CSE5120Fall2009
RTrees: Index structure for Spatial Searching
Guttman, SIGMOD 1984
Rtree :
•
a heightbalanced tree
•
has some similarity to a Btree
•
records in its leaf nodes pointing to data
objects.
An Rtree for spatial objects
A
B
C
D
E
F
G
H
J
K
I
N
M
L
A
B
C
D
E
F
G
H
I
J
K
L
M
N
. . .
. . .
xl
xh
yl
yh
. . .
pointer
to
child
An Rtree for data points
A
B
C
D
E
F
G
H
J
K
I
N
M
L
A
B
C
D
E
F
G
H
I
J
K
L
M
N
point x
point
x
is
under
A
only
point
y
point
y
is
under
B
only
Leaf nodes contain
(
I, object
identifier
)
I
is an
n
dimensional rectangle
which is the
bounding box
of the spatial
object
I
= (
I
0
, I
1
, ..., I
n

1
)
I
i
is an interval [a,b]
Nonleaf nodes contain
(
I, child
pointer
)
where
I
covers all rectangles in the lower
node’s entries.
Node size:
maximum
M
,
minimum
m
≤
l
M
2
m
47
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Some properties of an
R
tree:
1. In a leaf node,
I
is the smallest rectangle
that contains the
n
dimensional data ob
ject.
2. For each entry in a nonleaf node,
I
is the
smallest rectangle that contains the rectan
gles in the child node.
3. The root node has at least two children un
less it is a leaf, and at most
M
children.
4. All leaves appear on the same level.
The height of an Rtree containing
N
index
records is at most

log
m
N
 
1
The maximum number of nodes is
»
N
m
…
+
»
N
m
2
…
+
...
+ 1
Worst case space utilization for all nodes ex
cept the root is
m
M
.
Range Search
A
B
C
D
E
F
G
H
J
K
I
N
M
L
A
B
C
D
E
F
G
H
I
J
K
L
M
N
Query
Range Search
Similar to
B
tree in some way.
However,
more than one
subtree under a
node may be visited,
not possible to guarantee good perfor
mance
.
Algorithm Search
:
Given: an Rtree,
Query: a search rectangle Q.
1.
search subtrees
check each entry
E
to see if
E.I
overlaps
Q
.
For all overlapping entries
,
Search the subtree pointed to by its pointer.
2.
search leaf node
check all entries
E
to see if
E.I
overlaps
Q
.
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 Fall '09
 AdaFu
 Graph Theory, leaf node, Tree structure, F D J G

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