1. Many slides taken from George Kollios, Boston University
Spatial Indexing
Many slides taken from George Kollios, Boston University

2. Spatial Indexing
Point Access Methods can index only points. What about regions?
Use the transformation technique
Z-ordering and quadtrees
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. R-tree z-ordering: cuts regions to pieces -> dup. elim.
how could we avoid that?
Idea: try to extend/merge B-trees and k-d trees
R-tree: a generalization of the B+-tree for multidimensional spaces

5. Kd-Btrees
[Robinson, 81]: if f is the fanout, split point-set in f parts; and so on, recursively

6. Kd-Btrees
But: insertions/deletions are tricky (splits may propagate downwards and upwards)
no guarantee on space utilization

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)

8. 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)

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

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

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

12. 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
...

13. 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

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

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

16. 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

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

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

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

20. 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)

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

22. R-trees:Split (A1: plane sweep, until 50% of rectangles) P1 K
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

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

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

25. 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

26. 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

27. R-Trees:Deletion Other method (later)
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)

28. 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

29. R-trees - variations what about static datasets (no ins/del/upd)?
Q: Best way to pack points?

30. R-trees - variations what about static datasets (no ins/del/upd)?
Q: Best way to pack points?
A1: plane-sweep
great for queries on ‘x’;
terrible for ‘y’

31. R-trees - variations what about static datasets (no ins/del/upd)?
Q: Best way to pack points?
A1: plane-sweep
great for queries on ‘x’;
bad for ‘y’

32. R-trees - variations what about static datasets (no ins/del/upd)?
Q: Best way to pack points?
A1: plane-sweep
great for queries on ‘x’;
terrible for ‘y’
Q: how to improve?

33. R-trees - variations
A: plane-sweep on HILBERT curve!

34. R-trees - variations A: plane-sweep on HILBERT curve!
In fact, it can be made dynamic (how?), as well as to handle regions (how?)

35. R-trees - variations Dynamic (‘Hilbert R-tree):
each point has an ‘h’-value (hilbert value)
insertions: like a B-tree on the h-value
but also store MBR, for searches

36. R-trees - variations what about other bounding shapes? (and why?)
A1: arbitrary-orientation lines (cell-tree, [Guenther]
A2: P-trees (polygon trees) (MB polygon: 0, 90, 45, 135 degree lines)

37. R-trees - variations A3: L-shapes; holes (hB-tree)
A4: TV-trees [Lin+, VLDB-Journal 1994]
A5: SR-trees [Katayama+, SIGMOD97] (used in Informedia)

38. R-trees - conclusions Popular method; like multi-d B-trees
guaranteed utilization
good search times (for low-dim. at least)
Informix ships DataBlade with R-trees

39. Spatial Queries

40. Spatial Queries
Given a collection of geometric objects (points, lines, polygons, ...)
organize them on disk, to answer
point queries
range queries
k-nn queries
spatial joins (‘all pairs’ queries)

41. Spatial Queries
Given a collection of geometric objects (points, lines, polygons, ...)
organize them on disk, to answer
point queries
range queries
k-nn queries
spatial joins (‘all pairs’ queries)

42. Spatial Queries
Given a collection of geometric objects (points, lines, polygons, ...)
organize them on disk, to answer
point queries
range queries
k-nn queries
spatial joins (‘all pairs’ queries)

43. Spatial Queries
Given a collection of geometric objects (points, lines, polygons, ...)
organize them on disk, to answer
point queries
range queries
k-nn queries
spatial joins (‘all pairs’ queries)

44. Spatial Queries
Given a collection of geometric objects (points, lines, polygons, ...)
organize them on disk, to answer
point queries
range queries
k-nn queries
spatial joins (‘all pairs’ queries)

45. 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)

46. R-trees - NN search A B C D E F G H I J P1 P2 P3 P4 q

47. R-trees - NN search Q: How? (find near neighbor; refine...) A B C D EJ P1 P2 P3 P4 q

48. R-trees - NN search A1: depth-first search; then, range query P1 P3 IB J E P4 q P2 D

49. R-trees - NN search A1: depth-first search; then, range query P1 P3 IB J E P4 q P2 D

50. R-trees - NN search A1: depth-first search; then, range query P1 P3 IB J E P4 q P2 D

51. R-trees - NN search A2: [Roussopoulos+, sigmod95]:
priority queue, with promising MBRs, and their best and worst-case distance
main idea: Every face of any MBR contains at least one point of an actual spatial object!

