<s>
In	O
computational	O
geometry	O
,	O
the	O
farthest-first	B-Algorithm
traversal	I-Algorithm
of	O
a	O
compact	O
metric	O
space	O
is	O
a	O
sequence	O
of	O
points	O
in	O
the	O
space	O
,	O
where	O
the	O
first	O
point	O
is	O
selected	O
arbitrarily	O
and	O
each	O
successive	O
point	O
is	O
as	O
far	O
as	O
possible	O
from	O
the	O
set	O
of	O
previously-selected	O
points	O
.	O
</s>
<s>
For	O
a	O
finite	O
metric	O
space	O
or	O
finite	O
set	O
of	O
geometric	O
points	O
,	O
the	O
resulting	O
sequence	O
forms	O
a	O
permutation	B-Algorithm
of	O
the	O
points	O
,	O
also	O
known	O
as	O
the	O
greedy	B-Algorithm
permutation	B-Algorithm
.	O
</s>
<s>
Every	O
prefix	O
of	O
a	O
farthest-first	B-Algorithm
traversal	I-Algorithm
provides	O
a	O
set	O
of	O
points	O
that	O
is	O
widely	O
spaced	O
and	O
close	O
to	O
all	O
remaining	O
points	O
.	O
</s>
<s>
In	O
part	O
because	O
of	O
these	O
properties	O
,	O
farthest-point	O
traversals	O
have	O
many	O
applications	O
,	O
including	O
the	O
approximation	O
of	O
the	O
traveling	B-Algorithm
salesman	I-Algorithm
problem	I-Algorithm
and	O
the	O
metric	O
-center	O
problem	O
.	O
</s>
<s>
A	O
farthest-first	B-Algorithm
traversal	I-Algorithm
is	O
a	O
sequence	O
of	O
points	O
in	O
a	O
compact	O
metric	O
space	O
,	O
with	O
each	O
point	O
appearing	O
at	O
most	O
once	O
.	O
</s>
<s>
If	O
the	O
space	O
is	O
finite	O
,	O
each	O
point	O
appears	O
exactly	O
once	O
,	O
and	O
the	O
traversal	O
is	O
a	O
permutation	B-Algorithm
of	O
all	O
of	O
the	O
points	O
in	O
the	O
space	O
.	O
</s>
<s>
A	O
given	O
space	O
may	O
have	O
many	O
different	O
farthest-first	B-Algorithm
traversals	I-Algorithm
,	O
depending	O
both	O
on	O
the	O
choice	O
of	O
the	O
first	O
point	O
in	O
the	O
sequence	O
(	O
which	O
may	O
be	O
any	O
point	O
in	O
the	O
space	O
)	O
and	O
on	O
ties	O
for	O
the	O
maximum	O
distance	O
among	O
later	O
choices	O
.	O
</s>
<s>
Fix	O
a	O
number	O
,	O
and	O
consider	O
the	O
prefix	O
formed	O
by	O
the	O
first	O
points	O
of	O
the	O
farthest-first	B-Algorithm
traversal	I-Algorithm
of	O
any	O
metric	O
space	O
.	O
</s>
<s>
Conversely	O
any	O
sequence	O
having	O
these	O
properties	O
,	O
for	O
all	O
choices	O
of	O
,	O
must	O
be	O
a	O
farthest-first	B-Algorithm
traversal	I-Algorithm
.	O
</s>
<s>
These	O
are	O
the	O
two	O
defining	O
properties	O
of	O
a	O
Delone	O
set	O
,	O
so	O
each	O
prefix	O
of	O
the	O
farthest-first	B-Algorithm
traversal	I-Algorithm
forms	O
a	O
Delone	O
set	O
.	O
</s>
<s>
used	O
the	O
farthest-first	B-Algorithm
traversal	I-Algorithm
to	O
define	O
the	O
farthest-insertion	O
heuristic	B-Algorithm
for	O
the	O
travelling	B-Algorithm
salesman	I-Algorithm
problem	I-Algorithm
.	O
</s>
<s>
This	O
heuristic	B-Algorithm
finds	O
approximate	O
solutions	O
to	O
the	O
travelling	B-Algorithm
salesman	I-Algorithm
problem	I-Algorithm
by	O
building	O
up	O
a	O
tour	O
on	O
a	O
subset	O
of	O
points	O
,	O
adding	O
one	O
point	O
at	O
a	O
time	O
to	O
the	O
tour	O
in	O
the	O
ordering	O
given	O
by	O
a	O
farthest-first	B-Algorithm
traversal	I-Algorithm
.	O
</s>
<s>
prove	O
only	O
a	O
logarithmic	O
approximation	B-Algorithm
ratio	I-Algorithm
for	O
this	O
method	O
,	O
they	O
show	O
that	O
in	O
practice	O
it	O
often	O
works	O
better	O
than	O
other	O
insertion	O
methods	O
with	O
better	O
provable	O
approximation	B-Algorithm
