<s>
In	O
the	O
design	O
and	O
analysis	O
of	O
algorithms	O
for	O
combinatorial	O
optimization	O
,	O
parametric	B-Algorithm
search	I-Algorithm
is	O
a	O
technique	O
invented	O
by	O
for	O
transforming	O
a	O
decision	O
algorithm	O
(	O
does	O
this	O
optimization	O
problem	O
have	O
a	O
solution	O
with	O
quality	O
better	O
than	O
some	O
given	O
threshold	O
?	O
)	O
</s>
<s>
The	O
basic	O
idea	O
of	O
parametric	B-Algorithm
search	I-Algorithm
is	O
to	O
simulate	O
a	O
test	O
algorithm	O
that	O
takes	O
as	O
input	O
a	O
numerical	O
parameter	O
,	O
as	O
if	O
it	O
were	O
being	O
run	O
with	O
the	O
(	O
unknown	O
)	O
optimal	O
solution	O
value	O
as	O
its	O
input	O
.	O
</s>
<s>
To	O
simulate	O
each	O
comparison	O
,	O
the	O
parametric	B-Algorithm
search	I-Algorithm
applies	O
a	O
second	O
algorithm	O
,	O
a	O
decision	O
algorithm	O
,	O
that	O
takes	O
as	O
input	O
another	O
numerical	O
parameter	O
,	O
and	O
that	O
determines	O
whether	O
is	O
above	O
,	O
below	O
,	O
or	O
equal	O
to	O
the	O
optimal	O
solution	O
value	O
.	O
</s>
<s>
However	O
,	O
many	O
applications	O
use	O
other	O
test	O
algorithms	O
(	O
often	O
,	O
comparison	B-Algorithm
sorting	I-Algorithm
algorithms	O
)	O
.	O
</s>
<s>
Advanced	O
versions	O
of	O
the	O
parametric	B-Algorithm
search	I-Algorithm
technique	O
use	O
a	O
parallel	B-Operating_System
algorithm	I-Operating_System
as	O
the	O
test	O
algorithm	O
,	O
and	O
group	O
the	O
comparisons	O
that	O
must	O
be	O
simulated	O
into	O
batches	O
,	O
in	O
order	O
to	O
significantly	O
reduce	O
the	O
number	O
of	O
instantiations	O
of	O
the	O
decision	O
algorithm	O
.	O
</s>
<s>
In	O
the	O
most	O
basic	O
form	O
of	O
the	O
parametric	B-Algorithm
search	I-Algorithm
technique	O
,	O
both	O
the	O
test	O
algorithm	O
and	O
the	O
decision	O
algorithms	O
are	O
sequential	O
(	O
non-parallel	O
)	O
algorithms	O
,	O
possibly	O
the	O
same	O
algorithm	O
as	O
each	O
other	O
.	O
</s>
<s>
Using	O
this	O
decision	O
algorithm	O
as	O
both	O
the	O
test	O
algorithm	O
and	O
the	O
decision	O
algorithm	O
of	O
a	O
parametric	B-Algorithm
search	I-Algorithm
leads	O
to	O
an	O
algorithm	O
for	O
finding	O
the	O
optimal	O
time	O
in	O
quadratic	O
total	O
time	O
.	O
</s>
<s>
As	O
already	O
observed	O
,	O
the	O
parametric	B-Algorithm
search	I-Algorithm
technique	O
can	O
be	O
substantially	O
sped	O
up	O
by	O
replacing	O
the	O
simulated	O
test	O
algorithm	O
by	O
an	O
efficient	O
parallel	B-Operating_System
algorithm	I-Operating_System
,	O
for	O
instance	O
in	O
the	O
parallel	B-Operating_System
random-access	I-Operating_System
machine	I-Operating_System
(	O
PRAM	O
)	O
model	O
of	O
parallel	O
computation	O
,	O
where	O
a	O
collection	O
of	O
processors	O
operate	O
in	O
synchrony	O
on	O
a	O
shared	B-Operating_System
memory	I-Operating_System
,	O
all	O
performing	O
the	O
same	O
sequence	O
of	O
operations	O
on	O
different	O
memory	O
addresses	O
