<s>
In	O
computer	B-General_Concept
science	I-General_Concept
,	O
a	O
K-D-B-tree	B-Data_Structure
(	O
k-dimensional	O
B-tree	B-Architecture
)	O
is	O
a	O
tree	O
data	O
structure	O
for	O
subdividing	O
a	O
k-dimensional	O
search	O
space	O
.	O
</s>
<s>
The	O
aim	O
of	O
the	O
K-D-B-tree	B-Data_Structure
is	O
to	O
provide	O
the	O
search	O
efficiency	O
of	O
a	O
balanced	O
k-d	B-Data_Structure
tree	I-Data_Structure
,	O
while	O
providing	O
the	O
block-oriented	O
storage	O
of	O
a	O
B-tree	B-Architecture
for	O
optimizing	O
external	O
memory	O
accesses	O
.	O
</s>
<s>
Much	O
like	O
the	O
k-d	B-Data_Structure
tree	I-Data_Structure
,	O
a	O
K-D-B-tree	B-Data_Structure
organizes	O
points	O
in	O
k-dimensional	O
space	O
,	O
useful	O
for	O
tasks	O
such	O
as	O
range-searching	O
and	O
multi-dimensional	O
database	O
queries	O
.	O
</s>
<s>
K-D-B-trees	B-Data_Structure
subdivide	O
space	O
into	O
two	O
subspaces	O
by	O
comparing	O
elements	O
in	O
a	O
single	O
domain	O
.	O
</s>
<s>
Using	O
a	O
2-D-B-tree	O
(	O
2-dimensional	O
K-D-B-tree	B-Data_Structure
)	O
as	O
an	O
example	O
,	O
space	O
is	O
subdivided	O
in	O
the	O
same	O
manner	O
as	O
a	O
k-d	B-Data_Structure
tree	I-Data_Structure
:	O
using	O
a	O
point	O
in	O
just	O
one	O
of	O
the	O
domains	O
,	O
or	O
axes	O
in	O
this	O
case	O
,	O
all	O
other	O
values	O
are	O
either	O
less	O
than	O
or	O
greater	O
than	O
the	O
current	O
value	O
,	O
and	O
fall	O
to	O
the	O
left	O
and	O
right	O
of	O
the	O
splitting	O
plane	O
respectively	O
.	O
</s>
<s>
Unlike	O
a	O
k-d	B-Data_Structure
tree	I-Data_Structure
,	O
each	O
half-space	O
is	O
not	O
its	O
own	O
node	O
.	O
</s>
<s>
Instead	O
,	O
as	O
in	O
a	O
B-tree	B-Architecture
,	O
nodes	O
in	O
the	O
K-D-B-tree	B-Data_Structure
are	O
stored	O
as	O
pages	O
and	O
the	O
tree	O
stores	O
a	O
pointer	O
to	O
the	O
root	O
page	O
.	O
</s>
<s>
The	O
K-D-B-tree	B-Data_Structure
contains	O
two	O
types	O
of	O
pages	O
:	O
</s>
<s>
Page	O
overflows	O
occur	O
when	O
inserting	O
an	O
element	O
into	O
a	O
K-D-B-tree	B-Data_Structure
results	O
in	O
the	O
size	O
of	O
a	O
node	O
exceeding	O
its	O
optimal	O
size	O
.	O
</s>
<s>
Since	O
the	O
purpose	O
of	O
the	O
K-D-B-tree	B-Data_Structure
is	O
to	O
optimize	O
external	O
memory	O
accesses	O
like	O
those	O
from	O
a	O
hard-disk	O
,	O
a	O
page	O
is	O
considered	O
to	O
have	O
overflowed	O
or	O
be	O
overfilled	O
if	O
the	O
size	O
of	O
the	O
node	O
exceeds	O
the	O
external	O
memory	O
page	O
size	O
.	O
</s>
<s>
Throughout	O
insertion/deletion	O
operations	O
,	O
the	O
K-D-B-tree	B-Data_Structure
maintains	O
a	O
certain	O
set	O
of	O
properties	O
:	O
</s>
<s>
Like	O
a	O
B-tree	B-Architecture
,	O
the	O
path	O
length	O
to	O
the	O
leaves	O
of	O
the	O
tree	O
is	O
the	O
same	O
for	O
all	O
queries	O
.	O
</s>
<s>
Queries	O
on	O
a	O
K-D-B-tree	B-Data_Structure
are	O
a	O
range	O
search	O
over	O
intervals	O
in	O
all	O
domains	O
or	O
axes	O
in	O
the	O
tree	O
.	O
</s>
<s>
Since	O
an	O
insertion	O
into	O
a	O
K-D-B-tree	B-Data_Structure
may	O
require	O
the	O
splitting	O
of	O
a	O
page	O
in	O
the	O
case	O
of	O
a	O
page	O
overflow	O
,	O
it	O
is	O
important	O
to	O
first	O
define	O
the	O
splitting	O
operation	O
.	O
</s>
<s>
Deletions	O
from	O
a	O
K-D-B-tree	B-Data_Structure
are	O
incredibly	O
simple	O
if	O
no	O
minimum	O
requirements	O
are	O
placed	O
on	O
storage	O
utilization	O
.	O
</s>
<s>
Since	O
deletions	O
can	O
result	O
in	O
pages	O
that	O
contain	O
very	O
little	O
data	O
,	O
it	O
may	O
be	O
necessary	O
to	O
reorganize	O
the	O
K-D-B-tree	B-Data_Structure
to	O
meet	O
some	O
minimum	O
storage	O
utilization	O
criteria	O
.	O
</s>
<s>
Like	O
in	O
a	O
k-d	B-Data_Structure
tree	I-Data_Structure
,	O
updates	O
in	O
a	O
K-D-B-tree	B-Data_Structure
may	O
result	O
in	O
the	O
requirement	O
for	O
the	O
splitting	O
of	O
several	O
nodes	O
recursively	O
.	O
</s>
<s>
Lomet	O
and	O
Salzberg	O
proposed	O
a	O
structure	O
called	O
the	O
hB-tree	O
(	O
holey	O
brick	O
tree	O
)	O
to	O
improve	O
performance	O
of	O
K-D-B-trees	B-Data_Structure
by	O
limiting	O
the	O
splits	O
that	O
occur	O
after	O
an	O
insertion	O
to	O
only	O
one	O
root-to-leaf	O
path	O
.	O
</s>
<s>
More	O
recently	O
,	O
the	O
Bkd-tree	O
was	O
proposed	O
as	O
a	O
means	O
to	O
provide	O
the	O
fast	O
queries	O
and	O
near	O
100%	O
space	O
utilization	O
of	O
a	O
static	O
K-D-B-tree	B-Data_Structure
.	O
</s>
<s>
Instead	O
of	O
maintaining	O
a	O
single	O
tree	O
and	O
re-balancing	O
,	O
a	O
set	O
of	O
K-D-B-trees	B-Data_Structure
are	O
maintained	O
and	O
rebuilt	O
at	O
regular	O
intervals	O
.	O
</s>
