<s>
In	O
computer	B-General_Concept
science	I-General_Concept
,	O
fractional	B-Data_Structure
cascading	I-Data_Structure
is	O
a	O
technique	O
to	O
speed	O
up	O
a	O
sequence	O
of	O
binary	O
searches	O
for	O
the	O
same	O
value	O
in	O
a	O
sequence	O
of	O
related	O
data	O
structures	O
.	O
</s>
<s>
The	O
original	O
version	O
of	O
fractional	B-Data_Structure
cascading	I-Data_Structure
,	O
introduced	O
in	O
two	O
papers	O
by	O
Chazelle	O
and	O
Guibas	O
in	O
1986	O
(	O
;	O
)	O
,	O
combined	O
the	O
idea	O
of	O
cascading	O
,	O
originating	O
in	O
range	B-Data_Structure
searching	I-Data_Structure
data	O
structures	O
of	O
and	O
,	O
with	O
the	O
idea	O
of	O
fractional	O
sampling	O
,	O
which	O
originated	O
in	O
.	O
</s>
<s>
Later	O
authors	O
introduced	O
more	O
complex	O
forms	O
of	O
fractional	B-Data_Structure
cascading	I-Data_Structure
that	O
allow	O
the	O
data	O
structure	O
to	O
be	O
maintained	O
as	O
the	O
data	O
changes	O
by	O
a	O
sequence	O
of	O
discrete	O
insertion	O
and	O
deletion	O
events	O
.	O
</s>
<s>
As	O
a	O
simple	O
example	O
of	O
fractional	B-Data_Structure
cascading	I-Data_Structure
,	O
consider	O
the	O
following	O
problem	O
.	O
</s>
<s>
Fractional	B-Data_Structure
cascading	I-Data_Structure
allows	O
this	O
same	O
searching	O
problem	O
to	O
be	O
solved	O
with	O
time	O
and	O
space	O
bounds	O
meeting	O
the	O
best	O
of	O
both	O
worlds	O
:	O
query	O
time	O
O( log	O
n+k	O
)	O
,	O
and	O
space	O
O(n )	O
.	O
</s>
<s>
The	O
fractional	B-Data_Structure
cascading	I-Data_Structure
solution	O
is	O
to	O
store	O
a	O
new	O
sequence	O
of	O
lists	O
Mi	O
.	O
</s>
<s>
In	O
general	O
,	O
fractional	B-Data_Structure
cascading	I-Data_Structure
begins	O
with	O
a	O
catalog	O
graph	O
,	O
a	O
directed	O
graph	O
in	O
which	O
each	O
vertex	O
is	O
labeled	O
with	O
an	O
ordered	O
list	O
.	O
</s>
<s>
In	O
dynamic	O
fractional	B-Data_Structure
cascading	I-Data_Structure
,	O
the	O
list	O
stored	O
at	O
each	O
node	O
of	O
the	O
catalog	O
graph	O
may	O
change	O
dynamically	O
,	O
by	O
a	O
sequence	O
of	O
updates	O
in	O
which	O
list	O
items	O
are	O
inserted	O
and	O
deleted	O
.	O
</s>
<s>
Rather	O
,	O
a	O
technique	O
closely	O
related	O
to	O
B-trees	B-Architecture
allows	O
the	O
selection	O
of	O
a	O
fraction	O
of	O
data	O
that	O
is	O
guaranteed	O
to	O
be	O
smaller	O
than	O
1/d	O
,	O
with	O
the	O
selected	O
items	O
guaranteed	O
to	O
be	O
spaced	O
a	O
constant	O
number	O
of	O
positions	O
apart	O
in	O
the	O
full	O
list	O
,	O
and	O
such	O
that	O
an	O
insertion	O
or	O
deletion	O
into	O
the	O
augmented	O
list	O
associated	O
with	O
a	O
node	O
causes	O
changes	O
to	O
propagate	O
to	O
other	O
nodes	O
for	O
a	O
fraction	O
of	O
the	O
operations	O
that	O
is	O
less	O
than	O
1/d	O
.	O
</s>
<s>
Third	O
,	O
and	O
most	O
critically	O
,	O
the	O
static	O
fractional	B-Data_Structure
cascading	I-Data_Structure
data	O
structure	O
maintains	O
,	O
for	O
each	O
element	O
x	O
of	O
the	O
augmented	O
list	O
at	O
each	O
node	O
of	O
the	O
catalog	O
graph	O
,	O
the	O
index	O
of	O
the	O
result	O
that	O
would	O
be	O
obtained	O
when	O
searching	O
for	O
x	O
among	O
the	O
input	O
items	O
from	O
that	O
node	O
and	O
among	O
the	O
augmented	O
lists	O
stored	O
at	O
neighboring	O
nodes	O
.	O
</s>
<s>
Instead	O
,	O
dynamic	O
versions	O
of	O
fractional	B-Data_Structure
cascading	I-Data_Structure
maintain	O
several	O
data	O
structures	O
for	O
each	O
node	O
:	O
</s>
<s>
The	O
order-maintenance	B-General_Concept
technique	O
of	O
allows	O
this	O
numbering	O
to	O
be	O
maintained	O
efficiently	O
.	O
</s>
<s>
An	O
integer	O
searching	O
data	O
structure	O
such	O
as	O
a	O
van	B-Application
Emde	I-Application
Boas	I-Application
tree	I-Application
