<s>
SUBCLU	B-Algorithm
is	O
an	O
algorithm	O
for	O
clustering	B-Algorithm
high-dimensional	I-Algorithm
data	I-Algorithm
by	O
Karin	O
Kailing	O
,	O
Hans-Peter	O
Kriegel	O
and	O
Peer	O
Kröger	O
.	O
</s>
<s>
It	O
is	O
a	O
subspace	O
clustering	B-Algorithm
algorithm	I-Algorithm
that	O
builds	O
on	O
the	O
density-based	O
clustering	B-Algorithm
algorithm	I-Algorithm
DBSCAN	B-Algorithm
.	O
</s>
<s>
SUBCLU	B-Algorithm
can	O
find	O
clusters	B-Algorithm
in	O
axis-parallel	O
subspaces	O
,	O
and	O
uses	O
a	O
bottom-up	O
,	O
greedy	B-Algorithm
strategy	O
to	O
remain	O
efficient	O
.	O
</s>
<s>
SUBCLU	B-Algorithm
uses	O
a	O
monotonicity	O
criteria	O
:	O
if	O
a	O
cluster	O
is	O
found	O
in	O
a	O
subspace	O
,	O
then	O
each	O
subspace	O
also	O
contains	O
a	O
cluster	O
.	O
</s>
<s>
However	O
,	O
a	O
cluster	O
in	O
subspace	O
is	O
not	O
necessarily	O
a	O
cluster	O
in	O
,	O
since	O
clusters	B-Algorithm
are	O
required	O
to	O
be	O
maximal	O
,	O
and	O
more	O
objects	O
might	O
be	O
contained	O
in	O
the	O
cluster	O
in	O
that	O
contains	O
.	O
</s>
<s>
However	O
,	O
a	O
density-connected	B-Algorithm
set	I-Algorithm
in	O
a	O
subspace	O
is	O
also	O
a	O
density-connected	B-Algorithm
set	I-Algorithm
in	O
.	O
</s>
<s>
This	O
downward-closure	O
property	O
is	O
utilized	O
by	O
SUBCLU	B-Algorithm
in	O
a	O
way	O
similar	O
to	O
the	O
Apriori	B-Algorithm
algorithm	I-Algorithm
:	O
first	O
,	O
all	O
1-dimensional	O
subspaces	O
are	O
clustered	O
.	O
</s>
<s>
All	O
clusters	B-Algorithm
in	O
a	O
higher-dimensional	O
subspace	O
will	O
be	O
subsets	O
of	O
the	O
clusters	B-Algorithm
detected	O
in	O
this	O
first	O
clustering	O
.	O
</s>
<s>
SUBCLU	B-Algorithm
hence	O
recursively	O
produces	O
-dimensional	O
candidate	O
subspaces	O
by	O
combining	O
-dimensional	O
subspaces	O
with	O
clusters	B-Algorithm
sharing	O
attributes	O
.	O
</s>
<s>
After	O
pruning	O
irrelevant	O
candidates	O
,	O
DBSCAN	B-Algorithm
is	O
applied	O
to	O
the	O
candidate	O
subspace	O
to	O
find	O
out	O
if	O
it	O
still	O
contains	O
clusters	B-Algorithm
.	O
</s>
<s>
In	O
order	O
to	O
improve	O
the	O
runtime	O
of	O
DBSCAN	B-Algorithm
,	O
only	O
the	O
points	O
known	O
to	O
belong	O
to	O
clusters	B-Algorithm
in	O
one	O
-dimensional	O
subspace	O
(	O
which	O
is	O
chosen	O
to	O
contain	O
as	O
little	O
clusters	B-Algorithm
as	O
possible	O
)	O
are	O
considered	O
.	O
</s>
<s>
SUBCLU	B-Algorithm
takes	O
two	O
parameters	O
,	O
and	O
,	O
which	O
serve	O
the	O
same	O
role	O
as	O
in	O
DBSCAN	B-Algorithm
.	O
</s>
<s>
In	O
a	O
first	O
step	O
,	O
DBSCAN	B-Algorithm
is	O
used	O
to	O
find	O
1D-clusters	O
in	O
each	O
subspace	O
spanned	O
by	O
a	O
single	O
attribute	O
:	O
</s>
<s>
//	O
In	O
a	O
second	O
step	O
,	O
-dimensional	O
clusters	B-Algorithm
are	O
built	O
from	O
-dimensional	O
ones	O
:	O
</s>
<s>
The	O
set	O
contains	O
all	O
the	O
-dimensional	O
subspaces	O
that	O
are	O
known	O
to	O
contain	O
clusters	B-Algorithm
.	O
</s>
<s>
The	O
set	O
contains	O
the	O
sets	O
of	O
clusters	B-Algorithm
found	O
in	O
the	O
subspaces	O
.	O
</s>
<s>
The	O
is	O
chosen	O
to	O
minimize	O
the	O
runs	O
of	O
DBSCAN	B-Algorithm
(	O
and	O
the	O
number	O
of	O
points	O
that	O
need	O
to	O
be	O
considered	O
in	O
each	O
run	O
)	O
for	O
finding	O
the	O
clusters	B-Algorithm
in	O
the	O
candidate	O
subspaces	O
.	O
</s>
<s>
Candidate	O
subspaces	O
are	O
generated	O
much	O
alike	O
the	O
Apriori	B-Algorithm
algorithm	I-Algorithm
generates	O
the	O
frequent	O
itemset	O
candidates	O
:	O
Pairs	O
of	O
the	O
-dimensional	O
subspaces	O
are	O
compared	O
,	O
and	O
if	O
they	O
differ	O
in	O
one	O
attribute	O
only	O
,	O
they	O
form	O
a	O
-dimensional	O
candidate	O
.	O
</s>
<s>
An	O
example	O
implementation	O
of	O
SUBCLU	B-Algorithm
is	O
available	O
in	O
the	O
ELKI	B-Language
framework	I-Language
.	O
</s>
