and Clustering using. Hierarchies), and demonstrates that it is especially suitable
for very large databases. BIRCH incrementally and clynami- call y clusters ...
BIRCH:
An
Efficient
Data
Clustering
Method
for
Very
Large
Databases Tian
Zhang
(“lornputer
Sciences
[Jniv.
Raghu Dept.
Ramakrishnan
Computer
of Wisconsin-Maciison
of
[Jniv.
zhang@cs. wise.edu
Sciences
useful
considerable
patterns
interest
problems
work
datasets
Hierarchies),
of 1/0
presents
a data
Iterative
for very large databases. incoming
to produce
able resources BIRCH ditioual
are not part
We evaluate order
sensitivity,
experiments. recently
and
for
large
1
Introduction this
paper,
a particular set
spare
of
is usually
identifies
NASA
data
through
a performance S11OW that
The
amount
proposed BIRCH
is
the
of
not
clatla
mining
the
and overall the
it
efficiently
dataset[Lee81, has
been
144-EC
and
effectively
can than
than
of be the
tzrne
by
NSF
Grant
We
present
Its
1/(>
a .szngle
order with
we argue
tering
method also
SIGMOD ’96 6/96 Montreal, Canada IQ 1996 ACM 0-89791 -794-4/96/0006 ...$3.50
103
or
=
X)
such
and that
is willing
to
linear
wait
is the
for
the
in
passes further.
triangular any
quality
eficieucy,
for
inequality XI
,XZ,X3,
an
clusar-
parallelism,
is the first
i.e., (there
and
based
the
whose self exists
on
course
clustering
attribote
ill-
com-
BIRCIPs
over
space,
can
experi-
available
tuning
gained
is
and
through best
the
data
quality,
databases.
Eucltdtan
of
a good
additional
performance
attribute
size
yields
large
the
is the
BIRCH
and
very
the
algorithms
large
BIRCH for
dataset
clustering
ciataset,
of
for
to
point,
account
named suitable
opportunities
Finally, metmc
ule rvant
A related
time/space
BIRCH
very
requirements
Statistics,
(t~yptcall~y,
and
into
more
or dynamic
a
1/0.
the
existing
the
in
to take
is of
and
offers
about
1 Informally,
[1(X1,X3)).
trying
constraint,:
.sw~)
to improve
that for
execution.
definition
cost
one
other
interactive
X
is typically
without
used
method
BIRCH’S
ments,
X,
that This
algorithm.
sensitivity,
paring
any
for
a user
scan
be used
the
clus-
problem.
there
M ltrntted
set
it is especially
ancl
satisfy
tn
optimization
available
a clustering
(optionally)
the
a
the uletghted
functton.
solution
data
clustering
that
knowledge
Permission to make digitalhard copy of part or all of this work for personal or classroom use is ranted without fee provided that copies are not made or distributed for pro 7tt or commercial advantage, the aopyright notice, the title of the publication and its date appear, and notice is given that copying is by permission of ACM, Inc. To copy otherwise, to republish, to post on servers, or to redistribute to lists, requires prior specific permission andlor a fee.
and
of the dataset
minima,
to Lre able
clustering,
for
IRI-9057562
(e.g.,
definition
required
ttme that
of
of the
chitecture
78.
rlum-
potnts,
database-oriented
the
is desirable
amount
put
D,J80]. supported
problem
By evaluating
and
prol>-
destred
N
measurement
minimal
of memory
results
clustering
patterns clusters
a
of
[KR90]
additional,
the
dataset:
data
places,
distribution derived
the
Data
crowded
is
(~iven
points,
occupied. the
which
problem. data
uniformly
sparse
more
(;rant
data clustering,
examine
Besides,
research
the
an
-smaller
that
comparisons
method
adopt,
with
rninzmt.ze
several
clustering
pazr-s of pozrtts
of local
a global
most,
partitions.
but
much input
to find
possible We
a few acl-
effectively.
quality
rnulti-clirnensional
discovers
*This and
all
databases. we
dataset.
original
no way
a single scan
with
of thf
ln
as in
the
a partatzon
dtscrete
demonstrate
kind
large
visualized
the avail-
betuleeri
to an abundance
the
G’tven functzon
to find
involved
nonmetrzri.
attributes,
a dataset
dwtunce
the value
and
where
measurement
is a nonconvm Due
metric
and
rue are asked
constraints).
efficiency,
and
K
of attributes
metrtc
as follows:
clusters
rrl~nrmizes
clustering algorithm “noise)’ (data points
NS, a clustering
datasets,
of
superior.