ratios	I-Algorithm
.	O
</s>
<s>
Later	O
,	O
the	O
same	O
sequence	O
of	O
points	O
was	O
popularized	O
by	O
,	O
who	O
used	O
it	O
as	O
part	O
of	O
greedy	B-Algorithm
approximation	B-Algorithm
algorithms	I-Algorithm
for	O
two	O
problems	O
in	O
clustering	O
,	O
in	O
which	O
the	O
goal	O
is	O
to	O
partition	O
a	O
set	O
of	O
points	O
into	O
clusters	O
.	O
</s>
<s>
For	O
both	O
clustering	O
problems	O
,	O
Gonzalez	O
chooses	O
a	O
set	O
of	O
cluster	O
centers	O
by	O
selecting	O
the	O
first	O
points	O
of	O
a	O
farthest-first	B-Algorithm
traversal	I-Algorithm
,	O
and	O
then	O
creates	O
clusters	O
by	O
assigning	O
each	O
input	O
point	O
to	O
the	O
nearest	O
cluster	O
center	O
.	O
</s>
<s>
Thus	O
,	O
Gonzalez	O
's	O
heuristic	B-Algorithm
gives	O
an	O
approximation	B-Algorithm
ratio	I-Algorithm
of2	O
for	O
both	O
clustering	O
problems	O
.	O
</s>
<s>
Gonzalez	O
's	O
heuristic	B-Algorithm
was	O
independently	O
rediscovered	O
for	O
the	O
metric	O
-center	O
problem	O
by	O
,	O
who	O
applied	O
it	O
more	O
generally	O
to	O
weighted	O
-center	O
problems	O
.	O
</s>
<s>
Another	O
paper	O
on	O
the	O
-center	O
problem	O
from	O
the	O
same	O
time	O
,	O
,	O
achieves	O
the	O
same	O
approximation	B-Algorithm
ratio	I-Algorithm
of2	O
,	O
but	O
its	O
techniques	O
are	O
different	O
.	O
</s>
<s>
Nevertheless	O
,	O
Gonzalez	O
's	O
heuristic	B-Algorithm
,	O
and	O
the	O
name	O
"	O
farthest-first	B-Algorithm
traversal	I-Algorithm
"	O
,	O
are	O
often	O
incorrectly	O
attributed	O
to	O
Hochbaum	O
and	O
Shmoys	O
.	O
</s>
<s>
For	O
both	O
the	O
min-max	O
diameter	O
clustering	O
problem	O
and	O
the	O
metric	O
-center	O
problem	O
,	O
these	O
approximations	O
are	O
optimal	O
:	O
the	O
existence	O
of	O
a	O
polynomial-time	O
heuristic	B-Algorithm
with	O
any	O
constant	O
approximation	B-Algorithm
ratio	I-Algorithm
less	O
than2	O
would	O
imply	O
that	O
P	O
=	O
NP	O
.	O
</s>
<s>
As	O
well	O
as	O
for	O
clustering	O
,	O
the	O
farthest-first	B-Algorithm
traversal	I-Algorithm
can	O
also	O
be	O
used	O
in	O
another	O
type	O
of	O
facility	O
location	O
problem	O
,	O
the	O
max-min	O
facility	O
dispersion	O
problem	O
,	O
in	O
which	O
the	O
goal	O
is	O
to	O
choose	O
the	O
locations	O
of	O
different	O
facilities	O
so	O
that	O
they	O
are	O
as	O
far	O
apart	O
from	O
each	O
other	O
as	O
possible	O
.	O
</s>
<s>
Again	O
,	O
this	O
can	O
be	O
approximated	O
by	O
choosing	O
the	O
first	O
points	O
of	O
a	O
farthest-first	B-Algorithm
traversal	I-Algorithm
.	O
</s>
<s>
Therefore	O
,	O
the	O
heuristic	B-Algorithm
solution	O
given	O
by	O
the	O
farthest-first	B-Algorithm
traversal	I-Algorithm
is	O
within	O
a	O
factor	O
of	O
two	O
of	O
optimal	O
.	O
</s>
<s>
Other	O
applications	O
of	O
the	O
farthest-first	B-Algorithm
traversal	I-Algorithm
include	O
color	B-Algorithm
quantization	I-Algorithm
(	O
clustering	O
the	O
colors	O
in	O
an	O
image	O
to	O
a	O
smaller	O
set	O
of	O
representative	O
colors	O
)	O
,	O
progressive	B-Device
scanning	I-Device
of	O
images	O
(	O
choosing	O
an	O
order	O
to	O
display	O
the	O
pixels	B-Algorithm
of	O
an	O
image	O
so	O
that	O
prefixes	O
of	O
the	O
ordering	O
produce	O
good	O
lower-resolution	O
versions	O
of	O
the	O
whole	O
image	O
rather	O
than	O
filling	O
in	O
the	O
image	O
from	O
top	O
to	O
bottom	O
)	O
,	O
</s>
<s>
point	O
selection	O
in	O
the	O
