<s>
In	O
the	O
theory	O
of	O
cluster	B-Algorithm
analysis	I-Algorithm
,	O
the	O
nearest-neighbor	B-Algorithm
chain	I-Algorithm
algorithm	I-Algorithm
is	O
an	O
algorithm	O
that	O
can	O
speed	O
up	O
several	O
methods	O
for	O
agglomerative	B-Algorithm
hierarchical	I-Algorithm
clustering	I-Algorithm
.	O
</s>
<s>
The	O
clustering	B-Algorithm
methods	O
that	O
the	O
nearest-neighbor	B-Algorithm
chain	I-Algorithm
algorithm	I-Algorithm
can	O
be	O
used	O
for	O
include	O
Ward	B-Algorithm
's	I-Algorithm
method	I-Algorithm
,	O
complete-linkage	B-Algorithm
clustering	I-Algorithm
,	O
and	O
single-linkage	B-Algorithm
clustering	I-Algorithm
;	O
these	O
all	O
work	O
by	O
repeatedly	O
merging	O
the	O
closest	O
two	O
clusters	O
but	O
use	O
different	O
definitions	O
of	O
the	O
distance	O
between	O
clusters	O
.	O
</s>
<s>
The	O
cluster	O
distances	O
for	O
which	O
the	O
nearest-neighbor	B-Algorithm
chain	I-Algorithm
algorithm	I-Algorithm
works	O
are	O
called	O
reducible	O
and	O
are	O
characterized	O
by	O
a	O
simple	O
inequality	O
among	O
certain	O
cluster	O
distances	O
.	O
</s>
<s>
In	O
order	O
to	O
save	O
work	O
by	O
re-using	O
as	O
much	O
as	O
possible	O
of	O
each	O
path	O
,	O
the	O
algorithm	O
uses	O
a	O
stack	B-Application
data	I-Application
structure	I-Application
to	O
keep	O
track	O
of	O
each	O
path	O
that	O
it	O
follows	O
.	O
</s>
<s>
By	O
following	O
paths	O
in	O
this	O
way	O
,	O
the	O
nearest-neighbor	B-Algorithm
chain	I-Algorithm
algorithm	I-Algorithm
merges	O
its	O
clusters	O
in	O
a	O
different	O
order	O
than	O
methods	O
that	O
always	O
find	O
and	O
merge	O
the	O
closest	B-Algorithm
pair	I-Algorithm
of	O
clusters	O
.	O
</s>
<s>
The	O
nearest-neighbor	B-Algorithm
chain	I-Algorithm
algorithm	I-Algorithm
constructs	O
a	O
clustering	B-Algorithm
in	O
time	O
proportional	O
to	O
the	O
square	O
of	O
the	O
number	O
of	O
points	O
to	O
be	O
clustered	O
.	O
</s>
<s>
The	O
algorithm	O
uses	O
an	O
amount	O
of	O
memory	O
proportional	O
to	O
the	O
number	O
of	O
points	O
,	O
when	O
it	O
is	O
used	O
for	O
clustering	B-Algorithm
methods	O
such	O
as	O
Ward	B-Algorithm
's	I-Algorithm
method	I-Algorithm
that	O
allow	O
constant-time	O
calculation	O
of	O
the	O
distance	O
between	O
clusters	O
.	O
</s>
<s>
However	O
,	O
for	O
some	O
other	O
clustering	B-Algorithm
methods	O
it	O
uses	O
a	O
larger	O
amount	O
of	O
memory	O
in	O
an	O
auxiliary	O
data	B-General_Concept
structure	I-General_Concept
with	O
which	O
it	O
keeps	O
track	O
of	O
the	O
distances	O
between	O
pairs	O
of	O
clusters	O
.	O
</s>
<s>
Many	O
problems	O
in	O
data	B-General_Concept
analysis	I-General_Concept
concern	O
clustering	B-Algorithm
,	O
grouping	O
data	O
items	O
into	O
clusters	O
of	O
closely	O
related	O
items	O
.	O
</s>
<s>
Hierarchical	B-Algorithm
clustering	I-Algorithm
is	O
a	O
version	O
of	O
cluster	B-Algorithm
analysis	I-Algorithm
in	O
which	O
the	O
clusters	O
form	O
a	O
