Slideshow:
How a kd-tree is stored (in memory)
The data record is stored in the internal nodes !!
Modifying the kd-tree for disk storage
Modification 1: internal nodes are for searching only
Modifying the kd-tree for disk storage
Modification 2: add the equality (=) case to the right sub-tree
Modifying the kd-tree for disk storage
Modification 3: only the leaf nodes will store data (that's what we want to achieve !)
Example of a kd-tree index for disk
Input data:
Example of a kd-tree index for disk
Input data:
The kd-tree:
Search property of a kd-tree index
All data records in the left subtree (shaded) has salary search key values < 150:
(It's similar to a Binary Search Tree)
Search property of a kd-tree index
Data records in this left subtree has salary < 150 and age < 60:
(It's similar to a Binary Search Tree)
Search property of a kd-tree index
Data records in this right subtree has salary < 150 and age ≥ 60:
(It's similar to a Binary Search Tree)
And so on...
Search space partitioning property of a kd-tree index
Search space partitioning property of a kd-tree index
Search space partitioning property of a kd-tree index
Partitioning of the search space:
- Memory kd-tree:
- The actual record (data)
are stored
in node
of the kd-tree:
The node y1 contains the data (record) for (x1,y1)
The node x2 contains the data (record) for (x2,y1)
The node x3 contains the data (record) for (x3,y1)
And so on..
- The actual record (data)
are stored
in node
of the kd-tree:
- Modifications to
the kd-tree for storage on
disk:
- Interior nodes do
not store
data
Interior node only stores:
- Attribute name (i.e.: X or Y)
- Dividing value (i.e.: x1 or y4) of the attribute
- Pointers to the (2) children nodes
- Modify the
search boundery in
the internal nodes to:
The equality is included in right branch of the kd-tree to avoid storing data in an internal node !
- A leaf node of the (modified) kd-tree (for disk storage) contains data records and is stored in one (1) data block
Note:
-
This structure is
similar to the
B-tree:
The equality is included in the right(most) branch.
- Interior nodes do
not store
data
- Data on people who
buy jewelry:
(age, salary (in $1,000)) A(25,60) D(45,60) G(50,75) J(50,100) B(50,120) E(70,110) H(85,140) K(30,260) C(25,400) F(45,350) I(50,275) L(60,260)
- A kd-tree for the
above data:
- The
structural properties of the
kd-tree that
allows us to
search
in the kd-tree:
- All
data records in
the left (shaded) subtree
in the figure below has
salary search key values
<
150 (for salary) :
- Data records in
this left subtree has
salary < 150
and
age < 60:
- This right subtree has
salary < 150
and
age ≥ 60:
And so on
The search property is similar to the search property of the Binary Search Tree.
- All
data records in
the left (shaded) subtree
in the figure below has
salary search key values
<
150 (for salary) :
- The internal nodes of the
kd-tree
partitions the
search space
into disjoint search areas
The following figures shows how a kd-tree partitions the search space:
- The root node:
partitions the salary search space in half:
- The age nodes at
level 2:
partitions each sub-space in half again:
And so on...
- The root node:
- This kd-tree:
will divide the search (data) space up as follows:
(taken from the text book)