for	O
the	O
numbers	O
associated	O
with	O
the	O
input	O
list	O
of	O
the	O
node	O
.	O
</s>
<s>
These	O
data	O
structures	O
allow	O
dynamic	O
fractional	B-Data_Structure
cascading	I-Data_Structure
to	O
be	O
performed	O
at	O
a	O
time	O
of	O
O(logn )	O
per	O
insertion	O
or	O
deletion	O
,	O
and	O
a	O
sequence	O
of	O
k	O
binary	O
searches	O
following	O
a	O
path	O
of	O
length	O
k	O
in	O
the	O
catalog	O
graph	O
to	O
be	O
performed	O
in	O
time	O
O( logn+kloglogn	O
)	O
.	O
</s>
<s>
Typical	O
applications	O
of	O
fractional	B-Data_Structure
cascading	I-Data_Structure
involve	O
range	B-Data_Structure
search	I-Data_Structure
data	O
structures	O
in	O
computational	O
geometry	O
.	O
</s>
<s>
For	O
example	O
,	O
consider	O
the	O
problem	O
of	O
half-plane	O
range	B-Data_Structure
reporting	I-Data_Structure
:	O
that	O
is	O
,	O
intersecting	O
a	O
fixed	O
set	O
of	O
n	O
points	O
with	O
a	O
query	O
half-plane	O
and	O
listing	O
all	O
the	O
points	O
in	O
the	O
intersection	O
.	O
</s>
<s>
Within	O
a	O
single	O
layer	O
,	O
the	O
points	O
inside	O
the	O
query	O
half-plane	O
may	O
be	O
found	O
by	O
performing	O
a	O
binary	O
search	O
for	O
the	O
half-plane	O
boundary	O
line	O
's	O
slope	O
among	O
the	O
sorted	O
sequence	O
of	O
convex	O
polygon	O
edge	O
slopes	O
,	O
leading	O
to	O
the	O
polygon	O
vertex	O
that	O
is	O
inside	O
the	O
query	O
half-plane	O
and	O
farthest	O
from	O
its	O
boundary	O
,	O
and	O
then	O
sequentially	B-Algorithm
searching	I-Algorithm
along	O
the	O
polygon	O
edges	O
to	O
find	O
all	O
other	O
vertices	O
inside	O
the	O
query	O
half-plane	O
.	O
</s>
<s>
The	O
whole	O
half-plane	O
range	B-Data_Structure
reporting	I-Data_Structure
problem	O
may	O
be	O
solved	O
by	O
repeating	O
this	O
search	O
procedure	O
starting	O
from	O
the	O
outermost	O
layer	O
and	O
continuing	O
inwards	O
until	O
reaching	O
a	O
layer	O
that	O
is	O
disjoint	O
from	O
the	O
query	O
halfspace	O
.	O
</s>
<s>
Fractional	B-Data_Structure
cascading	I-Data_Structure
speeds	O
up	O
the	O
successive	O
binary	O
searches	O
among	O
the	O
sequences	O
of	O
polygon	O
edge	O
slopes	O
in	O
each	O
layer	O
,	O
leading	O
to	O
a	O
data	O
structure	O
for	O
this	O
problem	O
with	O
space	O
O(n )	O
and	O
query	O
time	O
O( logn+h	O
)	O
.	O
</s>
<s>
Another	O
application	O
of	O
fractional	B-Data_Structure
cascading	I-Data_Structure
in	O
geometric	O
data	O
structures	O
concerns	O
point	B-Data_Structure
location	I-Data_Structure
in	O
a	O
monotone	O
subdivision	O
,	O
that	O
is	O
,	O
a	O
partition	O
of	O
the	O
plane	O
into	O
polygons	O
such	O
that	O
any	O
vertical	O
line	O
intersects	O
any	O
polygon	O
in	O
at	O
most	O
two	O
points	O
.	O
</s>
<s>
Thus	O
,	O
each	O
point	B-Data_Structure
location	I-Data_Structure
query	O
can	O
be	O
solved	O
as	O
an	O
outer	O
layer	O
of	O
binary	O
search	O
among	O
the	O
paths	O
,	O
each	O
step	O
of	O
which	O
itself	O
performs	O
a	O
binary	O
search	O
among	O
x	O
coordinates	O
of	O
vertices	O
.	O
</s>
<s>
Fractional	B-Data_Structure
cascading	I-Data_Structure
can	O
be	O
used	O
to	O
speed	O
up	O
the	O
time	O
for	O
the	O
inner	O
binary	O
searches	O
,	O
reducing	O
the	O
total	O
time	O
per	O
query	O
to	O
O(logn )	O
using	O
a	O
data	O
structure	O
with	O
space	O
O(n )	O
.	O
</s>
<s>
Beyond	O
computational	O
geometry	O
,	O
and	O
apply	O
fractional	B-Data_Structure
cascading	I-Data_Structure
in	O
the	O
design	O
of	O
data	O
structures	O
for	O
fast	O
packet	O
filtering	O
in	O
internet	B-Protocol
routers	I-Protocol
.	O
</s>
<s>
use	O
fractional	B-Data_Structure
cascading	I-Data_Structure
as	O
a	O
model	O
for	O
data	O
distribution	O
and	O
retrieval	O
in	O
sensor	B-Architecture
networks	I-Architecture
.	O
</s>
