1. Jeremy Iverson & Zhang Yun 1

2.  Chapter 6 Key Concepts ◦ Structures and access methods ◦ R-Tree  R*-Tree  Mobile Object Indexing  Questions 2

3.  Indexes are used to efficiently locate data on hard disk  1D ◦ Indexes that are based on one key value  B and B + -trees  2D ◦ Indexes based on two key values  Ordered tilings 3

4.  Structures store data for efficient modification and querying  Types of data to store ◦ Raster (Region quadtrees) ◦ Point Object (2D trees) ◦ Linear (PM quadtrees) ◦ Collections of objects (R-trees) ◦ Spherical (QTM region quadtrees) 4

5. R-tree  A balanced tree to index spatial objects  Shape of objects is approximated by minimum bounding rectangle  Rectangles at any level may overlap 5 R-tree review

6. 6 R-tree and its limitations How to build R-tree  Given a set of spatial objects, build a R-tree is based on heuristic  R-tree is designed to minimize the area of containing rectangles Limitations  Heuristic of R-tree may cause much overlap  Cause other problems like uneven distribution Spatial objects setOne split methodAnother split method R-tree prefer this

7. New index  Better than R-tree performance  Support Multi spatial object types (e.g. point, polygon) Possible applications  Support spatial query processing (e.g. online map service)  Support imagine processing 7 Motivations

8. More heuristics  H1:The area covered by directory rectangles should be minimized  H2:The overlap between directory rectangles should be minimized  H3:Make bounding rectangles as square as possible  H4:The storage utilization should be optimized—reduce height of tree 8 R*-tree Heuristics may conflict Choose best design from experiments Spatial objects set H1: area minimum H2: overlap minimum

9. Insert new object  Minimize the overlap  Choose the entry in R*-tree whose rectangle needs least overlap enlargement to include the new object 9 R*-tree Operations R-tree: minimize area enlargement R*-tree: minimize overlap enlargement Spatial objects set R-tree

10.  R*-tree (also R-tree) suffer from the sequences of insertions  Reorganization of tree is necessary  Compute the distance between the centers of their rectangles and the center of the bounding rectangle, remove top k rectangle with maximum distance  Invoke insert operation for removed rectangles 10 R*-tree Reinsertion Spatial Objects Calculate distance Remove object A Reinsert object A

11. R*-tree highlights  Use more heuristics, design validated from experiments  Perform significantly better than R-tree Limitations  No concept for moving object  Not designed for spatio-temporal objects 11 R*-tree and its limitations

12.  Naïve Approach ◦ y(t)=vt+a ◦ v: velocity ◦ t: time ◦ a: intercept ◦ Query is expressed as 2D interval [(y1q,y2q),(t1q,t2q)] 12

13.  Benefits ◦ Intuitive representation  Drawbacks ◦ Length of lines is infinite  A lot of redundancy  High overhead for updates 13

14.  Time-Parameterized R-Tree ◦ Actually extends the R*-tree  A moving object o is represented with ◦ MBR ◦ Velocity Bounding Rectangle (VBR) of the form  o V ={o V1-,o V1+,o V2-,o V2+ }  o vi- represents the lower bound for velocity in dimension i  o vi+ represents the upper bound for velocity in dimension i 14

15. a v ={1,1,1,1} b v ={-2,-2,-2,-2} c v ={-2,0,0,-2} d v ={-1,-1,1,1} 15 *MBRs for non-leaf nodes are not required to always be minimum, only minimum at some time step.

16.  TPR-Tree allows one to index and query moving objects  TPR-Tree creates index structures much worse than optimal [Tao et al.] ◦ Thus, the TPR*-Tree is introduced, which considers multiple paths when inserting an object into the index structure, creating an index much closer to optimal 16

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