52. R-trees - NN search consider only P2 and P4, for illustration A B C DG H I J P1 P2 P3 P4 q

53. R-trees - NN search best of P4 => P4 is useless for 1-nn
worst of P2 H J E P4 q P2 D

54. R-trees - NN search what is really the worst of, say, P2? worst of P2q P2 D

55. R-trees - NN search what is really the worst of, say, P2?
A: the smallest of the two red segments! q P2

56. Nearest-neighbor searching
Branch and bound strategy
Compute MINDIST and MINMAXDIST [RKV95]
MINDIST(p,R) is the minimum distance between p and R with corner points l and u
the closest point in R is at least this distance away R u
Sqrt sum (pi-ri)2 where p
ri = li if pi < li
= ui if pi > ui
= pi otherwise p
MINDIST = 0 l p

57. Nearest-neighbor searching
MINMAXDIST(p,R) is the minimum of the maximum distance to each pair of faces of R
MaxDistanceToFace(p,R,k) = distance between p and Maxk= (M1,M2,..,Mk-1mk,MK+1,..,Mn)
mi = closer of the two boundary points along i axis
Mi = farther of the two boundary points along i axis
Max2 u p l
Max1

58. Nearest-neighbor searching
MINMAXDIST(p,R) is the minimum of the maximum distance to each pair of faces of R
MaxDistanceToFace(p,R,k) = distance between p and Maxk= (M1,M2,..,Mk-1mk,MK+1,..,Mn)
mi = closer of the two boundary points along i axis
Mi = farther of the two boundary points along i axis
Max1 u l p
Max2

59. Pruning ESTIMATE := smallest MINMAXDIST(p,R)
Prune an MBR R’ for which MINDIST(p,R’) is greater than ESTIMATE.
Generalize to k-nearest neighbor searching
Maintain kth largest MINMAXDIST
Prune an MBR if MINDIST to it is larger than the current estimate of kth MINMAXDIST
Can use objects to refine estimate

60. Order of searching Depth first order Inspect children in MINDIST order
For each node in the tree keep a list of nodes to be visited
Prune some of these nodes in the list
Continue until the lists are empty

61. Another NN search Global order [HS99]
Maintain distance to all entries in a common list
Order the list by MINDIST
Repeat
Inspect the next MBR in the list
Add the children to the list and reorder
Until all remaining MBRs can be pruned

62. Spatial Join Find all parks in a city
Find all trails that go through a forest
Basic operation
find all pairs of objects that overlap
Single-scan queries
nearest neighbor queries, range queries
Multiple-scan queries
spatial join

63. Algorithms No existing index structures
Transform data into 1-d space [O89]
z-transform; sensitive to size of pixel
Partition-based spatial-merge join [PW96]
partition into tiles that can fit into memory
plane sweep algorithm on tiles
Spatial hash joins [LR96, KS97]
Sort data [BBKK01]
With index structures [BKS93, HJR97]
k-d trees and grid files
R-trees

64. R-tree based Join [BKS93]

65. Join1(R,S)
Repeat
Find a pair of intersecting entries E in R and F in S
If R and S are leaf pages then add (E,F) to result-set
Else Join1(E,F)
Until all pairs are examined
CPU and I/O bottleneck

