<s>
In	O
computer	B-General_Concept
science	I-General_Concept
,	O
the	O
range	B-Data_Structure
searching	I-Data_Structure
problem	O
consists	O
of	O
processing	O
a	O
set	O
S	O
of	O
objects	O
,	O
in	O
order	O
to	O
determine	O
which	O
objects	O
from	O
S	O
intersect	O
with	O
a	O
query	O
object	O
,	O
called	O
the	O
range	O
.	O
</s>
<s>
The	O
range	B-Data_Structure
searching	I-Data_Structure
problem	O
and	O
the	O
data	B-General_Concept
structures	I-General_Concept
that	O
solve	O
it	O
are	O
a	O
fundamental	O
topic	O
of	O
computational	O
geometry	O
.	O
</s>
<s>
Applications	O
of	O
the	O
problem	O
arise	O
in	O
areas	O
such	O
as	O
geographical	B-Application
information	I-Application
systems	I-Application
(	O
GIS	B-Application
)	O
,	O
computer-aided	B-Application
design	I-Application
(	O
CAD	B-Application
)	O
and	O
databases	O
.	O
</s>
<s>
There	O
are	O
several	O
variations	O
of	O
the	O
problem	O
,	O
and	O
different	O
data	B-General_Concept
structures	I-General_Concept
may	O
be	O
necessary	O
for	O
different	O
variations	O
.	O
</s>
<s>
Object	O
types	O
:	O
Algorithms	O
depend	O
on	O
whether	O
S	O
consists	O
of	O
points	O
,	O
lines	O
,	O
line	O
segments	O
,	O
boxes	O
,	O
polygons	B-General_Concept
....	O
</s>
<s>
Some	O
well-studied	O
sets	O
of	O
ranges	O
,	O
and	O
the	O
names	O
of	O
the	O
respective	O
problems	O
are	O
axis-aligned	O
rectangles	O
(	O
orthogonal	O
range	B-Data_Structure
searching	I-Data_Structure
)	O
,	O
simplices	O
,	O
halfspaces	O
,	O
and	O
spheres/circles	O
.	O
</s>
<s>
Query	O
types	O
:	O
If	O
the	O
list	O
of	O
all	O
objects	O
that	O
intersect	O
the	O
query	O
range	O
must	O
be	O
reported	O
,	O
the	O
problem	O
is	O
called	O
range	B-Data_Structure
reporting	I-Data_Structure
,	O
and	O
the	O
query	O
is	O
called	O
a	O
reporting	O
query	O
.	O
</s>
<s>
Dynamic	B-Data_Structure
range	B-Data_Structure
searching	I-Data_Structure
vs.	O
static	O
range	B-Data_Structure
searching	I-Data_Structure
:	O
In	O
the	O
static	O
setting	O
the	O
set	O
S	O
is	O
known	O
in	O
advance	O
.	O
</s>
<s>
In	O
dynamic	B-Data_Structure
setting	O
objects	O
may	O
be	O
inserted	O
or	O
deleted	O
between	O
queries	O
.	O
</s>
<s>
Offline	O
range	B-Data_Structure
searching	I-Data_Structure
:	O
Both	O
the	O
set	O
of	O
objects	O
and	O
the	O
whole	O
set	O
of	O
queries	O
are	O
known	O
in	O
advance	O
.	O
</s>
<s>
In	O
orthogonal	O
range	B-Data_Structure
searching	I-Data_Structure
,	O
the	O
set	O
S	O
consists	O
of	O
points	O
in	O
dimensions	O
,	O
and	O
the	O
query	O
consists	O
of	O
intervals	O
in	O
each	O
of	O
those	O
dimensions	O
.	O
</s>
<s>
With	O
an	O
output	O
size	O
of	O
,	O
Jon	O
Bentley	O
used	O
a	O
k-d	B-Data_Structure
tree	I-Data_Structure
to	O
achieve	O
(	O
in	O
Big	O
O	O
notation	O
)	O
space	O
and	O
query	O
time	O
.	O
</s>
<s>
Bentley	O
also	O
proposed	O
using	O
range	B-Data_Structure
trees	I-Data_Structure
,	O
which	O
improved	O
query	O
time	O
to	O
but	O
increased	O
space	O
to	O
.	O
</s>
<s>
Dan	O
Willard	O
used	O
downpointers	O
,	O
a	O
special	O
case	O
of	O
fractional	B-Data_Structure
cascading	I-Data_Structure
to	O
reduce	O
the	O
query	O
time	O
further	O
to	O
.	O
</s>
<s>
While	O
the	O
above	O
results	O
were	O
achieved	O
in	O
the	O
pointer	B-Application
machine	I-Application
model	O
,	O
further	O
improvements	O
have	O
been	O
made	O
in	O
the	O
word	B-Application
RAM	I-Application
model	O
of	O
computation	O
