1. Spatial Data Management
The slides for this text are organized by chapter. This lecture covers Chapter 28 on spatial data management. However, the slides do not cover space-filling curves, quad trees, or grid files. 1

2. Summary
Spatial data requires special data structures, similar to B-trees

3. Lecture outline Types of spatial data
Example of spatial and geometric data – splines and Voronoi diagrams
One-dimensional index – interval trees
R-trees
T-tree variants

4. Types of Spatial Data Point Data Region Data
Points in a multidimensional space
E.g., Raster data such as satellite imagery, where each pixel stores a measured value
E.g., Feature vectors extracted from text
Region Data
Objects have spatial extent with location and boundary
DB typically uses geometric approximations constructed using line segments, polygons, etc., called vector data.

5. Examples of GIS data
Different types of sampling are used to collect data

6. Splines

7. Examples of Voronoi regions

8. Terrain from splines and Voronoi diagrams

9. Spatial filtering

10. Spatial filtering
Low-pass filter – the value for the cell is computed as average of other cells
High-pass-continuous surface –low pass

11. Spatial filtering
Window size has effect on filtering

12. Smoothing of maps

13. Types of Spatial Queries
Spatial Range Queries
Find all cities within 50 miles of Madison
Query has associated region (location, boundary)
Answer includes ovelapping or contained data regions
Nearest-Neighbor Queries
Find the 10 cities nearest to Madison
Results must be ordered by proximity
Spatial Join Queries
Find all cities near a lake
Expensive, join condition involves regions and proximity

14. Applications of Spatial Data
Geographic Information Systems (GIS)
E.g., ESRI’s ArcInfo; OpenGIS Consortium
Geospatial information
All classes of spatial queries and data are common
Computer-Aided Design/Manufacturing
Store spatial objects such as surface of airplane fuselage
Range queries and spatial join queries are common
Multimedia Databases
Images, video, text, etc. stored and retrieved by content
First converted to feature vector form; high dimensionality
Nearest-neighbor queries are the most common

15. Single-Dimensional Indexes
B+ trees are fundamentally single-dimensional indexes.
When we create a composite search key B+ tree, e.g., an index on <age, sal>, we effectively linearize the 2- dimensional space since we sort entries first by age and then by sal. 80 70 60
Consider entries:
<11, 80>, <12, 10>
<12, 20>, <13, 75> 50 40
B+ tree
order 30 20 10 2

16. Multidimensional Indexes
A multidimensional index clusters entries so as to exploit “nearness” in multidimensional space.
Keeping track of entries and maintaining a balanced index structure presents a challenge!
Spatial
clusters 70 60 50 40 30 20 10 80
B+ tree
order
Consider entries:
<11, 80>, <12, 10>
<12, 20>, <13, 75> 2

17. Motivation for Multidimensional Indexes
Spatial queries (GIS, CAD).
Find all hotels within a radius of 5 miles from the conference venue.
Find the city with population 500,000 or more that is nearest to Kalamazoo, MI.
Find all cities that lie on the Nile in Egypt.
Find all parts that touch the fuselage (in a plane design).
Multidimensional range queries.
50 < age < 55 AND 80K < sal < 90K

18. Motivation Similarity queries (content-based retrieval).
Given a face, find the five most similar faces/expressions.

19. Interval trees Geometric, 1-dimensional tree
Interval is defined by (x1,x2)
Split at the middle (5), again at the middle (3,7), again at the middle (2,8)
All intervals intersecting a middle point are stored at the corresponding root.
(4,6) (4,8) 1 2 3 4 5 6 7 8 9
(6,9)
(2,4)
(7.5,8.5)

20. Interval trees Finding intervals – by finding x1, x2 against the nodes
Find interval containing specific value – from the root
Sort intervals within each node of the tree according to their coorsinates
Cost of the “stabbing query”– finding all intervals containing the specified value is O(log n + k), where k is the number of reported intervals.

21. SAM (Spatial Access Method)
Constructs the minimal bounding box (mbb)
Check validity (predicate) on mbb
Refinement step verifies if actual objects satisfy the predicate.

22. What’s the difficulty? An index based on spatial location needed.
One-dimensional indexes don’t support multidimensional searching efficiently.
Hash indexes only support point queries; want to support range queries as well.
Must support inserts and deletes gracefully.
Ideally, want to support non-point data as well (e.g., lines, shapes).
The R-tree meets these requirements, and variants are widely used today.

23. The R-Tree
The R-tree is a tree-structured index that remains balanced on inserts and deletes.
Each key stored in a leaf entry is intuitively a box, or collection of intervals, with one interval per dimension.
Example in 2-D: X Y
Root of
R Tree
Leaf
level

24. R-Tree Properties Leaf entry = < n-dimensional box, rid >
This is Alternative (2), with key value being a box.
Box is the tightest bounding box for a data object.
Non-leaf entry = < n-dim box, ptr to child node >
Box covers all boxes in child node (in fact, subtree).
All leaves at same distance from root.
Nodes can be kept 50% full (except root).
Can choose a parameter m that is <= 50%, and ensure that every node is at least m% full.

25. Example of an R-Tree Leaf entry Index entry Spatial object
approximated by
bounding box R8 R3 R5
R13 R9 R8
R14
R10
R12 R7
R18
R17 R6
R16
R19
R15 R2

26. Example R-Tree (Contd.)

27. Search for Objects Overlapping Box Q
Start at root.
1. If current node is non-leaf, for each
entry <E, ptr>, if box E overlaps Q,
search subtree identified by ptr.
2. If current node is leaf, for each entry
<E, rid>, if E overlaps Q, rid identifies
an object that might overlap Q.
Note: May have to search several subtrees at each node!
(In contrast, a B-tree equality search goes to just one leaf.)