hierarchy	O
or	O
tree-like	O
structure	O
rather	O
than	O
a	O
strict	O
partition	O
of	O
the	O
data	O
items	O
.	O
</s>
<s>
In	O
some	O
cases	O
,	O
this	O
type	O
of	O
clustering	B-Algorithm
may	O
be	O
performed	O
as	O
a	O
way	O
of	O
performing	O
cluster	B-Algorithm
analysis	I-Algorithm
at	O
multiple	O
different	O
scales	O
simultaneously	O
.	O
</s>
<s>
Both	O
of	O
these	O
kinds	O
of	O
analysis	O
can	O
be	O
seen	O
,	O
for	O
instance	O
,	O
in	O
the	O
application	O
of	O
hierarchical	B-Algorithm
clustering	I-Algorithm
to	O
biological	O
taxonomy	O
.	O
</s>
<s>
The	O
input	O
to	O
a	O
clustering	B-Algorithm
problem	O
consists	O
of	O
a	O
set	O
of	O
points	O
.	O
</s>
<s>
A	O
cluster	O
is	O
any	O
proper	O
subset	O
of	O
the	O
points	O
,	O
and	O
a	O
hierarchical	B-Algorithm
clustering	I-Algorithm
is	O
a	O
maximal	O
family	O
of	O
clusters	O
with	O
the	O
property	O
that	O
any	O
two	O
clusters	O
in	O
the	O
family	O
are	O
either	O
nested	O
or	O
disjoint	B-Algorithm
.	O
</s>
<s>
Alternatively	O
,	O
a	O
hierarchical	B-Algorithm
clustering	I-Algorithm
may	O
be	O
represented	O
as	O
a	O
binary	O
tree	O
with	O
the	O
points	O
at	O
its	O
leaves	O
;	O
the	O
clusters	O
of	O
the	O
clustering	B-Algorithm
are	O
the	O
sets	O
of	O
points	O
in	O
subtrees	O
descending	O
from	O
each	O
node	O
of	O
the	O
tree	O
.	O
</s>
<s>
In	O
agglomerative	B-Algorithm
clustering	I-Algorithm
methods	O
,	O
the	O
input	O
also	O
includes	O
a	O
distance	O
function	O
defined	O
on	O
the	O
points	O
,	O
or	O
a	O
numerical	O
measure	O
of	O
their	O
dissimilarity	O
.	O
</s>
<s>
Different	O
clustering	B-Algorithm
methods	O
perform	O
this	O
extension	O
in	O
different	O
ways	O
.	O
</s>
<s>
For	O
instance	O
,	O
in	O
the	O
single-linkage	B-Algorithm
clustering	I-Algorithm
method	O
,	O
the	O
distance	O
between	O
two	O
clusters	O
is	O
defined	O
to	O
be	O
the	O
minimum	O
distance	O
between	O
any	O
two	O
points	O
from	O
each	O
cluster	O
.	O
</s>
<s>
Given	O
this	O
distance	O
between	O
clusters	O
,	O
a	O
hierarchical	B-Algorithm
clustering	I-Algorithm
may	O
be	O
defined	O
by	O
a	O
greedy	B-Algorithm
algorithm	I-Algorithm
that	O
initially	O
places	O
each	O
point	O
in	O
its	O
own	O
single-point	O
cluster	O
and	O
then	O
repeatedly	O
forms	O
a	O
new	O
cluster	O
by	O
merging	O
the	O
closest	B-Algorithm
pair	I-Algorithm
of	O
clusters	O
.	O
</s>
<s>
The	O
bottleneck	O
of	O
this	O
greedy	B-Algorithm
algorithm	I-Algorithm
is	O
the	O
subproblem	O
of	O
finding	O
which	O
two	O
clusters	O
to	O
merge	O
in	O
each	O
step	O
.	O
</s>
<s>
Known	O
methods	O
for	O
repeatedly	O
finding	O
the	O
closest	B-Algorithm
pair	I-Algorithm
of	O
clusters	O
in	O
a	O
dynamic	O
set	O
of	O
clusters	O
either	O
require	O
superlinear	O
space	O
to	O
maintain	O
a	O
data	B-General_Concept
structure	I-General_Concept
that	O
can	O
find	O
closest	B-Algorithm
pairs	I-Algorithm
quickly	O
,	O
or	O
they	O
take	O
greater	O
than	O