In
hence
CLARA
with
pattern)
clustering
We also present
consistently
with
further
time/space
suitable
data points
and time
find a goocl clustering
BIRCH’S
ber
wise.eclu
types
literature,
is formalized
ters),
are two
consider
Statistics
lern
totrd/average
using
and clynami-
metric
clustering
of the underlying
of BIR (;’H versus
the
incrementally
the quality
named
Clustering
it is especially
memory
we
Dept.
of Wisconsin-Maclison
to be clustered:
paper,
of the
of large
method
and
that
quality
and improve
widely
of clusters,
there
data
dzstance-based
clustering
scans. BIRCH is also the first in the database area to handle
proposerl
that
the best
(i. e., available
this
costs.
multi-dimensional
can typically
of the data,
most
the problem
Reducing
BIRCH
attracted
Sciences
mironf~cs.
in the
nensional clataset.
address
and demonstrates
call y clusters
has
one of the
in a multi-dir
does not adequately
paper
datasets
and
regions,
(;”H (Balanced
to try
large
recently,
and minimization
This Bfll
in
in this area is the identification
or deusel y populated Prior
LJniv.
Wisconsin-Maciison
Generally,
Finding
Livny”
Computer
raghuf~cs.wise.edu
Abstract
st,udied
Miron
Dept.
identity
of al-
values (for
a distance
d(XI , X2) + d(X2, X3) ~
goritlhm
proposed
in
the
datlahase
outl~~rs (intuitively, data a,s “noispi’ ) and proposes 1.1
Outline
The
rest
of
of the
surveys
4 introduces
au(i
( ~F tree,
of BIll(’H Sec. ture
2
which
Finally
rmearch
D.J80,
Fis87,
Fis95,
study
to
has
of
been
Lel>87]
I)ased
(like
and
most
in
consider
the case that, memory.
that,
to work
the
is typirally,
mu[-h
smaller
do the clustering the
1/()
than
on
independent
of this
for.
make
eslxv-ially
The
for
often
(e. g.,
hllilt
to
skpwed
points
also
A
related
the
data,
what
very
is not,
this
may
far
on
we are
the
on
number
of
problem tree
that
height, -balanced,
cause
the
bor
which
performance
is,
for
l)oints how
in
each close
ignore are eclually
purpose,
ancl d;nse
the
the
numlocol
or
far
decision, away
they
that
are of
data
[EK.X95h]
that
data
to
objects
a sample
dat, a
data
global
or
page;
and
may
that,
time
that
ch-awbacks
as the
addition,
it, may
In
to
1s improved
the
Later
techniques
on clisks is drawn
searching [EKX9,5aj (based
to deal
by
each
data
Their
experiments
with
a small
on with
( 1) clustering
from
on relevant
updates,
node stops
loccd nLznz71w
N,S’s ability
reside
(2) focusing
quality
selected
R*-tree
points
for show
loss of quality,
points.
they
inspect,
all
2.1
Cent
eclually
no
An
important
and
ancl
clist, ance the
focusing
of the ciat, aset
clusters are,
that,
same
due
CLARA
contiulles;
best, of these,
mmmezghbor.
propose improve
ancl
neigh-
as a 10CCZ1T7LZnZ-
of the so-called
minimum by
For
n~om~e~ghbor
{“?LARAN,S
the
two
medoid,
if a better
randomly
the
is a Ii”and
node.
the
node
efficiency
local
uocle
selectecl
data
is formali-
by one
mtntmum,
from
wrt.