.	O
</s>
<s>
Thus	O
,	O
the	O
total	O
time	O
for	O
parametric	B-Algorithm
search	I-Algorithm
in	O
this	O
case	O
becomes	O
(	O
for	O
the	O
simulation	O
itself	O
)	O
plus	O
the	O
time	O
for	O
calls	O
to	O
the	O
decision	O
algorithm	O
(	O
for	O
batches	O
of	O
comparisons	O
,	O
taking	O
calls	O
per	O
batch	O
)	O
.	O
</s>
<s>
Often	O
,	O
for	O
a	O
problem	O
that	O
can	O
be	O
solved	O
in	O
this	O
way	O
,	O
the	O
time-processor	O
product	O
of	O
the	O
PRAM	O
algorithm	O
is	O
comparable	O
to	O
the	O
time	O
for	O
a	O
sequential	O
decision	O
algorithm	O
,	O
and	O
the	O
parallel	O
time	O
is	O
polylogarithmic	O
,	O
leading	O
to	O
a	O
total	O
time	O
for	O
the	O
parametric	B-Algorithm
search	I-Algorithm
that	O
is	O
slower	O
than	O
the	O
decision	O
algorithm	O
by	O
only	O
a	O
polylogarithmic	O
factor	O
.	O
</s>
<s>
In	O
the	O
case	O
of	O
the	O
example	O
problem	O
of	O
finding	O
the	O
crossing	O
time	O
of	O
the	O
median	O
of	O
moving	O
particles	O
,	O
the	O
sequential	O
test	O
algorithm	O
can	O
be	O
replaced	O
by	O
a	O
parallel	O
sorting	B-Algorithm
algorithm	O
that	O
sorts	O
the	O
positions	O
of	O
the	O
particles	O
at	O
the	O
time	O
given	O
by	O
the	O
algorithm	O
's	O
parameter	O
,	O
and	O
then	O
uses	O
the	O
sorted	O
order	O
to	O
determine	O
the	O
median	O
particle	O
and	O
find	O
the	O
sign	O
of	O
its	O
position	O
.	O
</s>
<s>
The	O
best	O
choice	O
for	O
this	O
algorithm	O
(	O
according	O
to	O
its	O
theoretical	O
analysis	O
,	O
if	O
not	O
in	O
practice	O
)	O
is	O
the	O
sorting	B-Algorithm
network	I-Algorithm
of	O
.	O
</s>
<s>
Putting	O
these	O
time	O
bounds	O
together	O
gives	O
total	O
time	O
for	O
the	O
parametric	B-Algorithm
search	I-Algorithm
.	O
</s>
<s>
Because	O
of	O
the	O
large	O
constant	O
factors	O
arising	O
in	O
the	O
analysis	O
of	O
the	O
AKS	B-Algorithm
sorting	I-Algorithm
network	I-Algorithm
,	O
parametric	B-Algorithm
search	I-Algorithm
using	O
this	O
network	O
as	O
the	O
test	O
algorithm	O
is	O
not	O
practical	O
.	O
</s>
<s>
Instead	O
,	O
suggest	O
using	O
a	O
parallel	O
form	O
of	O
quicksort	B-Algorithm
(	O
an	O
algorithm	O
that	O
repeatedly	O
partitions	O
the	O
input	O
into	O
two	O
subsets	O
and	O
then	O
recursively	O
sorts	O
each	O
subset	O
)	O
.	O
</s>
<s>
The	O
resulting	O
parametric	O
algorithm	O
is	O
slower	O
in	O
the	O
worst	O
case	O
than	O
an	O
algorithm	O
based	O
on	O
the	O
AKS	B-Algorithm
sorting	I-Algorithm
network	I-Algorithm
.	O
</s>
<s>
Based	O
on	O
this	O
heuristic	O
analysis	O
,	O
and	O
on	O
experimental	O
results	O
with	O
an	O
implementation	O
of	O
the	O
algorithm	O
,	O
they	O
argue	O
that	O
a	O
quicksort-based	O
parametric	B-Algorithm
search	I-Algorithm
algorithm	O
will	O
be	O
more	O
practical	O
than	O
its	O
worst-case	O
analysis	O
would	O
suggest	O
.	O
</s>
<s>
further	O
optimized	O
the	O
parametric	B-Algorithm
search	I-Algorithm
technique	O
for	O
cases	O
(	O