linear	O
time	O
to	O
find	O
each	O
closest	B-Algorithm
pair	I-Algorithm
.	O
</s>
<s>
The	O
nearest-neighbor	B-Algorithm
chain	I-Algorithm
algorithm	I-Algorithm
uses	O
a	O
smaller	O
amount	O
of	O
time	O
and	O
space	O
than	O
the	O
greedy	B-Algorithm
algorithm	I-Algorithm
by	O
merging	O
pairs	O
of	O
clusters	O
in	O
a	O
different	O
order	O
.	O
</s>
<s>
In	O
this	O
way	O
,	O
it	O
avoids	O
the	O
problem	O
of	O
repeatedly	O
finding	O
closest	B-Algorithm
pairs	I-Algorithm
.	O
</s>
<s>
Nevertheless	O
,	O
for	O
many	O
types	O
of	O
clustering	B-Algorithm
problem	O
,	O
it	O
can	O
be	O
guaranteed	O
to	O
come	O
up	O
with	O
the	O
same	O
hierarchical	B-Algorithm
clustering	I-Algorithm
as	O
the	O
greedy	B-Algorithm
algorithm	I-Algorithm
despite	O
the	O
different	O
merge	O
order	O
.	O
</s>
<s>
Let	O
be	O
a	O
stack	B-Application
data	I-Application
structure	I-Application
,	O
initially	O
empty	O
,	O
the	O
elements	O
of	O
which	O
will	O
be	O
active	O
clusters	O
.	O
</s>
<s>
This	O
rule	O
prevents	O
certain	O
kinds	O
of	O
inconsistent	O
behavior	O
in	O
the	O
algorithm	O
;	O
for	O
instance	O
,	O
without	O
such	O
a	O
rule	O
,	O
the	O
neighboring	O
cluster	O
might	O
occur	O
earlier	O
in	O
the	O
stack	B-Application
than	O
as	O
the	O
predecessor	O
of	O
.	O
</s>
<s>
Each	O
iteration	O
of	O
the	O
loop	O
performs	O
a	O
single	O
search	O
for	O
the	O
nearest	O
neighbor	O
of	O
a	O
cluster	O
,	O
and	O
either	O
adds	O
one	O
cluster	O
to	O
the	O
stack	B-Application
or	O
removes	O
two	O
clusters	O
from	O
it	O
.	O
</s>
<s>
Every	O
cluster	O
is	O
only	O
ever	O
added	O
once	O
to	O
the	O
stack	B-Application
,	O
because	O
when	O
it	O
is	O
removed	O
again	O
it	O
is	O
immediately	O
made	O
inactive	O
and	O
merged	O
.	O
</s>
<s>
There	O
are	O
a	O
total	O
of	O
clusters	O
that	O
ever	O
get	O
added	O
to	O
the	O
stack	B-Application
:	O
single-point	O
clusters	O
in	O
the	O
initial	O
set	O
,	O
and	O
internal	O
nodes	O
other	O
than	O
the	O
root	O
in	O
the	O
binary	O
tree	O
representing	O
the	O
clustering	B-Algorithm
.	O
</s>
<s>
Since	O
the	O
only	O
data	B-General_Concept
structure	I-General_Concept
is	O
the	O
set	O
of	O
active	O
clusters	O
and	O
the	O
stack	B-Application
containing	O
a	O
subset	O
of	O
the	O
active	O
clusters	O
,	O
the	O
space	O
required	O
is	O
linear	O
in	O
the	O
number	O
of	O
input	O
points	O
.	O
</s>
<s>
For	O
the	O
algorithm	O
to	O
be	O
correct	O
,	O
it	O
must	O
be	O
the	O
case	O
that	O
popping	O
and	O
merging	O
the	O
top	O
two	O
clusters	O
from	O
the	O
algorithm	O
's	O
stack	B-Application
preserves	O
the	O
property	O
that	O
the	O
remaining	O
clusters	O
on	O
the	O
stack	B-Application
form	O
a	O
chain	O
of	O
nearest	O
neighbors	O
.	O
</s>
<s>
will	O
in	O
general	O
perform	O
its	O
merges	O
in	O
a	O
different	O
order	O
than	O
the	O
nearest-neighbor	B-Algorithm
chain	I-Algorithm
algorithm	I-Algorithm
.	O
</s>
<s>
This	O
property	O
was	O
