1. What we have covered? Indexing and Hashing Data warehouse and OLAP
Data Mining
Information Retrieval and Web Mining
XML and XQuery
Spatial Databases
Transaction Management

2. Lecture 6: Spatial Data Management

3. 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.

4. 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

5. Types of Spatial Queries
Spatial Range Queries
Find all cities within 50 miles of Madison
Query has associated region (location, boundary)
Answer includes overlapping 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

6. Spatial Indexing
Point Access Methods (PAMs) vs Spatial Access Methods (SAMs)
PAM: index only point data
Hierarchical (tree-based) structures
Multidimensional Hashing
Space filling curve
SAM: index both points and regions
Transformations
Overlapping regions
Clipping methods (non-overlapping)
Data partitioning vs Space partitioning

7. 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
SAL 40
B+ tree
order 30 20 10
AGE 2

8. 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

9. 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).
Similarity queries (content-based retrieval).
Given a face, find the five most similar faces.
Multidimensional range queries.
50 < age < 55 AND 80K < sal < 90K

10. What’s the difficulty? An index based on spatial location needed.
One-dimensional indexes don’t support multidimensional searching efficiently. (Why?)
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).

11. PAMs Point Access Methods Hierarchical methods: kd-tree based
Space Filling Curves: Z-ordering
Multidimensional Hashing: Grid File
Exponential growth of the directory

12. The problem
Given a point set and a rectangular query, find the points enclosed in the query
We allow insertions/deletions on line
Query

13. Tree-based PAMs Most of tb-PAMs are based on kd-tree
kd-tree is a main memory binary tree for indexing k-dimensional points
Needs to be adapted for the disk model
Levels rotate among the dimensions, partitioning the space based on a value for that dimension
kd-tree is not necessarily balanced

14. kd-tree At each level we use a different dimension x=5 C y=6 B y=3 E

15. Kd-tree properties Height of the tree O(log2 n)
Search time for exact match: O(log2 n)
Search time for range query: O(n1/2 + k)

16. kd-tree example
X=5
X=3
X=7
y=6
y=5
Y=6
x=8
x=7
x=3
y=2
Y=2
X=5
X=8

17. External memory kd-trees
Similar to B-tree, tree nodes split many ways instead of two ways
insertion becomes quite complex and expensive.
No storage utilization guarantee since when a higher level node splits, the split has to be propagated all the way to leaf level resulting in many empty blocks.
Pack many interior nodes (forming a subtree) into a block.
it may not be feasible to group nodes at lower level into a block productively.
Many interesting papers on how to optimally pack nodes into blocks recently published.

18. PAMs Point Access Methods Hierarchical methods: kd-tree based
Space Filling Curves: Z-ordering
Multidimensional Hashing: Grid File
Exponential growth of the directory

19. 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
SAL 40
B+ tree
order 30 20 10
AGE 2

20. Z-Curve What is a Z-curve? A space filling curve Fig 4.6 Fig 4.4
Generated from interleaving bits
x, y coordinate
See Fig. 4.6
Alternative generation method
see Fig. 4.5
Connecting points by z-order
see Fig. 4.4
looks like Ns or Zs
Implementing file operations
Fig 4.6
Fig 4.4

21. Example of Z-values Figure 4.7
Left part shows a map with spatial object A, B, C
Right part and Left bottom part Z-values within A, B and C
Note C gets z-values of 2 and 8, which are not close
Exercise: Compute z-values for B.
Fig 4.7

22. Hilbert Curve A space filling curve Fig 4.5 More complex to generate
Example: Fig. 4.5
More complex to generate
due to rotations
Illustration on next slide!
Implementing file operations
Fig 4.5

23. Calculating Hilbert Values (Optional Topic)
Fig 4.8

24. PAMs Point Access Methods Hierarchical methods: kd-tree based
Space Filling Curves: Z-ordering
Multidimensional Hashing: Grid File
Exponential growth of the directory

25. Grid File
Hashing methods for multidimensional points (extension of Extensible hashing)
Idea: Use a grid to partition the space each cell is associated with one page
Two disk access principle (exact match)

26. Grid File Start with one bucket for the whole space.
Select dividers along each dimension. Partition space into cells
Dividers cut all the way.
Each cell corresponds to 1 disk page.
Many cells can point to the same page.
Cell directory potentially exponential in the number of dimensions

27. Grid File Implementation
Dynamic structure using a grid directory
Grid array: a 2 dimensional array with pointers to buckets (this array can be large, disk resident) G(0,…, nx-1, 0, …, ny-1)
Linear scales: Two 1 dimensional arrays that used to access the grid array (main memory) X(0, …, nx-1), Y(0, …, ny-1)

28. Example Buckets/Disk Blocks Grid Directory Linear scale Y
Linear scale X

29. Grid File Search
Exact Match Search: at most 2 I/Os assuming linear scales fit in memory.
First use liner scales to determine the index into the cell directory
access the cell directory to retrieve the bucket address (may cause 1 I/O if cell directory does not fit in memory)
access the appropriate bucket (1 I/O)
Range Queries:
use linear scales to determine the index into the cell directory.
Access the cell directory to retrieve the bucket addresses of buckets to visit.
Access the buckets.

30. Grid File Insertions
Determine the bucket into which insertion must occur.
If space in bucket, insert.
Else, split bucket
how to choose a good dimension to split?
If bucket split causes a cell directory to split do so and adjust linear scales.
insertion of these new entries potentially requires a complete reorganization of the cell directory--- expensive!!!