probabilistic	O
roadmap	O
method	O
for	O
motion	O
planning	O
,	O
simplification	O
of	O
point	B-Algorithm
clouds	I-Algorithm
,	O
generating	O
masks	O
for	O
halftone	O
images	O
,	O
hierarchical	B-Algorithm
clustering	I-Algorithm
,	O
finding	O
the	O
similarities	O
between	O
polygon	B-Algorithm
meshes	I-Algorithm
of	O
similar	O
surfaces	O
,	O
choosing	O
diverse	O
and	O
high-value	O
observation	O
targets	O
for	O
underwater	O
robot	O
exploration	O
,	O
fault	O
detection	O
in	O
sensor	B-Architecture
networks	I-Architecture
,	O
modeling	O
phylogenetic	O
diversity	O
,	O
matching	O
vehicles	O
in	O
a	O
heterogenous	O
fleet	O
to	O
customer	O
delivery	O
requests	O
,	O
uniform	O
distribution	O
of	O
geodetic	O
observatories	O
on	O
the	O
Earth	O
's	O
surface	O
or	O
of	O
other	O
types	O
of	O
sensor	B-Architecture
network	I-Architecture
,	O
generation	O
of	O
virtual	O
point	O
lights	O
in	O
the	O
instant	O
radiosity	O
computer	O
graphics	O
rendering	O
method	O
,	O
and	O
geometric	O
range	B-Data_Structure
searching	I-Data_Structure
data	B-General_Concept
structures	I-General_Concept
.	O
</s>
<s>
The	O
farthest-first	B-Algorithm
traversal	I-Algorithm
of	O
a	O
finite	O
point	O
set	O
may	O
be	O
computed	O
by	O
a	O
greedy	B-Algorithm
algorithm	I-Algorithm
that	O
maintains	O
the	O
distance	O
of	O
each	O
point	O
from	O
the	O
previously	O
selected	O
points	O
,	O
performing	O
the	O
following	O
steps	O
:	O
</s>
<s>
A	O
faster	O
approximation	B-Algorithm
algorithm	I-Algorithm
,	O
given	O
by	O
,	O
applies	O
to	O
any	O
subset	O
of	O
points	O
in	O
a	O
metric	O
space	O
with	O
bounded	O
doubling	O
dimension	O
,	O
a	O
class	O
of	O
spaces	O
that	O
include	O
the	O
Euclidean	O
spaces	O
of	O
bounded	O
dimension	O
.	O
</s>
<s>
Instead	O
,	O
a	O
different	O
approximation	O
method	O
based	O
on	O
the	O
Johnson	O
–	O
Lindenstrauss	O
lemma	O
and	O
locality-sensitive	B-Algorithm
hashing	I-Algorithm
has	O
running	O
time	O
For	O
metrics	O
defined	O
by	O
shortest	O
paths	O
on	O
weighted	O
undirected	O
graphs	O
,	O
a	O
randomized	O
incremental	O
construction	O
based	O
on	O
Dijkstra	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
achieves	O
time	O
,	O
where	O
and	O
are	O
the	O
numbers	O
of	O
vertices	O
and	O
edges	O
of	O
the	O
input	O
graph	O
,	O
respectively	O
.	O
</s>
<s>
Instead	O
,	O
each	O
new	O
point	O
should	O
be	O
selected	O
as	O
the	O
center	O
of	O
the	O
largest	B-Algorithm
empty	I-Algorithm
circle	I-Algorithm
defined	O
by	O
the	O
previously-selected	O
point	O
set	O
.	O
</s>
<s>
This	O
center	O
will	O
always	O
lie	O
on	O
a	O
vertex	O
of	O
the	O
Voronoi	B-Architecture
diagram	I-Architecture
of	O
the	O
already	O
selected	O
points	O
,	O
or	O
at	O
a	O
point	O
where	O
an	O
edge	O
of	O
the	O
Voronoi	B-Architecture
diagram	I-Architecture
crosses	O
the	O
domain	O
boundary	O
.	O
</s>
<s>
In	O
this	O
formulation	O
the	O
method	O
for	O
constructing	O
farthest-first	B-Algorithm
traversals	I-Algorithm
has	O
also	O
been	O
called	O
incremental	O
Voronoi	O
insertion	O
.	O
</s>
<s>
It	O
is	O
similar	O
to	O
Delaunay	B-Algorithm
refinement	I-Algorithm
for	O
finite	B-Application
element	I-Application
mesh	B-Architecture
generation	I-Architecture
,	O
but	O
differs	O
in	O
the	O
choice	O
of	O
which	O
Voronoi	O
vertex	O
to	O
insert	O
at	O
each	O
step	O
.	O
</s>
