Chapter 6: Structures and Access Methods

1. Chapter 6: Structures and Access Methods Jeremy Iverson & Zhang Yun

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

3. Access Methods • 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

4. Structures • 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)

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

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 R-tree prefer this Spatial objects set One split method Another split method

7. Motivations 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

8. R*-tree Heuristics may conflict Choose best design from experiments 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 H1: area minimum H2: overlap minimum Spatial objects set

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

10. R*-tree Reinsertion • 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 Spatial Objects Calculate distance Remove object A Reinsert object A

11. R*-tree and its limitations 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

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

13. Conclusions about Naïve Approach • Benefits • Intuitive representation • Drawbacks • Length of lines is infinite • A lot of redundancy • High overhead for updates

14. TPR-Tree • Time-Parameterized R-Tree • Actually extends the R*-tree • A moving object o is represented with • MBR • Velocity Bounding Rectangle (VBR) of the form • oV={oV1-,oV1+,oV2-,oV2+} • ovi-represents the lower bound for velocity in dimension i • ovi+represents the upper bound for velocity in dimension i

15. av={1,1,1,1} bv={-2,-2,-2,-2} cv={-2,0,0,-2} dv={-1,-1,1,1} *MBRs for non-leaf nodes are not required to always be minimum, only minimum at some time step.

16. Conclusions about TPR-Tree • 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

17. QUESTIONS?