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Spatial Indexing

Spatial Indexing. SAMs. Spatial Indexing. Point Access Methods can index only points. What about regions? Z-ordering and quadtrees Use the transformation technique and a PAM New methods: Spatial Access Methods SAMs R-tree and variations. Problem.

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Spatial Indexing

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  1. Spatial Indexing SAMs

  2. Spatial Indexing • Point Access Methods can index only points. What about regions? • Z-ordering and quadtrees • Use the transformation technique and a PAM • New methods: Spatial Access Methods SAMs • R-tree and variations

  3. Problem • Given a collection of geometric objects (points, lines, polygons, ...) • organize them on disk, to answer spatial queries (range, nn, etc)

  4. Transformation Technique • Map an d-dim MBR into a point: ex. [(xmin, xmax) (ymin, ymax)] => (xmin, xmax, ymin, ymax) • Use a PAM to index the 2d points • Given a range query, map the query into the 2d space and use the PAM to answer it

  5. R-trees • [Guttman 84] Main idea: allow parents to overlap! • => guaranteed 50% utilization • => easier insertion/split algorithms. • (only deal with Minimum Bounding Rectangles - MBRs)

  6. R-trees • A multi-way external memory tree • Index nodes and data (leaf) nodes • All leaf nodes appear on the same level • Every node contains between m and M entries • The root node has at least 2 entries (children)

  7. Example • eg., w/ fanout 4: group nearby rectangles to parent MBRs; each group -> disk page I C A G H F B J E D

  8. A H D F G B E I C J Example • F=4 P1 P3 I C A G H F B J E P4 P2 D

  9. P1 A H D F P2 G B E I P3 C J P4 Example • F=4 P1 P3 I C A G H F B J E P4 P2 D

  10. P1 A P2 B P3 C P4 R-trees - format of nodes • {(MBR; obj_ptr)} for leaf nodes x-low; x-high y-low; y-high ... obj ptr ...

  11. P1 A P2 B P3 C P4 R-trees - format of nodes • {(MBR; node_ptr)} for non-leaf nodes x-low; x-high y-low; y-high ... node ptr ...

  12. P1 A H D F P2 G B E I P3 C J P4 R-trees:Search P1 P3 I C A G H F B J E P4 P2 D

  13. P1 A H D F P2 G B E I P3 C J P4 R-trees:Search P1 P3 I C A G H F B J E P4 P2 D

  14. R-trees:Search • Main points: • every parent node completely covers its ‘children’ • a child MBR may be covered by more than one parent - it is stored under ONLY ONE of them. (ie., no need for dup. elim.) • a point query may follow multiple branches. • everything works for any(?) dimensionality

  15. P1 A H D F P2 G B E I P3 C J P4 R-trees:Insertion Insert X P1 P3 I C A G H F B X J E P4 P2 D X

  16. P1 A H D F P2 G B E I P3 C J P4 R-trees:Insertion Insert Y P1 P3 I C A G H F B J Y E P4 P2 D

  17. P1 A H D F P2 G B E I P3 C J P4 R-trees:Insertion • Extend the parent MBR P1 P3 I C A G H F B J Y E P4 P2 D Y

  18. R-trees:Insertion • How to find the next node to insert the new object? • Using ChooseLeaf: Find the entry that needs the least enlargement to include Y. Resolve ties using the area (smallest) • Other methods (later)

  19. P1 A H D F P2 G B E I P3 C J P4 R-trees:Insertion • If node is full then Split : ex. Insert w P1 P3 K I C A G W H F B J K E P4 P2 D

  20. Q1 P3 P1 D H A C F Q2 P5 G K B E I P2 W J P4 R-trees:Insertion • If node is full then Split : ex. Insert w P3 P5 I K C A P1 G W H F B J E P4 P2 D Q2 Q1

  21. R-trees:Split • Split node P1: partition the MBRs into two groups. • (A1: plane sweep, • until 50% of rectangles) • A2: ‘linear’ split • A3: quadratic split • A4: exponential split: • 2M-1 choices P1 K C A W B

  22. seed2 R R-trees:Split • pick two rectangles as ‘seeds’; • assign each rectangle ‘R’ to the ‘closest’ ‘seed’ seed1

  23. seed2 R R-trees:Split • pick two rectangles as ‘seeds’; • assign each rectangle ‘R’ to the ‘closest’ ‘seed’: • ‘closest’: the smallest increase in area seed1

  24. R-trees:Split • How to pick Seeds: • Linear:Find the highest and lowest side in each dimension, normalize the separations, choose the pair with the greatest normalized separation • Quadratic: For each pair E1 and E2, calculate the rectangle J=MBR(E1, E2) and d= J-E1-E2. Choose the pair with the largest d

  25. R-trees:Insertion • Use the ChooseLeaf to find the leaf node to insert an entry E • If leaf node is full, then Split, otherwise insert there • Propagate the split upwards, if necessary • Adjust parent nodes

  26. R-Trees:Deletion • Find the leaf node that contains the entry E • Remove E from this node • If underflow: • Eliminate the node by removing the node entries and the parent entry • Reinsert the orphaned (other entries) into the tree using Insert • Other method (later)

  27. R-trees: Variations • R+-tree: DO not allow overlapping, so split the objects (similar to z-values) • R*-tree: change the insertion, deletion algorithms (minimize not only area but also perimeter, forced re-insertion ) • Hilbert R-tree: use the Hilbert values to insert objects into the tree

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