such	O
as	O
the	O
example	O
)	O
in	O
which	O
the	O
test	O
algorithm	O
is	O
a	O
comparison	B-Algorithm
sorting	I-Algorithm
algorithm	O
.	O
</s>
<s>
For	O
the	O
AKS	B-Algorithm
sorting	I-Algorithm
network	I-Algorithm
and	O
some	O
other	O
sorting	B-Algorithm
algorithms	O
that	O
can	O
be	O
used	O
in	O
its	O
place	O
,	O
Cole	O
observes	O
that	O
it	O
is	O
not	O
necessary	O
to	O
keep	O
the	O
simulated	O
processors	O
synchronized	O
with	O
each	O
other	O
:	O
instead	O
,	O
one	O
can	O
allow	O
some	O
of	O
them	O
to	O
progress	O
farther	O
through	O
the	O
sorting	B-Algorithm
algorithm	O
while	O
others	O
wait	O
for	O
the	O
results	O
of	O
their	O
comparisons	O
to	O
be	O
determined	O
.	O
</s>
<s>
Based	O
on	O
this	O
principle	O
,	O
Cole	O
modifies	O
the	O
simulation	O
of	O
the	O
sorting	B-Algorithm
algorithm	O
,	O
so	O
that	O
it	O
maintains	O
a	O
collection	O
of	O
unresolved	O
simulated	O
comparisons	O
that	O
may	O
not	O
all	O
come	O
from	O
the	O
same	O
parallel	O
time	O
step	O
of	O
the	O
test	O
algorithm	O
.	O
</s>
<s>
As	O
in	O
the	O
synchronized	O
parallel	O
version	O
of	O
parametric	B-Algorithm
search	I-Algorithm
,	O
it	O
is	O
possible	O
to	O
resolve	O
half	O
of	O
these	O
comparisons	O
by	O
finding	O
the	O
median	O
comparison	O
value	O
and	O
calling	O
the	O
decision	O
algorithm	O
on	O
this	O
value	O
.	O
</s>
<s>
Using	O
this	O
method	O
,	O
Cole	O
shows	O
that	O
a	O
parametric	B-Algorithm
search	I-Algorithm
algorithm	O
in	O
which	O
the	O
test	O
algorithm	O
is	O
sorting	B-Algorithm
may	O
be	O
completed	O
using	O
only	O
a	O
logarithmic	O
number	O
of	O
calls	O
to	O
the	O
decision	O
algorithm	O
,	O
instead	O
of	O
the	O
calls	O
made	O
by	O
Megiddo	O
's	O
original	O
version	O
of	O
parametric	B-Algorithm
search	I-Algorithm
.	O
</s>
<s>
Instead	O
of	O
using	O
the	O
AKS	B-Algorithm
sorting	I-Algorithm
network	I-Algorithm
,	O
it	O
is	O
also	O
possible	O
to	O
combine	O
this	O
technique	O
with	O
a	O
parallel	O
merge	B-Algorithm
sort	I-Algorithm
algorithm	O
of	O
,	O
resulting	O
in	O
time	O
bounds	O
with	O
smaller	O
constant	O
factors	O
that	O
,	O
however	O
,	O
are	O
still	O
not	O
small	O
enough	O
to	O
be	O
practical	O
.	O
</s>
<s>
A	O
similar	O
speedup	O
can	O
be	O
obtained	O
for	O
any	O
problem	O
that	O
can	O
be	O
computed	O
on	O
a	O
distributed	O
computing	O
network	O
of	O
bounded	O
degree	O
(	O
as	O
the	O
AKS	B-Algorithm
sorting	I-Algorithm
network	I-Algorithm
is	O
)	O
,	O
either	O
by	O
Cole	O
's	O
technique	O
or	O
by	O
a	O
related	O
technique	O
of	O
simulating	O
multiple	O
computation	O
paths	O
.	O
</s>
<s>
Instead	O
of	O
using	O
a	O
parallel	B-Algorithm
quicksort	I-Algorithm
,	O
as	O
van	O
Oostrum	O
and	O
Veltkamp	O
did	O
,	O
they	O
use	O
boxsort	O
,	O
a	O
variant	O
of	O
quicksort	B-Algorithm
developed	O
by	O
in	O
which	O
the	O
partitioning	O
step	O
partitions	O
the	O
input	O
randomly	O
into	O
smaller	O