28. Improving Search Using Constraints
It is convenient to store boxes in the R-tree as approximations of arbitrary regions, because boxes can be represented compactly.
But why not use convex polygons to approximate query regions more accurately?
Will reduce overlap with nodes in tree, and reduce the number of nodes fetched by avoiding some branches altogether.
Cost of overlap test is higher than bounding box intersection, but it is a main-memory cost, and can actually be done quite efficiently. Generally a win.

29. Insert Entry <B, ptr>
Start at root and go down to “best-fit” leaf L.
Go to child whose box needs least enlargement to cover B; resolve ties by going to smallest area child.
If best-fit leaf L has space, insert entry and stop. Otherwise, split L into L1 and L2.
Adjust entry for L in its parent so that the box now covers (only) L1.
Add an entry (in the parent node of L) for L2. (This could cause the parent node to recursively split.)

30. Splitting a Node During Insertion
The entries in node L plus the newly inserted entry must be distributed between L1 and L2.
Goal is to reduce likelihood of both L1 and L2 being searched on subsequent queries.
Idea: Redistribute so as to minimize area of L1 plus area of L2.
Exhaustive algorithm is too slow;
quadratic and linear heuristics are
popular in research.
GOOD SPLIT!
BAD!

31. R-Tree Variants
The R* tree uses the concept of forced reinserts to reduce overlap in tree nodes. When a node overflows, instead of splitting:
Remove some (say, 30% of the) entries and reinsert them into the tree.
Could result in all reinserted entries fitting on some existing pages, avoiding a split.
R* trees also use a different heuristic, minimizing box perimeters rather than box areas during insertion.
Another variant, the R+ tree, avoids overlap by inserting an object into multiple leaves if necessary.
Searches now take a single path to a leaf, at cost of redundancy.

32. GiST
The Generalized Search Tree (GiST) abstracts the “tree” nature of a class of indexes including B+ trees and R-tree variants.
Striking similarities in insert/delete/search and even concurrency control algorithms make it possible to provide “templates” for these algorithms that can be customized to obtain the many different tree index structures.
B+ trees are so important (and simple enough to allow further specialization) that they are implemented specially in all DBMSs.
GiST provides an alternative for implementing other tree indexes.

33. Indexing High-Dimensional Data
Typically, high-dimensional datasets are collections of points, not regions.
E.g., Feature vectors in multimedia applications.
Very sparse
Nearest neighbor queries are common.
R-tree becomes worse than sequential scan for most datasets with more than a dozen dimensions.
As dimensionality increases contrast (ratio of distances between nearest and farthest points) usually decreases; “nearest neighbor” is not meaningful.

34. Comments on R-trees
Spatial data management has many applications, including GIS, CAD/CAM, multimedia indexing.
Point and region data
Overlap/containment and nearest-neighbor queries
Many approaches to indexing spatial data
R-tree approach is widely used in GIS systems
Other approaches include Grid Files, Quad trees, and techniques based on “space-filling” curves.
For high-dimensional datasets, unless data has good “contrast”, nearest-neighbor may not be well-separated

35. Comments on R-Trees
Deletion consists of searching for the entry to be deleted, removing it, and if the node becomes under-full, deleting the node and then re-inserting the remaining entries.
Overall, works quite well for 2 and 3 D datasets. Several variants (notably, R+ and R* trees) have been proposed; widely used.
Can improve search performance by using a convex polygon to approximate query shape (instead of a bounding box) and testing for polygon-box intersection.

36. The grid
"Print Gallery," by M.C. Escher. Curious about the blank spot in the middle of Escher’s 1956 lithograph, Hendrik Lenstra set out to learn whether the artist had encountered a mathematical problem he couldn’t solve. ©2002 Cordon Art B.V., Baarn, Holland. All rights reserved.

37. The grid structure Fixed grid:
Stored as a 2D array, each entry contains a link to a list of points (object) stored in a grid.
a,b

38. Page overflow Too many points in one grid cell:
Solution A –overflow (linked list)
Solution B- Split the cell and increase index!

39. Rectangle indexing with grids
Rectangles may share different grid cells
Rectangle duplicates are stored
Grid cells are of fixed size

40. Grid file vs. grid
In a grid file, the index is dynamically increased in size when overflow happens.
The space is split by a vertical or a horizontal line, and then further subdivided when overflow happens!
Index is dynamically growing
Boundaries of cells of different sizes are stores, thus point and stabbing queries are easy

41. The quadtree Instead of using an array as an index, use tree!
Quadtree decomposition – cells are indexed by using quaternary B-tree.
All cells are squares, not polygons.
Search in a tree is faster!

42. First three levels of a quad tree
Quad-tree example
First three levels of a quad tree

43. Image stored in a quad-tree
8 x 8 pixel picture represented in a quad tree
Project #32: PICTURE REPRESENTATION USING QUAD TREES, McGill University:

44. Grid file
Example of a grid file

45. Linear quadtree
B+ index – actual references to rectangles are stored in the leaves, saving more space+ access time
Label nodes according to Z or “pi” order

46. Linear quadtree
Level of detail increases as the number of quadtree decompositions increases!
Decompositions have indexes of a form:
00,01,02,03,10,11,12,13, 2,300
301 ,302 ,303 ,31 ,32 ,33
Stores as Bplus tree

47. Z-order

49. Review questions What is spatial data structure?
What is the difference between grid and grid file?
Explain how z or p ordering works?
Define interval trees
Provide example of R-tree
List R-tree variants
How spatial index structure differs from regular B+ tree?

50. Resources Text 1 instructor’s resources McGill University web space
Wikepedia (z order images)
Face recognition research
SPARCS lab project on image processing