in	O
low	O
dimensions	O
(	O
2D	O
,	O
3D	O
,	O
4D	O
)	O
.	O
</s>
<s>
Bernard	O
Chazelle	O
used	O
compress	O
range	B-Data_Structure
trees	I-Data_Structure
to	O
achieve	O
query	O
time	O
and	O
space	O
for	O
range	O
counting	O
.	O
</s>
<s>
Joseph	O
JaJa	O
and	O
others	O
later	O
improved	O
this	O
query	O
time	O
to	O
for	O
range	O
counting	O
,	O
which	O
matches	O
a	O
lower	O
bound	O
and	O
is	O
thus	O
asymptotically	B-General_Concept
optimal	I-General_Concept
.	O
</s>
<s>
As	O
of	O
2015	O
,	O
the	O
best	O
results	O
(	O
in	O
low	O
dimensions	O
(	O
2D	O
,	O
3D	O
,	O
4D	O
)	O
)	O
for	O
range	B-Data_Structure
reporting	I-Data_Structure
found	O
by	O
Timothy	O
M	O
.	O
Chan	O
,	O
Kasper	O
Larsen	O
,	O
and	O
Mihai	O
Pătrașcu	O
,	O
also	O
using	O
compressed	O
range	B-Data_Structure
trees	I-Data_Structure
in	O
the	O
word	B-Application
RAM	I-Application
model	O
of	O
computation	O
,	O
are	O
one	O
of	O
the	O
following	O
:	O
</s>
<s>
In	O
the	O
orthogonal	O
case	O
,	O
if	O
one	O
of	O
the	O
bounds	O
is	O
infinity	B-Application
,	O
the	O
query	O
is	O
called	O
three-sided	O
.	O
</s>
<s>
If	O
two	O
of	O
the	O
bounds	O
are	O
infinity	B-Application
,	O
the	O
query	O
is	O
two-sided	O
,	O
and	O
if	O
none	O
of	O
the	O
bounds	O
are	O
infinity	B-Application
,	O
then	O
the	O
query	O
is	O
four-sided	O
.	O
</s>
<s>
While	O
in	O
static	O
range	B-Data_Structure
searching	I-Data_Structure
the	O
set	O
S	O
is	O
known	O
in	O
advance	O
,	O
dynamic	B-Data_Structure
range	B-Data_Structure
searching	I-Data_Structure
,	O
insertions	O
and	O
deletions	O
of	O
points	O
are	O
allowed	O
.	O
</s>
<s>
For	O
the	O
orthogonal	O
case	O
,	O
Kurt	O
Mehlhorn	O
and	O
Stefan	O
Näher	O
created	O
a	O
data	B-General_Concept
structure	I-General_Concept
for	O
dynamic	B-Data_Structure
range	B-Data_Structure
searching	I-Data_Structure
which	O
uses	O
dynamic	B-Data_Structure
fractional	B-Data_Structure
cascading	I-Data_Structure
to	O
achieve	O
space	O
and	O
query	O
time	O
.	O
</s>
<s>
Both	O
incremental	O
and	O
decremental	O
versions	O
of	O
the	O
problem	O
can	O
be	O
solved	O
with	O
query	O
time	O
,	O
but	O
it	O
is	O
unknown	O
whether	O
general	O
dynamic	B-Data_Structure
range	B-Data_Structure
searching	I-Data_Structure
can	O
be	O
done	O
with	O
that	O
query	O
time	O
.	O
</s>
<s>
Prosenjit	O
Gupta	O
and	O
others	O
described	O
a	O
data	B-General_Concept
structure	I-General_Concept
in	O
1995	O
which	O
solved	O
2D	O
orthogonal	O
colored	O
range	O
counting	O
in	O
space	O
and	O
query	O
time	O
.	O
</s>
<s>
In	O
addition	O
to	O
being	O
considered	O
in	O
computational	O
geometry	O
,	O
range	B-Data_Structure
searching	I-Data_Structure
,	O
and	O
orthogonal	O
range	B-Data_Structure
searching	I-Data_Structure
in	O
particular	O
,	O
has	O
applications	O
for	O
range	B-General_Concept
queries	I-General_Concept
in	O
databases	O
.	O
</s>
<s>
Colored	O
range	B-Data_Structure
searching	I-Data_Structure
is	O
also	O
used	O
for	O
and	O
motivated	O
by	O
searching	O
through	O
categorical	O
data	O
.	O
</s>
<s>
For	O
example	O
,	O
determining	O
the	O
rows	O
in	O
a	O
database	O
of	O
bank	O
accounts	O
which	O
represent	O
people	O
whose	O
age	O
is	O
between	O
25	O
and	O
40	O
and	O
who	O
have	O
between	O
$10000	O
and	O
$20000	O
might	O
be	O
an	O
orthogonal	O
range	B-Data_Structure
reporting	I-Data_Structure
problem	O
where	O
age	O
and	O
money	O
are	O
two	O
dimensions	O
.	O
</s>