subproblems	O
instead	O
of	O
only	O
two	O
subproblems	O
.	O
</s>
<s>
As	O
in	O
Cole	O
's	O
technique	O
,	O
they	O
use	O
a	O
desynchronized	O
parametric	B-Algorithm
search	I-Algorithm
,	O
in	O
which	O
each	O
separate	O
thread	O
of	O
execution	O
of	O
the	O
simulated	O
parallel	O
sorting	B-Algorithm
algorithm	O
is	O
allowed	O
to	O
progress	O
until	O
it	O
needs	O
to	O
determine	O
the	O
result	O
of	O
another	O
comparison	O
,	O
and	O
in	O
which	O
the	O
number	O
of	O
unresolved	O
comparisons	O
is	O
halved	O
by	O
finding	O
the	O
median	O
comparison	O
value	O
and	O
calling	O
the	O
decision	O
algorithm	O
with	O
that	O
value	O
.	O
</s>
<s>
As	O
they	O
show	O
,	O
the	O
resulting	O
randomized	O
parametric	B-Algorithm
search	I-Algorithm
algorithm	O
makes	O
only	O
a	O
logarithmic	O
number	O
of	O
calls	O
to	O
the	O
decision	O
algorithm	O
with	O
high	O
probability	O
,	O
matching	O
Cole	O
's	O
theoretical	O
analysis	O
.	O
</s>
<s>
An	O
extended	O
version	O
of	O
their	O
paper	O
includes	O
experimental	O
results	O
from	O
an	O
implementation	O
of	O
the	O
algorithm	O
,	O
which	O
show	O
that	O
the	O
total	O
running	O
time	O
of	O
this	O
method	O
on	O
several	O
natural	O
geometric	O
optimization	O
problems	O
is	O
similar	O
to	O
that	O
of	O
the	O
best	O
synchronized	O
parametric	B-Algorithm
search	I-Algorithm
implementation	O
(	O
the	O
quicksort-based	O
method	O
of	O
van	O
Oostrum	O
and	O
Veltkamp	O
)	O
:	O
a	O
little	O
faster	O
on	O
some	O
problems	O
and	O
a	O
little	O
slower	O
on	O
some	O
others	O
.	O
</s>
<s>
Instead	O
,	O
when	O
applicable	O
,	O
parametric	B-Algorithm
search	I-Algorithm
finds	O
strongly	O
polynomial	O
algorithms	O
,	O
whose	O
running	O
time	O
is	O
a	O
function	O
only	O
of	O
the	O
input	O
size	O
and	O
is	O
independent	O
of	O
numerical	O
precision	O
.	O
</s>
<s>
However	O
,	O
parametric	B-Algorithm
search	I-Algorithm
leads	O
to	O
an	O
increase	O
in	O
time	O
complexity	O
(	O
compared	O
to	O
the	O
decision	O
algorithm	O
)	O
that	O
may	O
be	O
larger	O
than	O
logarithmic	O
.	O
</s>
<s>
Because	O
they	O
are	O
strongly	O
rather	O
than	O
weakly	O
polynomial	O
,	O
algorithms	O
based	O
on	O
parametric	B-Algorithm
search	I-Algorithm
are	O
more	O
satisfactory	O
from	O
a	O
theoretical	O
point	O
of	O
view	O
.	O
</s>
<s>
In	O
practice	O
,	O
binary	O
search	O
is	O
fast	O
and	O
often	O
much	O
simpler	O
to	O
implement	O
,	O
so	O
algorithm	B-Algorithm
engineering	I-Algorithm
efforts	O
are	O
needed	O
to	O
make	O
parametric	B-Algorithm
search	I-Algorithm
practical	O
.	O
</s>
<s>
Nevertheless	O
,	O
write	O
that	O
"	O
while	O
a	O
simple	O
binary-search	O
approach	O
is	O
often	O
advocated	O
as	O
a	O
practical	O
replacement	O
for	O
parametric	B-Algorithm
search	I-Algorithm
,	O
it	O
is	O
outperformed	O
by	O
our	O
[	O
parametric	O
search ]	O
algorithm	O
"	O
in	O
the	O
experimental	O
comparisons	O
that	O
they	O
