<s>
In	O
computer	B-General_Concept
science	I-General_Concept
,	O
the	O
Bx	O
tree	O
is	O
a	O
query	O
that	O
is	O
used	O
to	O
update	O
efficient	O
B+	O
tree-based	O
index	O
structures	O
for	O
moving	O
objects	O
.	O
</s>
<s>
The	O
base	O
structure	O
of	O
the	O
Bx-tree	B-Data_Structure
is	O
a	O
B+	O
tree	O
in	O
which	O
the	O
internal	O
nodes	O
serve	O
as	O
a	O
directory	O
,	O
each	O
containing	O
a	O
pointer	O
to	O
its	O
right	O
sibling	O
.	O
</s>
<s>
In	O
the	O
earlier	O
version	O
of	O
the	O
Bx-tree	B-Data_Structure
,	O
the	O
leaf	B-Data_Structure
nodes	I-Data_Structure
contained	O
the	O
moving-object	O
locations	O
being	O
indexed	O
and	O
corresponding	O
index	O
time	O
.	O
</s>
<s>
In	O
the	O
optimized	O
version	O
,	O
each	O
leaf	B-Data_Structure
node	I-Data_Structure
entry	O
contains	O
the	O
id	O
,	O
velocity	O
,	O
single-dimensional	O
mapping	B-Algorithm
value	O
and	O
the	O
latest	O
update	O
time	O
of	O
the	O
object	O
.	O
</s>
<s>
The	O
fanout	O
is	O
increased	O
by	O
not	O
storing	O
the	O
locations	O
of	O
moving	O
objects	O
,	O
as	O
these	O
can	O
be	O
derived	O
from	O
the	O
mapping	B-Algorithm
values	O
.	O
</s>
<s>
In	O
order	O
to	O
adopt	O
the	O
B+	O
tree	O
as	O
a	O
moving	O
object	O
index	O
,	O
the	O
Bx-tree	B-Data_Structure
uses	O
a	O
linearization	O
technique	O
which	O
helps	O
to	O
integrate	O
objects	O
 '	O
location	O
at	O
time	O
t	O
into	O
single	O
dimensional	O
value	O
.	O
</s>
<s>
For	O
objects	O
within	O
the	O
same	O
partition	O
,	O
the	O
Bx-tree	B-Data_Structure
stores	O
their	O
locations	O
at	O
a	O
given	O
time	O
which	O
are	O
estimated	O
by	O
linear	O
interpolation	O
.	O
</s>
<s>
By	O
doing	O
so	O
,	O
the	O
Bx-tree	B-Data_Structure
keeps	O
a	O
consistent	O
view	O
of	O
all	O
objects	O
within	O
the	O
same	O
partition	O
without	O
storing	O
the	O
update	O
time	O
of	O
an	O
objects	O
.	O
</s>
<s>
Secondly	O
,	O
the	O
space	O
is	O
partitioned	O
by	O
a	O
grid	O
and	O
the	O
location	O
of	O
an	O
object	O
is	O
linearized	O
within	O
the	O
partitions	O
according	O
to	O
a	O
space-filling	B-Algorithm
curve	I-Algorithm
,	O
e.g.	O
,	O
the	O
Peano	B-Algorithm
or	O
Hilbert	B-Algorithm
curves	I-Algorithm
.	O
</s>
<s>
Finally	O
,	O
with	O
the	O
combination	O
of	O
the	O
partition	O
number	O
(	O
time	O
information	O
)	O
and	O
the	O
linear	O
order	O
(	O
location	O
information	O
)	O
,	O
an	O
object	O
is	O
indexed	O
in	O
Bx-tree	B-Data_Structure
with	O
a	O
one-dimensional	O
index	O
key	O
Bxvalue	O
:	O
</s>
<s>
Here	O
index-partition	O
is	O
an	O
index	O
partition	O
determined	O
by	O
the	O
update	O
time	O
and	O
xrep	O
is	O
the	O
space-filling	B-Algorithm
curve	I-Algorithm
value	O
of	O
the	O
object	O
position	O
at	O
the	O
indexed	O
time	O
,	O
denotes	O
the	O
binary	O
value	O
of	O
x	O
,	O
and	O
“	O
+	O
”	O
means	O
concatenation	O
.	O
</s>
<s>
Given	O
a	O
new	O
object	O
,	O
its	O
index	O
key	O
is	O
computed	O
and	O
then	O
the	O
object	O
is	O
inserted	O
into	O
the	O
Bx-tree	B-Data_Structure
as	O
in	O
the	O
B+	O
tree	O
.	O
</s>
<s>
In	O
this	O
way	O
,	O
the	O
Bx-tree	B-Data_Structure
directly	O
inherits	O
the	O
good	O