controlled
differ most
al-
is repre-
loc-ated
neighbor
returns
proposes
rnedoi(ls,
ancl
the
number
method
R*-trees)
at
a new
, and
a real
only
current,
local
RA N,S suffers
find
to
with
and
each Ii
a randomly
the
another
found
IO
of
randomly,
restarts
been
which
data
(’L,4BAN,$’
process
in
checks
moves
have
and
he considered
They
existing
all
irrrportant
and
granularity
clustering currently
that
it
after
not
it
records
for
(YA
be scanned the fact
shoulci
of individually, at
assume can
clataset
clustering
are close
and
or partially
in the
the
They
advance
totally
points
or all
and
shove
to
it
trimming
given
insteacl
is founcl,
to search
is
with
node,
spatial
centrally
a set
if they
of neighbors
mum,
For
by
Scj it is
a rluster
clustering
a graph
starts
current
number
is that
The
neighbors
otherwise
only
or the most,
a
com-
clustering
CLARANS,
medotd,
representeci are
search, traditional
cluster
as searching
the
of values
not,
the
CLARANS
and
expensive,
number
dependent
apprc)aclles:
to
by its
time
presents
ranclornizecl
Witlh
best
as a useful
outperforms
“lwst,”
(or splitting
N.
[NH94]
In
clustertn[g to find
is 0(N2).
large
to
case tirnp
clusters,
algorithm with
of
sensitivp
pair
the
recognized
in Statistics.
in
zed
on
try
to form
recently.
based
partition
of c-lusters
probability-based
clusters
methods
matter
a large
but
.se771z-globol
data
have
clusters
attribute.
collectively
That,
the
attributes,
They
respect
(_’LARAN,$
nodes
is
exists,
representations
storing
[Fis87]),
this
is exactly
of
are
is
that
sented
statistically
has been
that
not
value
minimum,
worst,
the closest,
HC
well
to
the
Hierarchtml
measurement,,
to scale
method
gorithms
drarnatlically,
not, all clata points
to
typically
attributes
each
identify
freclueutly. with
between
probability
Distance-based data
of that,
probahilit,y
reality,
are
input
(Iqiyade
In
complexities
number
values
are
of correlation
and
their
attributes
other,
if the attributes
because the
each kind
updating
that
of objects
of a practical
mining
keeping
They
assumption
separate
( correlation
sometimes looking
the
pair) distance
unable
point
make
distributions true.
while
approaches:
( ‘KSt38]
from
in terms
merging
Clustering
do not
memory
as possible
plexity
keeps
or
is vpry
does
with
to another
a local
ancl the
is
moving
improves
It can find
Mur83]
it small.
starts
group
minimum
So in
extremely
possible
one
exponential.
KR90,
subsets.
KR90]
all
partition,
is still
farthest,
still
low.
Probability-based [FisH7,
or
the size of the dataset)
as accurately
costs
the
can he too they
(e.g.,
of the local
but
[DH73,
(EE),
of par-
minimum,
are
or swapping
selected
[DH73,
K
from
hut, the c!uality
K
of
quality,
ways
global
tries
function,
reasonable
, C1O not,
be viewed
resources
dif-
Learning)
Statistics)
the clataset
must
a limited
and
a moving
initially
into
(10)
data
none
eTmnltratzort
the
IV and
points
stal)le
points
then
of data
clusters,
probability-
In particular,
problem
with
methods
when
partition,
with
! [DH73]
find
all
Hence
IiN/I< data
can
the measurement
(HC)
EKX95a,
it
scanning
clusters.
exhausttvt
optzmwatzon
initial
complexity
[CKS88,
of N
except,
see if such
fw
Statistics
[NH94,
in
in main
rerognize
the
Learning
Machine
work
though
the
approaches,
large
how
in
different,
adequately to fit
for
7,
Database
approaches (like
directions
a set
swapping
in
using
titioning
require
scalability
approximately
practice,
an
a pre-
time
example, are
Iterattve
details
Research
Previous
most
The
currently
linear
infeasible
((~F)
is presented
Machine
with
contri-
5, and
BIRCH
studied
and
communities
distauce-basecl
have
there
2
material.
