Query Processing and Advanced Queries. Advanced Queries (2): R-TreeR

Transcription

1 Query Processing and Advanced Queries Advanced Queries (2): R-TreeR

2 Review: PAM Given a point set and a rectangular query, find the points enclosed in the query We allow insertions/deletions online Query CMPT 454: Database Systems II Advanced Queries (2) 2 / 29

3 Review: kd-tree X=7 X=5 y=5 y=6 Y=5 Y=6 x=3 x=8 x=7 Y=2 y=2 X=3 X=5 X=8 Each leaf node can hold up to 2 points. CMPT 454: Database Systems II Advanced Queries (2) 3 / 29

4 Query Answering using kd-tree X=7 X=5 y=5 y=6 Y=5 Y=6 Y=4.5 x=3 x=8 x=7 Y=2 Query y=2 Y=1.1 X=3 X=5 X=8 X=3.1 X=6.2 Each leaf node can hold up to 2 points. CMPT 454: Database Systems II Advanced Queries (2) 4 / 29

5 Review: Spatial Indexing Point Access Methods can index only points. What about regions? Use the transformation technique and a PAM New methods: Spatial Access Methods SAMs R-tree and variations CMPT 454: Database Systems II Advanced Queries (2) 5 / 29

6 Minimum Bounding Rectangle Approximate each region with a simple shape: usually Minimum Bounding Rectangle (MBR) = [(x1, x2), (y1, y2)] y2 y1 x1 x2 CMPT 454: Database Systems II Advanced Queries (2) 6 / 29

7 Transformation Technique Map an d-dim MBR into a point: ex. [(x min, x max ) (y min, y max )] => (x min, x max, y min, y max ) 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 CMPT 454: Database Systems II Advanced Queries (2) 7 / 29

8 SAM: The Problem Given a collection of geometric objects (points, lines, polygons,...) Organize them on disk, to answer spatial queries (e.g., range query, NN query, etc) CMPT 454: Database Systems II Advanced Queries (2) 8 / 29

9 Two steps: Indexing using SAMs Filtering step: Find all the MBRs (using the SAM) that satisfy the query Refinement step: For each qualified MBR, check the original object against the query CMPT 454: Database Systems II Advanced Queries (2) 9 / 29

10 R-Trees [Guttman 84] Main idea: allow parents to overlap! => guaranteed 50% utilization => easier insertion/split algorithms. (only deal with Minimum Bounding Rectangles - MBRs) CMPT 454: Database Systems II Advanced Queries (2) 10 / 29

11 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) CMPT 454: Database Systems II Advanced Queries (2) 11 / 29

12 Example Fan-out = 4: group nearby rectangles to parent MBRs; each group -> disk page B A C D E F G J H I CMPT 454: Database Systems II Advanced Queries (2) 12 / 29

13 Example F=4 P1 A B P2 C D E P3 G F P4 J H I A B C D E H I J F G CMPT 454: Database Systems II Advanced Queries (2) 13 / 29

14 Example F=4 P1 A B P2 C D E P3 G F P4 J H I A B C D E P1 P2 P3 P4 H I J F G CMPT 454: Database Systems II Advanced Queries (2) 14 / 29

15 R-Trees - Format of Nodes {(MBR; obj_ptr)} for leaf nodes P1 P2 P3 P4 x-low; x-high y-low; y-high... obj ptr... A B C CMPT 454: Database Systems II Advanced Queries (2) 15 / 29

16 R-Trees - Format of Nodes {(MBR; node_ptr)} for non-leaf nodes x-low; x-high y-low; y-high... node ptr... P1 P2 P3 P4 A B C CMPT 454: Database Systems II Advanced Queries (2) 16 / 29

17 R-Trees: Search P1 A B P2 C D E P3 G F P4 J H I A B C D E P1 P2 P3 P4 H I J F G CMPT 454: Database Systems II Advanced Queries (2) 17 / 29

18 R-Trees: Search P1 A B P2 C D E P3 G F P4 J H I A B C D E P1 P2 P3 P4 H I J F G CMPT 454: Database Systems II Advanced Queries (2) 18 / 29

19 Main points: R-Trees: Search 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. A point query may follow multiple branches. Everything works for any(?) dimensionality CMPT 454: Database Systems II Advanced Queries (2) 19 / 29

20 Spatial Queries Given a collection of geometric objects (points, lines, polygons,...) organize them on disk, to answer efficiently range queries k-nn queries CMPT 454: Database Systems II Advanced Queries (2) 20 / 29

21 Spatial Queries Given a collection of geometric objects (points, lines, polygons,...) organize them on disk, to answer efficiently range queries k-nn queries CMPT 454: Database Systems II Advanced Queries (2) 21 / 29

22 Spatial Queries Given a collection of geometric objects (points, lines, polygons,...) organize them on disk, to answer efficiently range queries k-nn queries CMPT 454: Database Systems II Advanced Queries (2) 22 / 29

23 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) CMPT 454: Database Systems II Advanced Queries (2) 23 / 29

24 R-Trees - NN Search q P1 A B P2 C D E P3 G F P4 J H I CMPT 454: Database Systems II Advanced Queries (2) 24 / 29

25 Two Metrics to Ordering the NN Search MINDIST (P, R) is the minimum distance between a point P and a rectangle R. If the point is inside the rectangle, MINDIST = 0; If the point is outside the rectangle, MINDIST is the minimal possible distance from the point to any object in or on the perimeter of the rectangle. o R, MINDIST( P, R) ( P, o) CMPT 454: Database Systems II Advanced Queries (2) 25 / 29

26 MINMAXDIST MINMAXDIST(P,R): for each dimension, find the closest face, compute the distance to the furthest point on this face and take the minimum of all these (d) distances MINMAXDIST(P,R) is the smallest possible upper bound of distances from P to R MINMAXDIST guarantees that there is at least one object in R with a distance to P smaller or equal to it. o R, ( P, o) MINMAXDIST( P, R) CMPT 454: Database Systems II Advanced Queries (2) 26 / 29

27 MINDIST and MINMAXDIST MINDIST(P, R) <= NN(P) <=MINMAXDIST(P,R) R1 MINMAXDIST R3 R4 MINDIST MINDIST MINDIST MINMAXDIST R2 MINMAXDIST CMPT 454: Database Systems II Advanced Queries (2) 27 / 29

28 R-Trees - NN Search Q: How? (find near neighbor; refine...) q P1 A B P2 C D E P3 G F P4 J H I CMPT 454: Database Systems II Advanced Queries (2) 28 / 29

29 R-Trees - NN Search Q: How? (find near neighbor; refine...) q P1 A B P2 C D E P3 G F P4 J H I CMPT 454: Database Systems II Advanced Queries (2) 29 / 29