identified	O
by	O
in	O
connection	O
with	O
an	O
earlier	O
clustering	B-Algorithm
method	O
that	O
used	O
mutual	O
nearest	O
neighbor	O
pairs	O
but	O
not	O
chains	O
of	O
nearest	O
neighbors	O
.	O
</s>
<s>
A	O
distance	O
function	O
on	O
clusters	O
is	O
defined	O
to	O
be	O
reducible	O
if	O
,	O
for	O
every	O
three	O
clusters	O
,	O
and	O
in	O
the	O
greedy	O
hierarchical	B-Algorithm
clustering	I-Algorithm
such	O
that	O
and	O
are	O
mutual	O
nearest	O
neighbors	O
,	O
the	O
following	O
inequality	O
holds	O
:	O
</s>
<s>
First	O
,	O
it	O
can	O
be	O
shown	O
using	O
this	O
property	O
that	O
,	O
at	O
each	O
step	O
of	O
the	O
algorithm	O
,	O
the	O
clusters	O
on	O
the	O
stack	B-Application
form	O
a	O
valid	O
chain	O
of	O
nearest	O
neighbors	O
,	O
because	O
whenever	O
a	O
nearest	O
neighbor	O
becomes	O
invalidated	O
it	O
is	O
immediately	O
removed	O
from	O
the	O
stack	B-Application
.	O
</s>
<s>
Second	O
,	O
and	O
even	O
more	O
importantly	O
,	O
it	O
follows	O
from	O
this	O
property	O
that	O
,	O
if	O
two	O
clusters	O
and	O
both	O
belong	O
to	O
the	O
greedy	O
hierarchical	B-Algorithm
clustering	I-Algorithm
,	O
and	O
are	O
mutual	O
nearest	O
neighbors	O
at	O
any	O
point	O
in	O
time	O
,	O
then	O
they	O
will	O
be	O
merged	O
by	O
the	O
greedy	O
clustering	B-Algorithm
,	O
for	O
they	O
must	O
remain	O
mutual	O
nearest	O
neighbors	O
until	O
they	O
are	O
merged	O
.	O
</s>
<s>
It	O
follows	O
that	O
each	O
mutual	O
nearest	O
neighbor	O
pair	O
found	O
by	O
the	O
nearest	O
neighbor	O
chain	O
algorithm	O
is	O
also	O
a	O
pair	O
of	O
clusters	O
found	O
by	O
the	O
greedy	B-Algorithm
algorithm	I-Algorithm
,	O
and	O
therefore	O
that	O
the	O
nearest	O
neighbor	O
chain	O
algorithm	O
computes	O
exactly	O
the	O
same	O
clustering	B-Algorithm
(	O
although	O
in	O
a	O
different	O
order	O
)	O
as	O
the	O
greedy	B-Algorithm
algorithm	I-Algorithm
.	O
</s>
<s>
Ward	B-Algorithm
's	I-Algorithm
method	I-Algorithm
is	O
an	O
agglomerative	B-Algorithm
clustering	I-Algorithm
method	O
in	O
which	O
the	O
dissimilarity	O
between	O
two	O
clusters	O
and	O
is	O
measured	O
by	O
the	O
amount	O
by	O
which	O
merging	O
the	O
two	O
clusters	O
into	O
a	O
single	O
larger	O
cluster	O
would	O
increase	O
the	O
average	O
squared	O
distance	O
of	O
a	O
point	O
to	O
its	O
cluster	O
centroid	O
.	O
</s>
<s>
Although	O
highly	O
sensitive	O
to	O
outliers	O
,	O
Ward	B-Algorithm
's	I-Algorithm
method	I-Algorithm
is	O
the	O
most	O
popular	O
variation	O
of	O
agglomerative	B-Algorithm
clustering	I-Algorithm
both	O
because	O
of	O
the	O
round	O
shape	O
of	O
the	O
clusters	O
it	O
typically	O
forms	O
and	O
because	O
of	O
its	O
principled	O
definition	O
as	O
the	O
clustering	B-Algorithm
that	O
at	O
each	O
step	O
has	O
the	O
smallest	O
variance	O
within	O
its	O
clusters	O
.	O
</s>
<s>
Because	O
Ward	O
's	O
distance	O
is	O
reducible	O
,	O
the	O
nearest-neighbor	B-Algorithm
chain	I-Algorithm
algorithm	I-Algorithm