BIRCH.
in Sec.
Mur83],
Sec.
feature
in Sec.
conclusions
Lee81,
emphases,
them
BIRCH’S
of clustering
of Relevant
ferent
be regarded
existing
background
are central
our
clustering
E1iX951]]
some
are presented
[DHi’3,
which
or all
as follows.
is described
Summary
Data
measurements,
points
For
the concepts
performance 6,
glol]al
solution.
ancl summarizes
algorithm
liminary
a plausible
3 presents
Ser.
should
is organized
work
Sec.
that,
addresses
that
Paper paper
relat,ed
l)utlunh
points
area
they
use
clustering,
104
ributions
of
BIRCH
contribution
is
problem
in
a way
our that,
formulation is
appropriate
of
the for
very
large
ry
clatasets,
constraints
following
by
making
explicit.
In
advantages
over
the
time
addition,
and
memo-
BIRCH
previous
has
~he average mtra-cluster distance D4
the
distance-based
ap-
in$er.clust distance of the two
er distancs D%, average D3 and variance increase clusters are defined as:
proaches. (6)
●BIRCH
is
local
clustering points
(as
decision or
all
currently
rneasnrements points,
that
and
at
maintained ●
is
every
data
are treated . BIRt~H finest
while
clustering
and
balanced
. If
by and
ornit
an
time
optional
method
and
does only
space not
clustering
D3
is
treated
in sparse
regions
of
accuracy)
is organized
not
scans
5,
Due
the
clataset
two
different
4 The
to
is
at
A
an
f,.
Ztl
whole
N
(1)
(3)
N(N–1)
centroid.
D
a cluster.
the
They
tightness
of
between
two
for
is
average
are
the
two
cluster
clusters,
measuring
their
(iiven the the centroid Manhattan defined as:
pairwise
alternative around
we define
points distance
the
of
centroid.
i =
(CF)
vector
of
=
Ixbl
the
vector
two
disjoint
the
Next
cluster:
NI d-dimensional i = 1,2, . . . . N], {X-’}
where
j
=
proof
N1 +
l,N1
that
a
number
of
of the
square
sum
❑
X-iz.
dw~oint
as
sum
S’5’ is the
=
Assu7ne
(N2,
L3’2, ,$,$’z)
clusters.
is formed
Then
by merging
+ N2, L~l
+ L~2,
the the
,$,$1 + ,5’,5’2) (9)
D1,
the
(5)
+ N2,
can
summary less
105
is
than
accurate
that
corresponding
all
in BIRCH.
as clusters
are
the
CF
D,
DO,
R,
quality
diameter
rnet,rics
of clusters)
easily.
vector not
that
data it
as a set
stored
only
the
because
measurements
and
given XO,
as the usual
of a cluster
CF
we
can be stored
prove
as well
[]
theorem,
accurately
to
total/average
think the
and
the
D4,
algebra.
additivity
of clusters
easy
be calculated
decisions
and
vectors
also
as weighted all
CF
the
is
D3 and
only
much
CF
of clusters,
but
also
definition
incrementally
D2,
(such can
of straightforward
CF
It
vectors
in a cluster: {.~i } points in another + 2, . ..)Nl
is the
Theorem);
that
in
is:
(Nl
consists the
One
data points and N2 data
points
Clustering
linear
a91~ CF2
of two cluster
clusters,
calculated
(4) – xb2(t)\
~~=1
the
is defined
N
and
, ,$,$1),
vectors
=
cluster
Additivity
, L~l
of the
+ CF2
know
of two clusters: X~l ancl X_62, distance DO and centroid D1 of the two clusters are
[Xhl(t)
i.e.,
(CF
(Nl
the
L~$ is the
~ ~~,
points,
data
1, 2, . . . . N,
where
tree
a cluster.
d-dimensional
the
CF
clustering.
summarizing
about
cluster, ~~
and
a triple
L%’, S’S),
i.e.,
CF
merged.
