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Spatial Query Processing using the R-tree

Spatial Query Processing using the R-tree. Donghui Zhang CCIS, Northeastern University Feb 8, 2005. Content. The R-tree Range Query Aggregation Query NN Query RNN Query Closest Pair Query Close Pair Query Skyline Query. R-Tree Motivation. y axis. 10. m. g. h. l. 8. k. f. e.

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Spatial Query Processing using the R-tree

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  1. Spatial Query Processing using the R-tree Donghui Zhang CCIS, Northeastern University Feb 8, 2005

  2. Content • The R-tree • Range Query • Aggregation Query • NN Query • RNN Query • Closest Pair Query • Close Pair Query • Skyline Query

  3. R-Tree Motivation y axis 10 m g h l 8 k f e 6 i j d 4 b a 2 c x axis 10 0 8 2 4 6 Range query: find the objects in a given range. E.g. find all hotels in Boston. No index: scan through all objects. NOT EFFICIENT!

  4. R-Tree: Clustering by Proximity

  5. R-Tree

  6. R-Tree

  7. Range Query y axis 10 m g h l 8 k f e E 2 6 i j E d 1 4 b a 2 c x axis 10 0 8 2 4 6 Root E E 1 2 E E E E E E 1 E 3 4 5 6 7 2 e a c d g b f j m i l h k E E E E E 4 5 3 6 7

  8. Range Query y axis 10 m g h l 8 k f e E 2 6 i j E d 1 4 b a 2 c x axis 10 0 8 2 4 6 Root E E 1 2 E E E E E E 1 E 3 4 5 6 7 2 e a c d g b f j m i l h k E E E E E 4 5 3 6 7

  9. Aggregation Query • Given a range, find some aggregate value of objects in this range. • COUNT, SUM, AVG, MIN, MAX • E.g. find the total number of hotels in Massachusetts. • Straightforward approach: reduce to a range query. • Better approach: along with each index entry, store aggregate of the sub-tree.

  10. Aggregation Query y axis 10 m g h l 8 k f e E 2 6 i j E d 1 4 b a 2 c x axis 10 0 8 2 4 6 Root E :8 E :5 1 2 E E :3 E :2 E :3 E :3 E :2 1 E 3 4 5 6 7 2 e a c d g b f j m i l h k E E E E E 4 5 3 6 7

  11. Aggregation Query y axis 10 m g h l 8 k f e E 2 6 i j E d 1 4 b a Subtree pruned! 2 c x axis 10 0 8 2 4 6 Root E :8 E :5 1 2 E E :3 E :2 E :3 E :3 E :2 1 E 3 4 5 6 7 2 e a c d g b f j m i l h k E E E E E 4 5 3 6 7

  12. Content • The R-tree • Range Query • Aggregation Query • NN Query • RNN Query • Closest Pair Query • Close Pair Query • Skyline Query

  13. Nearest Neighbor (NN) Query • Given a query location q, find the nearest object. • E.g.: given a hotel, find its nearest bar. a q

  14. A Useful Metric: MINDIST • Minimum distance between q and an MBR. • It is an lower bound of d(o, q) for every object o in E1. • MINDIST(o, q) = d(o, q). E1 q

  15. NN Basic Algorithm E1 q • Keep a heap H of index entries and objects, ordered by MINDIST. • Initially, H contains the root. • While H  • Extract the element with minimum MINDIST • If it is an index entry, insert its children into H. • If it is an object, return it as NN. • End while

  16. NN Query Example y axis 10 m E E 7 2 g Action Heap h l 8 E 6 Visit Root E E k 1 2 1 2 f E e 5 6 E i j 4 E 1 query d 4 E 3 b a 2 c x axis 10 0 8 2 4 6 Root E E 1 2 1 2 E E E E E E 3 4 5 6 7 1 E 2 9 5 13 5 2 a c b j i k e d g f m l h 2 10 13 E E E E E 4 5 3 6 7