31. Grid File Deletions
Deletions may decrease the space utilization. Merge buckets
We need to decide which cells to merge and a merging threshold
Buddy system and neighbor system
A bucket can merge with only one buddy in each dimension
Merge adjacent regions if the result is a rectangle

32. Grid File Example
(N=6) A 1 6 2 A 1 2 3 4 5 6 5 3 4

33. Grid File Example (N=6) A B A A A B 1 2 3 4 5 6 1 3 5 7 2 4 6 8 10 911 12 12 3 10 11 4

34. Grid File Example (N=6) A B A B C A B C A B 1 7 8 13 2 4 6 9 11 12 3 514 7 8 9 10 11 12 1 2 3 4 5 6 13 15 1 7 8 13 A 2 4 6 9 11 12 B 3 5 10 C 1 3 5 7 8 10 A 2 4 6 9 11 12 B 14 15

35. Grid File Example (N=6) A B C A D B C A B C A B C D 1 3 5 7 8 10 2 4 69 10 11 12 1 2 3 4 5 6 13 14 15 16 1 3 5 7 8 10 A 2 4 6 9 11 12 B 13 C 14 15 1 2 3 4 5 6 A 7 B 8 13 9 11 12 10 C 16 14 15 D

36. Grid File Example
(N=6) x1 x2 x3 x4 y4 y2 y1 A B C D E F G H I y3

37. 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

38. R-Tree Properties Leaf entry = < n-dimensional box, rid >
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.

39. 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

40. Example R-Tree (Contd.)

41. 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.)

42. 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.

43. 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.)

44. 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.
GOOD SPLIT!
BAD!

45. Spatial Data Warehousing
Spatial data warehouse: Integrated, subject-oriented, time-variant, and nonvolatile spatial data repository for data analysis and decision making
Spatial data integration: a big issue
Structure-specific formats (raster- vs. vector-based, OO vs. relational models, different storage and indexing, etc.)
Vendor-specific formats (ESRI, MapInfo, Integraph, etc.)
Spatial data cube: multidimensional spatial database
Both dimensions and measures may contain spatial components

46. Dimensions and Measures in Spatial Data Warehouse
numerical
distributive (e.g. count, sum)
algebraic (e.g. average)
holistic (e.g. median, rank)
spatial
collection of spatial pointers (e.g. pointers to all regions with degrees in July)
Dimension modeling
nonspatial
e.g. temperature: degrees generalizes to hot
spatial-to-nonspatial
e.g. region “B.C.” generalizes to description “western provinces”
spatial-to-spatial
e.g. region “Burnaby” generalizes to region “Lower Mainland”

47. Example: BC weather pattern analysis
Input
A map with about 3,000 weather probes scattered in B.C.
Daily data for temperature, precipitation, wind velocity, etc.
Concept hierarchies for all attributes
Output
A map that reveals patterns: merged (similar) regions
Goals
Interactive analysis (drill-down, slice, dice, pivot, roll-up)
Fast response time
Minimizing storage space used
Challenge
A merged region may contain hundreds of “primitive” regions (polygons)

48. Star Schema of the BC Weather Warehouse
Spatial data warehouse
Dimensions
region_name
time
temperature
precipitation
Measurements
region_map
area
count
Dimension table
Fact table

49. Spatial Merge Precomputing all: too much storage space
On-line merge: very expensive

50. Methods for Computation of Spatial Data Cube
On-line aggregation: collect and store pointers to spatial objects in a spatial data cube
expensive and slow, need efficient aggregation techniques
Precompute and store all the possible combinations
huge space overhead
Precompute and store rough approximations in a spatial data cube
accuracy trade-off
Selective computation: only materialize those which will be accessed frequently
a reasonable choice

51. Spatial Association Analysis
Spatial association rule: A  B [s%, c%]
A and B are sets of spatial or nonspatial predicates
Topological relations: intersects, overlaps, disjoint, etc.
Spatial orientations: left_of, west_of, under, etc.
Distance information: close_to, within_distance, etc.
s% is the support and c% is the confidence of the rule
Examples
is_a(x, large_town) ^ intersect(x, highway) ® adjacent_to(x, water)
[7%, 85%]
is_a(x, large_town) ^adjacent_to(x, georgia_strait) ® close_to(x, u.s.a.) [1%, 78%]

52. Progressive Refinement Mining of Spatial Association Rules
Hierarchy of spatial relationship:
g_close_to: near_by, touch, intersect, contain, etc.
First search for rough relationship and then refine it
Two-step mining of spatial association:
Step 1: Rough spatial computation (as a filter)
Using MBR or R-tree for rough estimation
Step2: Detailed spatial algorithm (as refinement)
Apply only to those objects which have passed the rough spatial association test (no less than min_support)

53. Spatial Classification and Spatial Trend Analysis
Analyze spatial objects to derive classification schemes, such as decision trees in relevance to certain spatial properties (district, highway, river, etc.)
Example: Classify regions in a province into rich vs. poor according to the average family income
Spatial trend analysis
Detect changes and trends along a spatial dimension
Study the trend of nonspatial or spatial data changing with space
Example: Observe the trend of changes of the climate or vegetation with the increasing distance from an ocean

54. LSD-tree Local Split Decision – tree
Use kd-tree to partition the space. Each partition contains up to B points. The kd-tree is stored in main-memory.
If the kd-tree (directory) is large, we store a sub-tree on disk
Goal: the structure must remain balanced: external balancing property

55. Example: LSD-tree

56. LSD-tree: main points Split strategies: Paging algorithm
Data dependent
Distribution dependent
Paging algorithm
Two types of splits: bucket splits and internal node splits

57. Handling Regions with Z-curve
Fig 4.9