– X7121 = ~
(N,
in the
=
are
From
,=1
(iiven where
=
data
CF
is
we maintain
4.1
CF1
d 1)1
N
incremental
N
points,
that
The
distances
Do = ((X731 – xi12)’)+
[Jsers
the
within
measures
5 alternative
to
closeness.
centroids Euclidian distance
data
Feature
BIRCH’S
Given
CF
CF1 member
a
Tree
Clustering
where
Theorem
~,
from
of
the relative
CF
of
{ii}
points
of the
D of the cluster
Jx’mt-m’)+ distance
sake
or shifting
affecting
and
Feature that 4.1
triple: data
(2)
average
the
separately.
by weighting
Feature
Clustering
Definition
once.
AT
R is the
without
core
information
Assume that readers are familiar with the terminology of vector spaces, we begin by defining centroid, radius and diameter for a cluste~. Given N d-dimensional data points in a cluster: {.Xi} where i = 1,2, . . . . N, the diameter
For
as properties
them
dimensions
of
the
Feature
R and
state data
concepts
are
Background
centroid XO, radius are defined as:
and
preprocess
Clustering
a cluster:
3
D
and
1, D2, D3 and D4 as properties
DO, D
clusters
optionally
cluster.
R
and
scalable.
require the
and
merged
X-O,
placement.
height-
BIRCH
treat
cluster,
along
efficiency).
structure.
4
can
D of the
we
between
ensure
is linearly
is actually
clarity,
single
to derive
in-memory,
Phase
that
in advance,
for
ensure
tree
its running the
incremental clataset
(to
of
of
optionally.
process
use
uses
hence
memory
(to
highly-occupied
features,
we
removed
reducing
It
data
points
Points
costs
the
the and
of
subclusters 1/0
characterized
that
use of available
possible
data
process.
region
and
each
all
incrementally
important
as outlters
that
closeness
be
occupied,
cluster.
minimizing
these
clustering
as a single
full
can
observation
dense
makes
natural
time,
is equally
in
scanning clusters.
the
uniformly
A
collectively
The
the
point
global)
existing
same
the
not
purposes.
the
the
exploits
usually
to without
reflect
during
BIRCH
opposed is made
as
efficient points
is sufficient we need
for
of data
points,
summary. because
in for
the
This it
stores
cluster,
hut
calculating
all
making
clustering
4.1
CF
A
CF
et(ers:
is a height-balanced
branching
nonleaf
node
[CFi,
rhddi],
to
i-th
its
clust, er
factor
node,
l>y its each
addition,
the
efficient
ter
made
scans. up
entries.
of
But
thrr.shold
all
tree
slllaller
size
the
of size
tree
is. the
the
sizes
of leaf
B and
Such
tree
the
the
same
il]tu
the
corrert
tree
is
a very
correct
will
into
we only
rep-
addition
leaf
at most,
“prev”
represents
by
must]
of
its
satisfy
T.
The
larger
‘T is, the
a node
to it,
in a pa~e
of T.
d of the
nonleaf
data
entries
by P.
space
are
So P can
dynamically to guide
for
is
compact entry
a leaf
just
ldentzf~ymg tile
the
closest
for
entry
data
metric:
inserting
finc]s
the
the
whether threshold
new leaf
entry,
it
an
pair
rluster
we
the
threshold cal}
say
“ Ent”
cluster
eomputecl
and
CF
the
in Sec.
L,,
as seeds,
the
a node
for
NJ,
postponing
later
entries
with
splits;
other
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“outsiders”
cluster.
(ranging
are close
Note
the
0.07
cluster
presentations
clue
to the
lack
of
space).
5 distance that
and
cluster and
in a BIll(”~H the
are
average
actual
are 0.17
from
the
and
an
BIR(~H
than
assigns
by
of location,
( 1.41).
to a proper
be
clusters
of points
rlusters
BIRCH
BIR(’H
of
an average
in
presented
in terms
cluster
of the
radius
of points
maximal
centroids
radii
centroid,
of DS 1 are
the
raclii.
number
of 1)S 1 hy plotting
is the
clusters
the
The
raclii
3
weight,wi
, ~)