  17. NN Query Example y axis 10 m E E 7 2 g Action Heap h l 8 E 6 Visit Root E E k 1 2 1 2 f E e 5 E follow E E E E 6 5 5 9 1 3 5 4 2 E 2 i j 4 E 1 query d 4 E 3 b a 2 c x axis 10 0 8 2 4 6 Root E E 1 2 1 2 E E E E E E 3 4 5 6 7 1 E 2 9 5 13 5 2 a c b j i k e d g f m l h 2 10 13 E E E E E 4 5 3 6 7

  18. NN Query Example y axis 10 m E E 7 2 g Action Heap h l 8 E 6 Visit Root E E k 1 2 1 2 f E e 5 E follow E E E E 6 5 5 9 1 3 5 4 2 E 2 i j 4 E E follow E E E E E 1 query 5 5 9 2 13 d 3 5 4 2 7 6 4 E 3 b a 2 c x axis 10 0 8 2 4 6 Root E E 1 2 1 2 E E E E E E 3 4 5 6 7 1 E 2 9 5 13 5 2 a c b j i k e d g f m l h 2 10 13 E E E E E 4 5 3 6 7

  19. NN Query Example y axis 10 m E E 7 2 g Action Heap h l 8 E 6 Visit Root E E k 1 2 1 2 f E e 5 E follow E E E E 6 5 5 9 1 3 5 4 2 E 2 i j 4 E E follow E E E E E 1 query 5 5 9 2 13 d 3 5 4 2 7 6 4 E E follow E j i E E k E 3 13 10 6 5 5 13 b 2 9 7 3 5 a 4 2 c x axis 10 0 8 2 4 6 Root E E 1 2 1 2 E E E E E E 3 4 5 6 7 1 E 2 9 5 13 5 2 a c b j i k e d g f m l h 2 10 13 E E E E E 4 5 3 6 7

  20. NN Query Example y axis 10 m E E 7 2 g Action Heap h l 8 E 6 Visit Root E E k 1 2 1 2 f E e 5 E follow E E E E 6 5 5 9 1 3 5 4 2 E 2 i j 4 E E follow E E E E E 1 query 5 5 9 2 13 d 3 5 4 2 7 6 4 E E follow E j i E E k E 3 13 10 6 5 5 13 b 2 9 7 3 5 a 4 Report i and terminate 2 c x axis 10 0 8 2 4 6 Root E E 1 2 1 2 E E E E E E 3 4 5 6 7 1 E 2 9 5 13 5 2 a c b j i k e d g f m l h 2 10 13 E E E E E 4 5 3 6 7

  21. Pruning 1 in NN Query • If we see an object o, prune every MBR whose MINDIST > d(o, q). • Side notice: at most one object in H! E1 E2 E3 o E4 q

  22. Pruning 2 using MINMAXDIST  Object o in sub-tree of E2 s.t. d(o, q)  MINMAXDIST(q, E2) • Prune even before we see an object! • Prune E1 if exists E2 s.t. MINDIST(q, E1) > MINMAXDIST(q, E2). • MINMAXDIST: compute max dist between q and each edge of E2, then take min. E1 E2 q

  23. NN Full-Blown Algorithm • Keep a heap H of index entries and objects, ordered by MINDIST. • Initially, H contains the root. • Setδ= +. • While H  • Extract the element e with minimum MINDIST. • If it is an object, return it as NN. • For every entry se in PAGE(e) whose MINDISTδ • Insert se into H. • Decreaseδto MINMAXDIST(q, se) if possible. • End while

  24. Content • The R-tree • Range Query • Aggregation Query • NN Query • RNN Query • Closest Pair Query • Close Pair Query • Skyline Query

  25. Reverse NN: Definition • Given a set of points, and a query location q. • Find the points whose NN is q. • RNN(q)={p1, p2}, NN(q)= p3. p p 4 1 p 3 q p 2

  26. Half-plane pruning p p 1 1 p 2 First NN N 1 First NN N 2 p' p ^ q ( ) , 1 ^ ( p , q ) p ^ q ( ) , q 2 q Second NN

  27. (TPL) Algorithm • Two logical steps: • Filter step: Find the set Scnd of candidate points • Find NN; • Prune space; • Find NN in unpruned space; • … • Till no more object left. • Refinement step: eliminate false positives • For each point p in Scnd, check whether its NN is not q. • The two steps are combined in a single tree traversal, by keeping all pruned MBRs/objects in Srfn.

