3.3.1 Representative Points

3.3.2 Collections of Representative Points

3.3.3 LSD Tree

3.3.4 Summary

3.4 Area-based Methods

3.4.1 MX-CIF Quadtree

3.4.2 Alternatives to the MX-CIF Quadtree (HV/VH Tree)

3.4.3 Multiple Quadtree Block Representations

Chapter 4: High-Dimensional Data

4.1 Best-first Incremental Nearest Neighbor Finding (Ranking)

4.1.1 Motivation

4.1.2 Search Hierarchy

4.1.3 Algorithm

4.1.4 Duplicate Objects

4.1.5 Algorithm Extensions (Farthest Neighbor, Skylines)

4.1.6 Related Work

4.2 The Depth-first K-nearest Neighbor Algorithm

4.2.1 Basic Algorithm

4.2.2 Pruning Rules

4.2.3 Effects of Clustering Methods on Pruning

4.2.4 Ordering the Processing of the Elements of the Active List

4.2.5 Improved Algorithm

4.2.6 Incorporating MaxNearestDist in a Best-first Algorithm

4.2.7 Example

4.2.8 Comparison

4.3 Approximate Nearest Neighbor Finding

4.4 Multidimensional Indexing Methods

4.4.1 X-tree

4.4.2 Bounding Sphere Methods: Sphere Tree, SS-tree, Balltree, and SR-tree

4.4.3 Increasing the Fanout: TV-tree, Hybrid Tree, and A-tree

4.4.4 Methods Based on the Voronoi Diagram: OS-tree

4.4.5 Approximate Voronoi Diagram (AVD)

4.4.6 Avoiding Overlapping All of the Leaf Blocks

4.4.7 Pyramid Technique

4.4.8 Sequential Scan Methods (VA-file, IQ-tree, VA+-file)

4.5 Distance-based Indexing Methods

4.5.1 Distance Metric and Search Pruning

4.5.2 Ball Partitioning Methods (VP-tree, MVP-tree)

4.5.3 Generalized Hyperplane Partitioning Methods (GH-tree, GNAT, MB-tree)

4.5.4 M-tree

4.5.5 Sa-tree

4.5.6 kNN Graph

4.5.7 Distance Matrix Methods

4.5.8 SASH - Indexing Without Using the Triangle Inequality

4.6 Dimension-reduction Methods

4.6.1 Searching in the Dimensionally-reduced Space

4.6.2 Using Only One Dimension

4.6.3 Representative Point Methods

4.6.4 Transformation into a Different and Smaller Feature Set (SVD,DFT)

4.6.5 Summary

4.7 Embedding Methods

4.7.1 Introduction

4.7.2 Lipschitz Embeddings

4.7.3 FastMap

4.7.4 Locality Sensitive Hashing (LSH)

Appendix 1: Overview of B-trees
Appendix 2: Linear Hashing
Appendix 3: Spiral Hashing
Appendix 4: Description of Pseudo-Code Language
Solutions to Exercises
Bibliography
Index