using	O
Ward	O
's	O
distance	O
calculates	O
exactly	O
the	O
same	O
clustering	B-Algorithm
as	O
the	O
standard	O
greedy	B-Algorithm
algorithm	I-Algorithm
.	O
</s>
<s>
Complete-linkage	B-Algorithm
or	O
furthest-neighbor	O
clustering	B-Algorithm
is	O
a	O
form	O
of	O
agglomerative	B-Algorithm
clustering	I-Algorithm
that	O
defines	O
the	O
dissimilarity	O
between	O
clusters	O
to	O
be	O
the	O
maximum	O
distance	O
between	O
any	O
two	O
points	O
from	O
the	O
two	O
clusters	O
.	O
</s>
<s>
Similarly	O
,	O
average-distance	O
clustering	B-Algorithm
uses	O
the	O
average	O
pairwise	O
distance	O
as	O
the	O
dissimilarity	O
.	O
</s>
<s>
Like	O
Ward	O
's	O
distance	O
,	O
these	O
two	O
forms	O
of	O
clustering	B-Algorithm
obey	O
a	O
formula	O
of	O
Lance	O
–	O
Williams	O
type	O
.	O
</s>
<s>
Unlike	O
Ward	B-Algorithm
's	I-Algorithm
method	I-Algorithm
,	O
these	O
two	O
forms	O
of	O
clustering	B-Algorithm
do	O
not	O
have	O
a	O
constant-time	O
method	O
for	O
computing	O
distances	O
between	O
pairs	O
of	O
clusters	O
.	O
</s>
<s>
Maintaining	O
this	O
array	O
over	O
the	O
course	O
of	O
the	O
clustering	B-Algorithm
algorithm	I-Algorithm
takes	O
time	O
and	O
space	O
.	O
</s>
<s>
The	O
nearest-neighbor	B-Algorithm
chain	I-Algorithm
algorithm	I-Algorithm
may	O
be	O
used	O
in	O
conjunction	O
with	O
this	O
array	O
of	O
distances	O
to	O
find	O
the	O
same	O
clustering	B-Algorithm
as	O
the	O
greedy	B-Algorithm
algorithm	I-Algorithm
for	O
these	O
cases	O
.	O
</s>
<s>
by	O
a	O
technique	O
that	O
overlays	O
a	O
quadtree-based	O
priority	B-Application
queue	I-Application
data	B-General_Concept
structure	I-General_Concept
on	O
top	O
of	O
the	O
distance	O
matrix	O
and	O
uses	O
it	O
to	O
perform	O
the	O
standard	O
greedy	O
clustering	B-Algorithm
algorithm	I-Algorithm
.	O
</s>
<s>
This	O
quadtree	B-Data_Structure
method	O
is	O
more	O
general	O
,	O
as	O
it	O
works	O
even	O
for	O
clustering	B-Algorithm
methods	O
that	O
are	O
not	O
reducible	O
.	O
</s>
<s>
However	O
,	O
the	O
nearest-neighbor	B-Algorithm
chain	I-Algorithm
algorithm	I-Algorithm
matches	O
its	O
time	O
and	O
space	O
bounds	O
while	O
using	O
simpler	O
data	B-General_Concept
structures	I-General_Concept
.	O
</s>
<s>
In	O
single-linkage	B-Algorithm
or	O
nearest-neighbor	O
clustering	B-Algorithm
,	O
the	O
oldest	O
form	O
of	O
agglomerative	B-Algorithm
hierarchical	I-Algorithm
clustering	I-Algorithm
,	O
the	O
dissimilarity	O
between	O
clusters	O
is	O
measured	O
as	O
the	O
minimum	O
distance	O
between	O
any	O
two	O
points	O
from	O
the	O
two	O
clusters	O
.	O
</s>
<s>
(	O
Single-linkage	B-Algorithm
also	O
obeys	O
a	O
Lance	O
–	O
Williams	O
formula	O
,	O
but	O
with	O
a	O
negative	O
coefficient	O
from	O
which	O
it	O
is	O
more	O
difficult	O
to	O
prove	O
reducibility	O
.	O
)	O
</s>
<s>
As	O
with	O
complete	O
linkage	O
and	O
average	O
distance	O
,	O
the	O
difficulty	O
of	O
calculating	O
cluster	O
