RTrees: A Dynamic Index Structure For Spatial Searching

1
R-Trees A Dynamic Index Structure For Spatial
Searching
CSED700 - Advanced Topics in Data Management

• Antonin Guttman
• Presented by Gae-won You

2
Introduction

• R-Tree
• A multi-dimensional index structure
• A height balanced tree like the B-Tree
• Used widely in spatial and multimedia databases

3
R-Tree Structure
M maximum number of entries m minimum number
of entries ( M/2)
R2
R1

• Every leaf node contains between m and M index
  records unless it is the root.
• Each leaf node has the smallest rectangle that
  spatially contains the n-dimensional data
  objects.
• Every non-leaf node has between m and M children
  unless it is the root.
• Each non-leaf node has the smallest rectangle
  that spatially contains the rectangles in the
  child node.
• The root node has at least two children unless
  it is a leaf.
• All leaves appear on the same level.

R3
A
B
ltMBR, Pointer to a child nodegt
ltMBR, Pointer to a spatial objectgt
4
Search and Update

• Search
• Update
• Insertion
• Deletion
• Node Split (with Insertion)
• A Quadratic-Cost Algorithm
• A Linear-Cost Algorithm

5
Search and Update

• Search
• Update
• Insertion
• Deletion
• Node Split (with Insertion)
• A Quadratic-Cost Algorithm
• A Linear-Cost Algorithm

6
R-Tree Search (1/7)
R2
Find all objects whose rectangles are overlapped
with a search rectangle S
R1
R3
A
B
S
7
R-Tree Search (2/7)
R2
R1
R3
A
B
S
8
R-Tree Search (3/7)
R2
R1
R3
A
B
S
9
R-Tree Search (4/7)
R2
R1
R3
A
B
S
10
R-Tree Search (5/7)
R2
R1
R3
A
B
S
11
R-Tree Search (6/7)
R2
R1
R3
A
B
S
12
R-Tree Search (7/7)

R1
R3
A
B
S
B and D ? overlapped objects with S
13
Search and Update

• Search
• Update
• Insertion
• Deletion
• Node Split (with Insertion)
• A Quadratic-Cost Algorithm
• A Linear-Cost Algorithm

14
R-Tree Insertion (1/7)
R2
Insert a new spatial object X
R1
R3
A
B
X
15
R-Tree Insertion (2/7)
R2
Find the proper child node - least enlargement
- smallest MBR if child nodes contains a
new object
R1
R3
A
B
X
16
R-Tree Insertion (3/7)
R2
R1
R3
A
B
X
17
R-Tree Insertion (4/7)
A
B
X
18
R-Tree Insertion (5/7)
R2
R1
R3
A
B
R4
X
E
D
F
19
R-Tree Insertion (6/7)
A
B
X
20
R-Tree Insertion (7/7)
Empty Spot
21
Search and Update

• Search
• Update
• Insertion
• Deletion
• Node Split (with Insertion)
• A Quadratic-Cost Algorithm
• A Linear-Cost Algorithm

22
R-Tree Deletion

• Performed unlike a B-Tree deletion
• eliminate the node if it has too few entries (
  m)
• propagate node elimination upward as necessary
• re-insert its entries using insertion method
• re-insertion
• (1) easier to implement
• (2) prevent gradual deterioration

23
Search and Update

• Search
• Update
• Insertion
• Deletion
• Node Split (with Insertion)
• A Quadratic-Cost Algorithm
• A Linear-Cost Algorithm

24
Split with insertion

Y
R4
R4
?
E
F
D
Y
25
Split

• The bad split may cause multiple paths for
  searching

A
A
VS.
B
B
E
E
F
F
? Minimize the total area of the two covering
rectangles
26
Quadratic split

• Divide S into S1 and S2
• Initial step choose two candidates
• Choose max MBR(a,b) area(a) area(b) for
  all a, b
• Iteration step
• Choose max MBR(S1, a) - MBR(S2, a) for
  the remaining entry a
• Add to the group whose covering rectangle will
  have to be enlarged least

A
B
E
F
27
Conclusion

• R-tree index structure
• 3 operations
• Search
• Insertion ( with split )
• Deletion

28
Q A