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Carnegie Mellon Univ. Dept. of Computer Science 15-415 - Database Applications

Carnegie Mellon Univ. Dept. of Computer Science 15-415 - Database Applications. C. Faloutsos Spatial Access Methods - R-trees. General Overview. Relational model – SQL; db design Indexing; Q-opt;Transaction processing Advanced topics Distributed Databases RAID Authorization / Stat. DB

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Carnegie Mellon Univ. Dept. of Computer Science 15-415 - Database Applications

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  1. Carnegie Mellon Univ.Dept. of Computer Science15-415 - Database Applications C. Faloutsos Spatial Access Methods - R-trees

  2. General Overview • Relational model – SQL; db design • Indexing; Q-opt;Transaction processing • Advanced topics • Distributed Databases • RAID • Authorization / Stat. DB • Spatial Access Methods (SAMs) • Multimedia Indexing 15-415 - C. Faloutsos

  3. SAMs - Detailed outline • spatial access methods • problem dfn • z-ordering • R-trees 15-415 - C. Faloutsos

  4. SAMs - more detailed outline • R-trees • main idea; file structure • (algorithms: insertion/split) • (deletion) • (search: range, nn, spatial joins) • variations (packed; hilbert;...) 15-415 - C. Faloutsos

  5. Reminder: problem • Given a collection of geometric objects (points, lines, polygons, ...) • organize them on disk, to answer spatial queries (range, nn, etc) 15-415 - C. Faloutsos

  6. R-trees z-ordering: cuts regions to pieces -> dup. elim. how could we avoid that? Idea: Minimum Bounding Rectangles 15-415 - C. Faloutsos

  7. R-trees [Guttman 84] Main idea: allow parents to overlap! => guaranteed 50% utilization => easier insertion/split algorithms. (only deal with Minimum Bounding Rectangles - MBRs) 15-415 - C. Faloutsos

  8. R-trees • eg., w/ fanout 4: group nearby rectangles to parent MBRs; each group -> disk page I C A G H F B J E D 15-415 - C. Faloutsos

  9. A H D F G B E I C J R-trees • eg., w/ fanout 4: P1 P3 I C A G H F B J E P4 P2 D 15-415 - C. Faloutsos

  10. P1 H A D F P2 G B E I P3 C J P4 R-trees • eg., w/ fanout 4: P1 P3 I C A G H F B J E P4 P2 D 15-415 - C. Faloutsos

  11. 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 ... 15-415 - C. Faloutsos

  12. P1 A P2 B P3 C P4 x-low; x-high y-low; y-high ... node ptr ... R-trees - format of nodes • {(MBR; node-ptr)} for non-leaf nodes 15-415 - C. Faloutsos

  13. P1 A H D F P2 G B E I P3 C J P4 R-trees - range search? P1 P3 I C A G H F B J E P4 P2 D 15-415 - C. Faloutsos

  14. P1 A H D F P2 G B E I P3 C J P4 R-trees - range search? P1 P3 I C A G H F B J E P4 P2 D 15-415 - C. Faloutsos

  15. R-trees - range search Observations: • 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.) 15-415 - C. Faloutsos

  16. R-trees - range search Observations - cont’d • a point query may follow multiple branches. • everything works for any dimensionality 15-415 - C. Faloutsos

  17. SAMs - more detailed outline • R-trees • main idea; file structure • algorithms: insertion/split • deletion • search: range, nn, spatial joins • performance analysis • variations (packed; hilbert;...) 15-415 - C. Faloutsos

  18. P1 A H D F P2 G B E I P3 C J P4 R-trees - insertion • eg., rectangle ‘X’ P1 P3 I C A G H F B X J E P4 P2 D 15-415 - C. Faloutsos

  19. P1 H A D F P2 G B E I P3 C J P4 R-trees - insertion • eg., rectangle ‘X’ P1 P3 I C A G H F B X J E P4 P2 D X 15-415 - C. Faloutsos

  20. P1 A H D F P2 G B E I P3 C J P4 skip R-trees - insertion • eg., rectangle ‘Y’ P1 P3 I C A G H F B J Y E P4 P2 D 15-415 - C. Faloutsos

  21. P1 A H D F P2 G B E I P3 C J P4 skip R-trees - insertion • eg., rectangle ‘Y’: extend suitable parent. P1 P3 I C A G H F B J Y E P4 P2 D Y 15-415 - C. Faloutsos

  22. skip R-trees - insertion • eg., rectangle ‘Y’: extend suitable parent. • Q: how to measure ‘suitability’? 15-415 - C. Faloutsos

  23. skip R-trees - insertion • eg., rectangle ‘Y’: extend suitable parent. • Q: how to measure ‘suitability’? • A: by increase in area (volume) (more details: later, under ‘performance analysis’) • Q: what if there is no room? how to split? 15-415 - C. Faloutsos

  24. P1 A H D F P2 G B E I P3 C J P4 skip R-trees - insertion • eg., rectangle ‘W’ P1 P3 K I C A G W H F B J K E P4 P2 D 15-415 - C. Faloutsos