distances	O
causes	O
the	O
nearest-neighbor	B-Algorithm
chain	I-Algorithm
algorithm	I-Algorithm
to	O
take	O
time	O
and	O
space	O
to	O
compute	O
the	O
single-linkage	B-Algorithm
clustering	I-Algorithm
.	O
</s>
<s>
However	O
,	O
the	O
single-linkage	B-Algorithm
clustering	I-Algorithm
can	O
be	O
found	O
more	O
efficiently	O
by	O
an	O
alternative	O
algorithm	O
that	O
computes	O
the	O
minimum	O
spanning	O
tree	O
of	O
the	O
input	O
distances	O
using	O
Prim	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
,	O
and	O
then	O
sorts	O
the	O
minimum	O
spanning	O
tree	O
edges	O
and	O
uses	O
this	O
sorted	O
list	O
to	O
guide	O
the	O
merger	O
of	O
pairs	O
of	O
clusters	O
.	O
</s>
<s>
Within	O
Prim	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
,	O
each	O
successive	O
minimum	O
spanning	O
tree	O
edge	O
can	O
be	O
found	O
by	O
a	O
sequential	B-Algorithm
search	I-Algorithm
through	O
an	O
unsorted	O
list	O
of	O
the	O
smallest	O
edges	O
connecting	O
the	O
partially	O
constructed	O
tree	O
to	O
each	O
additional	O
vertex	O
.	O
</s>
<s>
This	O
choice	O
saves	O
the	O
time	O
that	O
the	O
algorithm	O
would	O
otherwise	O
spend	O
adjusting	O
the	O
weights	O
of	O
vertices	O
in	O
its	O
priority	B-Application
queue	I-Application
.	O
</s>
<s>
Using	O
Prim	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
in	O
this	O
way	O
would	O
take	O
time	O
and	O
space	O
,	O
matching	O
the	O
best	O
bounds	O
that	O
could	O
be	O
achieved	O
with	O
the	O
nearest-neighbor	B-Algorithm
chain	I-Algorithm
algorithm	I-Algorithm
for	O
distances	O
with	O
constant-time	O
calculations	O
.	O
</s>
<s>
Another	O
distance	O
measure	O
commonly	O
used	O
in	O
agglomerative	B-Algorithm
clustering	I-Algorithm
is	O
the	O
distance	O
between	O
the	O
centroids	O
of	O
pairs	O
of	O
clusters	O
,	O
also	O
known	O
as	O
the	O
weighted	O
group	O
method	O
.	O
</s>
<s>
Therefore	O
,	O
the	O
nearest-neighbor	B-Algorithm
chain	I-Algorithm
algorithm	I-Algorithm
will	O
not	O
necessarily	O
find	O
the	O
same	O
clustering	B-Algorithm
as	O
the	O
greedy	B-Algorithm
algorithm	I-Algorithm
.	O
</s>
<s>
Nevertheless	O
,	O
writes	O
that	O
the	O
nearest-neighbor	B-Algorithm
chain	I-Algorithm
algorithm	I-Algorithm
provides	O
"	O
a	O
good	O
heuristic	O
"	O
for	O
the	O
centroid	O
method	O
.	O
</s>
<s>
A	O
different	O
algorithm	O
by	O
can	O
be	O
used	O
to	O
find	O
the	O
greedy	O
clustering	B-Algorithm
in	O
time	O
for	O
this	O
distance	O
measure	O
.	O
</s>
<s>
The	O
nearest-neighbor	B-Algorithm
chain	I-Algorithm
algorithm	I-Algorithm
was	O
developed	O
and	O
implemented	O
in	O
1982	O
by	O
Jean-Paul	O
Benzécri	O
and	O
J	O
.	O
Juan	O
.	O
</s>
<s>
They	O
based	O
this	O
algorithm	O
on	O
earlier	O
methods	O
that	O
constructed	O
hierarchical	B-Algorithm
clusterings	I-Algorithm
using	O
mutual	O
nearest	O
neighbor	O
pairs	O
without	O
taking	O
advantage	O
of	O
nearest	O
neighbor	O
chains	O
.	O
</s>
