Spatial Indexing

1
Spatial Indexing

• SAMs

2
Spatial Indexing

• Point Access Methods can index only points. What
  about regions?
• Z-ordering and quadtrees
• Use the transformation technique and a PAM
• 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
Transformation Technique

• Map an d-dim MBR into a point ex.
• (xmin, xmax) (ymin, ymax) gt
• (xmin, xmax, ymin, ymax)
• 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

5
R-trees

• Guttman 84 Main idea allow parents to overlap!
• gt guaranteed 50 utilization
• gt easier insertion/split algorithms.
• (only deal with Minimum Bounding Rectangles -
  MBRs)

6
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)

7
Example

• eg., w/ fanout 4 group nearby rectangles to
  parent MBRs each group -gt disk page

I
C
A
G
H
F
B
J
E
D
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Example

• F4

P1
P3
I
C
A
G
H
F
B
J
E
P4
D
P2
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Example

• F4

P1
P3
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A
G
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B
J
E
P4
D
P2
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R-trees - format of nodes

• (MBR obj_ptr) for leaf nodes

x-low x-high y-low y-high ...
obj ptr
...
11
R-trees - format of nodes

• (MBR node_ptr) for non-leaf nodes

x-low x-high y-low y-high ...
node ptr
...
12
R-treesSearch
P1
P3
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A
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B
J
E
P4
D
P2
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R-treesSearch
P1
P3
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A
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B
J
E
P4
D
P2
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R-treesSearch

• 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

15
R-treesInsertion
Insert X
P1
P3
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A
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F
B
X
J
E
P4
D
P2
X
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R-treesInsertion
Insert Y
P1
P3
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A
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B
J
E
P4
Y
D
P2
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R-treesInsertion

• Extend the parent MBR

P1
P3
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C
A
G
H
F
B
J
E
P4
Y
D
P2
Y
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R-treesInsertion

• 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)

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R-treesInsertion

• If node is full then Split ex. Insert w

P1
P3
K
I
C
A
G
W
H
F
B
J
K
E
P4
D
P2
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R-treesInsertion

• If node is full then Split ex. Insert w

P3
I
P5
K
C
A
G
P1
W
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B
J
E
P4
D
P2
Q2
Q1
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R-treesSplit

• 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
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R-treesSplit

• pick two rectangles as seeds
• assign each rectangle R to the closest seed

seed1
23
R-treesSplit

• pick two rectangles as seeds
• assign each rectangle R to the closest
  seed
• closest the smallest increase in area

seed1
24
R-treesSplit

• How to pick Seeds
• LinearFind 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 JMBR(E1, E2) and d J-E1-E2. Choose
  the pair with the largest d

25
R-treesInsertion

• 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

26
R-TreesDeletion

• 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)

27
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