<s>
In	O
mathematics	O
,	O
given	O
a	O
non-empty	O
set	O
of	O
objects	O
of	O
finite	O
extension	O
in	O
-dimensional	O
space	O
,	O
for	O
example	O
a	O
set	O
of	O
points	O
,	O
a	O
bounding	B-Algorithm
sphere	I-Algorithm
,	O
enclosing	O
sphere	O
or	O
enclosing	O
ball	O
for	O
that	O
set	O
is	O
an	O
-dimensional	O
solid	O
sphere	O
containing	O
all	O
of	O
these	O
objects	O
.	O
</s>
<s>
Used	O
in	O
computer	O
graphics	O
and	O
computational	O
geometry	O
,	O
a	O
bounding	B-Algorithm
sphere	I-Algorithm
is	O
a	O
special	O
type	O
of	O
bounding	B-Algorithm
volume	I-Algorithm
.	O
</s>
<s>
There	O
are	O
several	O
fast	O
and	O
simple	O
bounding	B-Algorithm
sphere	I-Algorithm
construction	O
algorithms	O
with	O
a	O
high	O
practical	O
value	O
in	O
real-time	O
computer	O
graphics	O
applications	O
.	O
</s>
<s>
In	O
statistics	O
and	O
operations	O
research	O
,	O
the	O
objects	O
are	O
typically	O
points	O
,	O
and	O
generally	O
the	O
sphere	O
of	O
interest	O
is	O
the	O
minimal	B-Algorithm
bounding	I-Algorithm
sphere	I-Algorithm
,	O
that	O
is	O
,	O
the	O
sphere	O
with	O
minimal	O
radius	O
among	O
all	O
bounding	B-Algorithm
spheres	I-Algorithm
.	O
</s>
<s>
The	O
problem	O
of	O
computing	O
the	O
center	O
of	O
a	O
minimal	B-Algorithm
bounding	I-Algorithm
sphere	I-Algorithm
is	O
also	O
known	O
as	O
the	O
"	O
unweighted	O
Euclidean	O
1-center	B-Algorithm
problem	I-Algorithm
"	O
.	O
</s>
<s>
Such	O
spheres	O
are	O
useful	O
in	O
clustering	B-Algorithm
,	O
where	O
groups	O
of	O
similar	O
data	B-Application
points	I-Application
are	O
classified	O
together	O
.	O
</s>
<s>
In	O
statistical	O
analysis	O
the	O
scattering	O
of	O
data	B-Application
points	I-Application
within	O
a	O
sphere	O
may	O
be	O
attributed	O
to	O
measurement	O
error	O
or	O
natural	O
(	O
usually	O
thermal	O
)	O
processes	O
,	O
in	O
which	O
case	O
the	O
cluster	O
represents	O
a	O
perturbation	O
of	O
an	O
ideal	O
point	O
.	O
</s>
<s>
In	O
operations	O
research	O
the	O
clustering	B-Algorithm
of	O
values	O
to	O
an	O
ideal	O
point	O
may	O
also	O
be	O
used	O
to	O
reduce	O
the	O
number	O
of	O
inputs	O
in	O
order	O
to	O
obtain	O
approximate	O
values	O
for	O
NP-hard	O
problems	O
in	O
a	O
reasonable	O
time	O
.	O
</s>
<s>
The	O
point	O
chosen	O
is	O
not	O
usually	O
the	O
center	O
of	O
the	O
sphere	O
,	O
as	O
this	O
can	O
be	O
biased	O
by	O
outliers	O
,	O
but	O
instead	O
some	O
form	O
of	O
average	O
location	O
such	O
as	O
a	O
least	B-Algorithm
squares	I-Algorithm
point	O
is	O
computed	O
to	O
represent	O
the	O
cluster	O
.	O
</s>
<s>
There	O
are	O
exact	O
and	O
approximate	O
algorithms	O
for	O
the	O
solving	O
bounding	B-Algorithm
sphere	I-Algorithm
problem	O
.	O
</s>
<s>
Nimrod	O
Megiddo	O
studied	O
the	O
1-center	B-Algorithm
problem	I-Algorithm
extensively	O
and	O
published	O
on	O
it	O
at	O
least	O
five	O
times	O
in	O
the	O
1980s	O
.	O
</s>
<s>
In	O
1983	O
,	O
he	O
proposed	O
a	O
"	O
prune	B-Algorithm
and	I-Algorithm
search	I-Algorithm
"	O
algorithm	O
which	O
finds	O
the	O
optimum	O
bounding	B-Algorithm
sphere	I-Algorithm
and	O
runs	O
in	O
linear	O
time	O
if	O
the	O
dimension	O
is	O
fixed	O
as	O
a	O
constant	O
.	O
</s>
<s>
In	O
1991	O
,	O
Emo	O
Welzl	O
proposed	O
a	O
much	O
simpler	O
randomized	B-General_Concept
algorithm	I-General_Concept
,	O
generalizing	O
a	O
randomized	O
linear	B-Algorithm
programming	I-Algorithm
algorithm	O
by	O
Raimund	O
Seidel	O
.	O
</s>
<s>
The	O
open-source	O
Computational	B-Language
Geometry	I-Language
Algorithms	I-Language
Library	I-Language
(	O
CGAL	B-Language
)	O
contains	O
an	O
implementation	O
of	O
Welzl	O
’s	O
algorithm	O
.	O
</s>
<s>
If	O
all	O
points	O
in	O
are	O
within	O
ball	O
,	O
then	O
we	O
get	O
a	O
bounding	B-Algorithm
sphere	I-Algorithm
.	O
</s>
<s>
presented	O
a	O
approximation	O
to	O
the	O
bounding	B-Algorithm
sphere	I-Algorithm
problem	O
,	O
where	O
a	O
approximation	O
means	O
that	O
the	O
constructed	O
sphere	O
has	O
radius	O
at	O
most	O
,	O
where	O
is	O
the	O
smallest	O
possible	O
radius	O
of	O
a	O
bounding	B-Algorithm
sphere	I-Algorithm
.	O
</s>
<s>
A	O
coreset	B-Algorithm
is	O
a	O
small	O
subset	O
,	O
that	O
a	O
expansion	O
of	O
the	O
solution	O
on	O
the	O
subset	O
is	O
a	O
bounding	B-Algorithm
sphere	I-Algorithm
of	O
the	O
whole	O
set	O
.	O
</s>
<s>
The	O
coreset	B-Algorithm
is	O
constructed	O
incrementally	O
by	O
adding	O
the	O
farthest	O
point	O
into	O
the	O
set	O
in	O
each	O
iteration	O
.	O
</s>
<s>
The	O
algorithm	O
is	O
purely	O
combinatorial	O
and	O
implements	O
a	O
pivoting	O
scheme	O
similar	O
to	O
the	O
simplex	B-Algorithm
method	I-Algorithm
for	O
linear	B-Algorithm
programming	I-Algorithm
,	O
used	O
earlier	O
in	O
some	O
heuristics	O
.	O
</s>
<s>
proposed	O
the	O
"	O
extremal	O
points	O
optimal	O
sphere	O
"	O
method	O
with	O
controllable	O
speed	O
to	O
accuracy	O
approximation	O
to	O
solve	O
the	O
bounding	B-Algorithm
sphere	I-Algorithm
problem	O
.	O
</s>