performed	O
.	O
</s>
<s>
Parametric	B-Algorithm
search	I-Algorithm
has	O
been	O
applied	O
in	O
the	O
development	O
of	O
efficient	O
algorithms	O
for	O
optimization	O
problems	O
,	O
particularly	O
in	O
computational	O
geometry	O
.	O
</s>
<s>
In	O
three	O
dimensions	O
,	O
parametric	B-Algorithm
search	I-Algorithm
can	O
be	O
used	O
to	O
find	O
centerpoints	O
in	O
time	O
.	O
</s>
<s>
Parametric	B-Algorithm
search	I-Algorithm
can	O
be	O
used	O
as	O
the	O
basis	O
for	O
an	O
time	O
algorithm	O
for	O
the	O
Theil	O
–	O
Sen	O
estimator	O
,	O
a	O
method	O
in	O
robust	O
statistics	O
for	O
fitting	O
a	O
line	O
to	O
a	O
set	O
of	O
points	O
that	O
is	O
much	O
less	O
sensitive	O
to	O
outliers	O
than	O
simple	B-General_Concept
linear	I-General_Concept
regression	I-General_Concept
.	O
</s>
<s>
In	O
computer	O
graphics	O
,	O
the	O
ray	B-Algorithm
shooting	I-Algorithm
problem	O
(	O
determining	O
the	O
first	O
object	O
in	O
a	O
scene	O
that	O
is	O
intersected	O
by	O
a	O
given	O
line	O
of	O
sight	O
or	O
light	O
beam	O
)	O
can	O
be	O
solved	O
by	O
combining	O
parametric	B-Algorithm
search	I-Algorithm
with	O
a	O
data	O
structure	O
for	O
a	O
simpler	O
problem	O
,	O
testing	O
whether	O
any	O
of	O
a	O
given	O
set	O
of	O
obstacles	O
occludes	O
a	O
given	O
ray	O
.	O
</s>
<s>
The	O
biggest	O
stick	O
problem	O
involves	O
finding	O
the	O
longest	O
line	O
segment	O
that	O
lies	O
entirely	O
within	O
a	O
given	O
polygon	B-General_Concept
.	O
</s>
<s>
It	O
can	O
be	O
solved	O
in	O
time	O
,	O
for	O
-sided	O
polygons	B-General_Concept
and	O
any	O
,	O
using	O
an	O
algorithm	O
based	O
on	O
parametric	B-Algorithm
search	I-Algorithm
.	O
</s>
<s>
Again	O
,	O
this	O
problem	O
can	O
be	O
solved	O
in	O
time	O
,	O
for	O
-sided	O
polygons	B-General_Concept
and	O
any	O
,	O
using	O
an	O
algorithm	O
based	O
on	O
parametric	B-Algorithm
search	I-Algorithm
.	O
</s>
<s>
The	O
Hausdorff	O
distance	O
between	O
translates	B-Algorithm
of	O
two	O
polygons	B-General_Concept
,	O
with	O
the	O
translation	O
chosen	O
to	O
minimize	O
the	O
distance	O
,	O
can	O
be	O
found	O
using	O
parametric	B-Algorithm
search	I-Algorithm
in	O
time	O
,	O
where	O
and	O
are	O
the	O
numbers	O
of	O
sides	O
of	O
the	O
polygons	B-General_Concept
.	O
</s>
<s>
The	O
Fréchet	O
distance	O
between	O
two	O
polygonal	O
chains	O
can	O
be	O
computed	O
using	O
parametric	B-Algorithm
search	I-Algorithm
in	O
time	O
,	O
where	O
and	O
are	O
the	O
numbers	O
of	O
segments	O
of	O
the	O
curves	O
.	O
</s>
<s>
Parametric	B-Algorithm
search	I-Algorithm
can	O
also	O
be	O
used	O
for	O
other	O
similar	O
problems	O
of	O
finding	O
the	O
time	O
at	O
which	O
some	O
measure	O
of	O
a	O
set	O
of	O
moving	O
points	O
obtains	O
its	O
minimum	O
value	O
,	O
for	O
measures	O
including	O
the	O
size	O
of	O
the	O
minimum	O
enclosing	O
ball	O
or	O
the	O
weight	O
of	O
the	O
maximum	O
spanning	O
tree	O
.	O
</s>