properties	O
of	O
the	O
B+	O
tree	O
,	O
and	O
achieves	O
efficient	O
update	O
performance	O
.	O
</s>
<s>
The	O
Bx-tree	B-Data_Structure
uses	O
query-window	O
enlargement	O
technique	O
to	O
answer	O
queries	O
.	O
</s>
<s>
Since	O
the	O
Bx-tree	B-Data_Structure
stores	O
an	O
object	O
's	O
location	O
as	O
of	O
sometime	O
after	O
its	O
update	O
time	O
,	O
the	O
enlargement	O
involves	O
two	O
cases	O
:	O
a	O
location	O
must	O
either	O
be	O
brought	O
back	O
to	O
an	O
earlier	O
time	O
or	O
forward	O
to	O
a	O
later	O
time	O
.	O
</s>
<s>
After	O
the	O
enlargement	O
,	O
the	O
partitions	O
of	O
the	O
Bx-tree	B-Data_Structure
need	O
to	O
be	O
traversed	O
to	O
find	O
objects	O
falling	O
in	O
the	O
enlarged	O
query	O
window	O
.	O
</s>
<s>
In	O
each	O
partition	O
,	O
the	O
use	O
of	O
a	O
space-filling	B-Algorithm
curve	I-Algorithm
means	O
that	O
a	O
range	O
query	O
in	O
the	O
native	O
,	O
two-dimensional	O
space	O
becomes	O
a	O
set	O
of	O
range	O
queries	O
in	O
the	O
transformed	O
,	O
one-dimensional	O
space	O
.	O
</s>
<s>
Another	O
possibility	O
is	O
to	O
employ	O
similar	O
querying	O
ideas	O
in	O
The	B-Algorithm
iDistance	I-Algorithm
Technique	I-Algorithm
.	O
</s>
<s>
Since	O
the	O
Bx-tree	B-Data_Structure
is	O
an	O
index	O
built	O
on	O
top	O
of	O
a	O
B+	O
tree	O
index	O
,	O
all	O
operations	O
in	O
the	O
Bx-tree	B-Data_Structure
,	O
including	O
the	O
insertion	O
,	O
deletion	O
and	O
search	O
,	O
are	O
the	O
same	O
as	O
those	O
in	O
the	O
B+	O
tree	O
.	O
</s>
<s>
The	O
only	O
difference	O
is	O
to	O
implement	O
the	O
procedure	O
of	O
deriving	O
the	O
indexing	O
key	O
as	O
a	O
stored	O
procedure	O
in	O
an	O
existing	O
DBMS	B-General_Concept
.	O
</s>
<s>
Therefore	O
,	O
the	O
Bx-tree	B-Data_Structure
can	O
be	O
easily	O
integrated	O
into	O
existing	O
DBMS	B-General_Concept
without	O
touching	O
the	O
kernel	B-Algorithm
.	O
</s>
<s>
SpADE	O
is	O
moving	O
object	O
management	O
system	O
built	O
on	O
top	O
of	O
a	O
popular	O
relational	O
database	O
system	O
MySQL	B-Application
,	O
which	O
uses	O
the	O
Bx-tree	B-Data_Structure
for	O
indexing	O
the	O
objects	O
.	O
</s>
<s>
In	O
the	O
implementation	O
,	O
moving	O
object	O
data	O
is	O
transformed	O
and	O
stored	O
directly	O
on	O
MySQL	B-Application
,	O
and	O
queries	O
are	O
transformed	O
into	O
standard	O
SQL	O
statements	O
which	O
are	O
efficiently	O
processed	O
in	O
the	O
relational	O
engine	O
.	O
</s>
<s>
Most	O
importantly	O
,	O
all	O
these	O
are	O
achieved	O
neatly	O
and	O
independently	O
without	O
infiltrating	O
into	O
the	O
MySQL	B-Application
core	O
.	O
</s>
<s>
The	O
Bx	O
tree	O
uses	O
a	O
grid	O
for	O
space	O
partitioning	O
while	O
mapping	B-Algorithm
two-dimensional	O
location	O
into	O
one-dimensional	O
key	O
.	O
</s>
<s>
The	O
ST2B-tree	O
introduces	O
a	O
self-tuning	O
framework	O
for	O
tuning	O
the	O
performance	O
of	O
the	O
Bx-tree	B-Data_Structure
while	O
dealing	O
with	O
data	O
skew	O
in	O
space	O
and	O
data	O
change	O
with	O
time	O
.	O
</s>
<s>
The	O
Bx-tree	B-Data_Structure
have	O
multiple	O
partitions	O
regarding	O
different	O
time	O
intervals	O
.	O
</s>