  28. Example

  29. Example

  30. Example

  31. Example

  32. Example

  33. Example

  34. Example (end of filter step)

  35. Example (end of filter step)

  36. Example (refinement)

  37. Example (refinement)

  38. Content • The R-tree • Range Query • Aggregation Query • NN Query • RNN Query • Closest Pair Query • Close Pair Query • Skyline Query

  39. Closest Pair (CP) Query • Given two sets of objects R and S, • Find the pair of objects (rR, sS) with minimum distance. • CP = (r2, s1) • E.g. find the closest pair of hotel-bar. s 3 r r r 3 s 2 1 1 r 5 r s 4 2

  40. CP Solution Idea • Assume R and S are indexed by R-trees with same height. • Similar to the NN query algorithm. • MINDIST, MINMAXDIST for a pair of MBRs: Sj Ri MINDIST MINMAXDIST

  41. CP Basic Algorithm • Keep a heap H of pairs of index entries and pairs of objects, ordered by MINDIST. • Initially, H contains the pair of roots. • While H  • Extract the pair (eR, eS) with minimum MINDIST. • If it is a pair of objects, return it as CP. • For every entry seR in PAGE(eR ) and every entry seS in PAGE(eS) • Insert(eR, eS) into H. • End while

  42. CP Full-Blown Algorithm • Keep a priority queue H of pairs of index entries and pairs of objects, ordered by MINDIST. • Initially, H contains the pair of roots. • Setδ= +. • While H  • Extract the pair (eR, eS) with minimum MINDIST. • If it is a pair of objects, return it as CP. • For every entry seR in PAGE(eR ) and every entry seS in PAGE(eS) whose MINDISTδ • Insert( seR, seS ) into H. • Decreaseδto MINMAXDIST( seR, seS) if possible. • End while

  43. Content • The R-tree • Range Query • Aggregation Query • NN Query • RNN Query • Closest Pair Query • Close Pair Query • Skyline Query

  44. Close Pair Query • Given two sets of objects R and S, plus a threshold α, • Find every pair of objects (rR, sS) with distance <α. • Close pairs = (r1, s1), (r2, s1), and (r3, s3). α s 3 r r r 3 s 2 1 1 r 5 r s 4 2

  45. Close Pair Solution Idea • Observation: if d(r, s) <α, mbrR, mbrSthat contain r and s, respectively, we have: MINDIST(mbrR, mbrS) <α. • Solution idea: • start with the pair of root nodes, • Join pairs of index entries whose MINDIST <α, • Till we reach leaf level.

  46. Close Pair Algorithm • Push the pair of root nodes into stack. • While stack  • Pop a pair (eR, eS) from stack. • For every entry seR in PAGE(eR) and seS in PAGE(eS) where MINDIST(seR, seS) <α • Push (seR, seS) into stack if seR is an index entry; • Otherwise report (seR, seS) as one close pair. • End while

  47. Content • The R-tree • Range Query • Aggregation Query • NN Query • RNN Query • Closest Pair Query • Close Pair Query • Skyline Query

  48. Skyline of Manhattan • Which buildings can we see? • Higher or nearer

  49. Finding A Hotel Close to the Beach • Which one is better? • i or h? (i, because its price and distance dominate those of h) • i or k?

  50. Skyline Queries • Retrieve points not dominated by any other point.

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