  25. skip R-trees - insertion • eg., rectangle ‘W’ - focus on ‘P1’ - how to split? P1 K C A W B 15-415 - C. Faloutsos

  26. skip R-trees - insertion • eg., rectangle ‘W’ - focus on ‘P1’ - how to split? P1 • (A1: plane sweep, • until 50% of rectangles) • A2: ‘linear’ split • A3: quadratic split • A4: exponential split K C A W B 15-415 - C. Faloutsos

  27. seed2 R skip R-trees - insertion & split • pick two rectangles as ‘seeds’; • assign each rectangle ‘R’ to the ‘closest’ ‘seed’ seed1 15-415 - C. Faloutsos

  28. skip R-trees - insertion & split • pick two rectangles as ‘seeds’; • assign each rectangle ‘R’ to the ‘closest’ ‘seed’ • Q: how to measure ‘closeness’? 15-415 - C. Faloutsos

  29. skip R-trees - insertion & split • pick two rectangles as ‘seeds’; • assign each rectangle ‘R’ to the ‘closest’ ‘seed’ • Q: how to measure ‘closeness’? • A: by increase of area (volume) 15-415 - C. Faloutsos

  30. seed2 R skip R-trees - insertion & split • pick two rectangles as ‘seeds’; • assign each rectangle ‘R’ to the ‘closest’ ‘seed’ seed1 15-415 - C. Faloutsos

  31. seed2 R skip R-trees - insertion & split • pick two rectangles as ‘seeds’; • assign each rectangle ‘R’ to the ‘closest’ ‘seed’ seed1 15-415 - C. Faloutsos

  32. skip R-trees - insertion & split • pick two rectangles as ‘seeds’; • assign each rectangle ‘R’ to the ‘closest’ ‘seed’ • smart idea: pre-sort rectangles according to delta of closeness (ie., schedule easiest choices first!) 15-415 - C. Faloutsos

  33. skip R-trees - insertion - pseudocode • decide which parent to put new rectangle into (‘closest’ parent) • if overflow, split to two, using (say,) the quadratic split algorithm • propagate the split upwards, if necessary • update the MBRs of the affected parents. 15-415 - C. Faloutsos

  34. skip R-trees - insertion - observations • many more split algorithms exist (next!) 15-415 - C. Faloutsos

  35. skip SAMs - more detailed outline • R-trees • main idea; file structure • algorithms: insertion/split • deletion • search: range, nn, spatial joins • performance analysis • variations (packed; hilbert;...) 15-415 - C. Faloutsos

  36. skip R-trees - deletion • delete rectangle • if underflow • ?? 15-415 - C. Faloutsos

  37. skip R-trees - deletion • delete rectangle • if underflow • temporarily delete all siblings (!); • delete the parent node and • re-insert them 15-415 - C. Faloutsos

  38. SAMs - more detailed outline • R-trees • main idea; file structure • algorithms: insertion/split • deletion • search: range, nn, spatial joins • performance analysis • variations (packed; hilbert;...) 15-415 - C. Faloutsos

  39. R-trees - range search pseudocode: check the root for each branch, if its MBR intersects the query rectangle apply range-search (or print out, if this is a leaf) 15-415 - C. Faloutsos

  40. P1 P3 I C A G H F B J q E P4 P2 D skip R-trees - nn search 15-415 - C. Faloutsos

  41. P1 P3 I C A G H F B J q E P4 P2 D skip R-trees - nn search • Q: How? (find near neighbor; refine...) 15-415 - C. Faloutsos

  42. skip R-trees - nn search • A1: depth-first search; then, range query P1 P3 I C A G H F B J E P4 q P2 D 15-415 - C. Faloutsos

  43. skip R-trees - nn search • A1: depth-first search; then, range query P1 P3 I C A G H F B J E P4 q P2 D 15-415 - C. Faloutsos

  44. skip R-trees - nn search • A1: depth-first search; then, range query P1 P3 I C A G H F B J E P4 q P2 D 15-415 - C. Faloutsos

  45. skip R-trees - nn search • A2: [Roussopoulos+, sigmod95]: • priority queue, with promising MBRs, and their best and worst-case distance • main idea: 15-415 - C. Faloutsos

  46. P1 P3 I C A G H F B J E P4 P2 D skip R-trees - nn search consider only P2 and P4, for illustration q 15-415 - C. Faloutsos

  47. skip R-trees - nn search best of P4 => P4 is useless for 1-nn worst of P2 H J E P4 q P2 D 15-415 - C. Faloutsos

  48. skip R-trees - nn search • what is really the worst of, say, P2? worst of P2 E q P2 D 15-415 - C. Faloutsos

  49. skip R-trees - nn search • what is really the worst of, say, P2? • A: the smallest of the two red segments! q P2 15-415 - C. Faloutsos

  50. skip R-trees - nn search • variations: [Hjaltason & Samet] incremental nn: • build a priority queue • scan enough of the tree, to make sure you have the k nn • to find the (k+1)-th, check the queue, and scan some more of the tree • ‘optimal’ (but, may need too much memory) 15-415 - C. Faloutsos

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