66. Reducing CPU bottleneckS R

67. Join2(R,S,IntersectedVol)
Repeat
Find a pair of intersecting entries E in R and F in S that overlap with IntersectedVol
If R and S are leaf pages then add (E,F) to result-set
Else Join2(E,F,CommonEF)
Until all pairs are examined
14+6 comparisons instead of 49
In general, number of comparisons equals
size(R) + size(S) + relevant(R)*relevant(S)
Reduce the product term

68. Using Plane Sweep S R s1 s2 r1 r2 r3 Consider the extents along x-axis
Start with the first entry r1
sweep a vertical line

69. Using Plane Sweep S R s1 s2 r1 r2 r3
Check if (r1,s1) intersect along y-dimension
Add (r1,s1) to result set

70. Using Plane Sweep S R s1 s2 r1 r2 r3
Check if (r1,s2) intersect along y-dimension
Add (r1,s2) to result set

71. Using Plane Sweep S R s1 s2 r1 r2 r3 Reached the end of r1
Start with next entry r2

72. Using Plane Sweep S R s1 s2 r1 r2 r3
Reposition sweep line

73. Using Plane Sweep S R s1 s2 r1 r2 r3
Check if r2 and s1 intersect along y
Do not add (r2,s1) to result

74. Using Plane Sweep S R s1 s2 r1 r2 r3 Reached the end of r2
Start with next entry s1

75. Using Plane Sweep S R s1 s2 r1 r2 r3
Total of 2(r1) + 1(r2) + 0 (s1)+ 1(s2)+ 0(r3) = 4 comparisons

76. Reducing I/O Read schedule r1, s1, s2, r3
Every subtree examined only once
Consider a slightly different layout

77. Reducing I/O S R s1 r2 r1 s2 r3
Read schedule is r1, s2, r2, s1, s2, r3
Subtree s2 is examined twice

78. Pinning of nodes
After examining a pair (E,F), compute the degree of intersection of each entry
degree(E) is the number of intersections between E and unprocessed rectangles of the other dataset
If the degrees are non-zero, pin the pages of the entry with maximum degree
Perform spatial joins for this page
Continue with plane sweep

79. Reducing I/O S R s1 r2 r1 s2 r3 After computing join(r1,s2),
degree(r1) = 0
degree(s2) = 1
So, examine s2 next
Read schedule = r1, s2, r3, r2, s1
Subtree s2 examined only once

80. References
[SK98] Optimal multi-step k-nearest neighbor search, T. Seidl and H. Kriegel, SIGMOD 1998:
[BBKK01] Epsilon Grid Order: An Algorithm for the Similarity Join on Massive High-Dimensional Data, C. Bohm, B. Braunmüller, F. Krebs and H.-P. Kriegel, SIGMOD 2001.
[RKV95] Roussopoulos N., Kelley S., Vincent F. Nearest Neighbor Queries. Proceedings of the ACM-SIGMOD International Conference on Management of Data, pages 71-79, 1995.

81. References
[HS99] G. R. Hjaltason and H. Samet, Distance browsing in spatial databases, ACM Transactions on Database Systems 24, 2 (June 1999),
[O89] Jack A. Orenstein: Redundancy in Spatial Databases. SIGMOD Conference 1989:
[PW96] Jignesh M. Patel, David J. DeWitt: Partition Based Spatial-Merge Join. SIGMOD Conference 1996:
[LR96] Ming-Ling Lo, Chinya V. Ravishankar: Spatial Hash-Joins. SIGMOD Conference 1996:

82. References
[KS97] Nick Koudas, Kenneth C. Sevcik: Size Separation Spatial Join. SIGMOD Conference 1997:
[HJR97] Yun-Wu Huang, Ning Jing, Elke A. Rundensteiner: Spatial Joins Using R-trees: Breadth-First Traversal with Global Optimizations. VLDB 1997,
[BKS93] Thomas Brinkhoff, Hans-Peter Kriegel, Bernhard Seeger: Efficient Processing of Spatial Joins Using R-Trees. SIGMOD Conference 1993: