Int
ern
at
i
onal
Journ
al of Ele
ctrical
an
d
Co
mput
er
En
gin
eeri
ng
(IJ
E
C
E)
Vo
l.
8
,
No.
6
,
D
ece
m
ber
201
8
, pp.
4477
~
44
85
IS
S
N: 20
88
-
8708
,
DOI: 10
.11
591/
ijece
.
v8
i
6
.
pp4477
-
44
85
4477
Journ
al h
om
e
page
:
http:
//
ia
es
core
.c
om/
journa
ls
/i
ndex.
ph
p/IJECE
Postdiffset
Algo
rithm in
Ra
re P
attern:
An
Impl
emen
tatio
n
via
Benchm
ar
k
Case S
tud
y
Musta
fa
Man
1
, Wan
Aez
w
an
i Wa
n
Ab
u
Bakar
2
,
Masi
t
a
M
asi
la
Ab
d
.
Jali
l
3
, J
ul
aily
Aida J
usoh
4
1,3
School
of
In
fo
rm
at
ic
s &
Appli
ed
Math
ematics
,
Univer
sit
i
Mal
a
y
sia
T
ere
ngg
anu
,
Ma
la
ysi
a
2,4
Facul
t
y
Infor
m
at
ic
and
Com
p
uti
ng
,
Univer
sit
i S
ult
an Za
in
al Abidi
n
,
Ma
la
ysi
a
Art
ic
le
In
f
o
ABSTR
A
CT
Art
ic
le
history:
Re
cei
ved
A
pr
10
, 201
8
Re
vised
Jun
12
, 201
8
Accepte
d
J
un
30
, 201
8
Freque
nt
and
i
nfre
quent
i
te
m
set
m
ini
ng
are
tre
nding
in
d
at
a
m
ini
n
g
te
chn
ique
s.
The
pat
t
ern
of
As
socia
ti
on
Rule
(AR)
gene
ra
te
d
will
he
lp
dec
ision
m
ake
r
or
business
poli
c
y
m
ake
r
to
proje
c
t
for
the
ne
xt
int
end
ed
it
ems
ac
ross
a
wide
var
i
ety
of
a
ppli
c
ations.
W
hil
e
fre
quen
t
it
ems
et
s
are
dea
l
ing
with
ite
m
s
tha
t
are
m
ost
purc
hase
d
or
used,
infre
qu
e
nt
it
ems
are
those
it
ems
tha
t
are
infre
quen
tly
o
cc
ur
or
al
so
ca
ll
ed
ra
re
ite
m
s.
The
A
R
m
ini
ng
stil
l
remai
ns
as
one
of
t
he
m
ost
pro
m
ine
nt
areas
in
dat
a
m
ini
ng
tha
t
ai
m
s
to
ext
r
act
int
er
esti
ng
co
rre
lations,
p
at
t
e
rns,
association
or
ca
sua
l
struct
ure
s
among
set
of
it
ems
in
the
tra
nsa
ct
i
on
dat
aba
ses
or
othe
r
data
rep
ositories.
Th
e
design
of
d
ataba
se
stru
ct
ur
e
in
associ
at
ion
r
ule
s
m
ini
ng
al
gorit
hm
s
ar
e
b
ase
d
upon
hor
izontal
o
r
ver
ti
c
al
dat
a
form
at
s.
Th
ese
two
d
at
a
form
at
s
have
b
e
en
widely
discu
ss
ed
b
y
show
ing
few
exa
m
pl
es
of
al
gori
thm
of
each
d
at
a
f
orm
at
s.
Th
e
ef
f
orts
on
hori
zon
ta
l
form
at
suffe
rs
in
huge
ca
ndid
at
e
g
ene
r
at
ion
and
m
ult
ip
le
databa
se
sca
n
s
which
result
in
g
in
highe
r
m
emory
consu
m
pti
ons.
To
ov
erc
om
e
th
e
issue,
the
soluti
ons
on
ver
ti
c
al
appr
oac
h
es
ar
e
proposed.
One
o
f
the
esta
b
li
shed
al
gor
it
hm
s
in
v
ert
i
ca
l
data
form
at
is
Eclat
.
ECL
AT
or
Equ
i
val
en
ce
Cl
ass
T
ran
sform
at
ion
a
l
gorit
hm
is
one
exa
m
ple
sol
uti
on
that
lies
in
ver
tical
d
at
ab
ase
form
at
.
Because
of
it
s
‘fa
st
int
erse
ct
ion’
,
in
thi
s
pape
r,
we
ana
l
y
z
e
the
fu
ndamenta
l
E
cl
a
t
and
Ec
lat
-
var
ia
n
ts
such
asdiff
seta
nd
sortd
i
ffset.
In
respons
e
to
ver
t
ical
da
ta
form
at
and
as
a
cont
inui
t
y
t
o
Ec
la
t
ex
te
nsio
n,
we
propose
a
postdiffset
al
go
rit
hm
as
a
new
m
ember
in
Ec
l
at
var
i
ant
s
tha
t
use
ti
dset
for
m
at
in
the
first
loopi
ng
and
diffse
t
in
the
l
a
te
r
loopi
ng
.
In
thi
s
pape
r,
we
pre
sent
the
pe
rf
orm
anc
e
of
Pos
tdi
ffset
al
gor
it
hm
prior
to
implem
ent
a
ti
on
in
m
ini
ng
of
infre
q
uent
or
r
ar
e
it
ems
et
.
Pos
tdi
f
fset
a
lgori
thm
outpe
rform
s
23%
and
84
%
to
diffse
t
an
d
sortdiffse
t
in
m
ushroom
and
94
%
and
99%
to
d
iffse
t
and
sortd
if
fset
in
re
ta
i
l
dat
ase
t.
Ke
yw
or
d:
Associ
at
ion
r
ule m
ining
Ecl
at
algorit
hm
Fr
e
qu
e
nt
it
em
s
et
Infr
e
quent it
e
m
se
t
Ver
ti
cal
d
at
a
bse
Copyright
©
201
8
Instit
ut
e
o
f Ad
vanc
ed
Engi
n
ee
r
ing
and
S
cienc
e
.
Al
l
rights re
serv
ed
.
Corres
pond
in
g
Aut
h
or
:
Wan Aez
wa
ni
Wan A
bu Ba
ka
r
,
Faculty
of Inf
orm
at
ic
an
d
Com
pu
ti
ng,
Unive
r
sit
i Sulta
n
Zai
na
l Ab
i
din
,
Be
su
t C
am
pu
s,
2220
0
Be
s
ut, Te
reng
ganu
,
Ma
la
ysi
a
.
Em
a
il
: wan
aezwani
@unisza.e
du.m
y
1.
INTROD
U
CTION
The
m
ai
n
ob
j
e
ct
ives
of
as
so
c
ia
ti
on
ru
le
s
m
i
ning
are
to
fin
d
the
c
orrelat
ions,
ass
ociat
io
ns
or
cas
ual
structu
res
am
on
g
set
s
of
it
em
s
in
the
data
re
posit
or
y.
In
ot
her
w
ords,
it
al
lows
non
disc
ov
e
ry
of
im
pli
cat
ive
and
inte
resti
ng
te
nd
encies
in
databases
.
Fr
e
qu
e
nt
it
e
m
set
and
in
fr
e
quent
it
e
m
se
t
m
ining
are
crit
ic
al
fiel
ds
in
associat
ion
r
ul
e
m
ining
.
Th
e
fiel
ds
a
re
widely
us
e
d
ac
r
os
s
a
va
riet
y
of
dom
ai
ns
suc
h
as
m
ark
e
t
bas
ket
analy
sis,
rem
e
dial,
bio
l
og
y,
bankin
g
or
r
et
ai
l
serv
ic
es
[1
]
,
[
21]
.
Fr
e
qu
e
nt
or
in
fr
e
qu
e
nt
it
e
m
set
s
m
ay
con
t
rib
ute
to
bi
g
data
ge
ner
at
ion
.
Und
oubte
dly,
the
crit
ic
al
issues
re
garding
m
e
m
or
y
sp
ace
con
s
um
pti
on
a
nd
data
stora
ge
capaci
ty
will
sign
ific
an
tl
y
eff
ect
pr
i
or
to
fr
e
qu
e
nt
or
in
fr
e
quent
gen
e
rati
on
of
Evaluation Warning : The document was created with Spire.PDF for Python.
IS
S
N
:
2088
-
87
08
In
t J
Elec
&
C
om
p
En
g,
V
ol.
8
, N
o.
6
,
Dece
m
ber
2
01
8
:
4477
-
4485
4478
it
e
m
set
s
[2
2],
[23],
[
24
]
.
T
he
obj
ect
ive
of
f
r
equ
e
nt
it
e
m
set
is
to
find
fr
e
quent
groupin
g
of
it
em
s
in
database
con
ta
ini
ng
s
eri
es
of
it
em
tran
sact
ion
s
wh
il
e
the
obj
ect
ive
of
inf
reque
nt
it
em
se
t
is
con
tradict
to
fr
e
quen
t.
All
it
e
m
set
s
wh
ic
h
has
value
that
is
gr
eat
er
t
han
m
ini
m
u
m
su
pp
ort
is
cal
le
d
f
reque
nt
it
e
m
se
ts.
Infr
e
quent
it
e
m
set
fin
ds
hi
dd
e
n
as
so
ci
at
ion
a
nd
c
orrelat
ion
am
on
g
rar
e
it
em
sets.
The
rar
e
c
on
so
li
datio
n
of
th
ese
it
e
m
set
s
may
be
interest
ing
an
d
gain
m
or
e
pr
of
it
m
aking
.
R
are
cases
hav
e
sp
eci
al
co
nc
e
rn
si
nce
they
r
epr
ese
nt
sig
nif
ic
ant
diff
ic
ulti
es
fo
r
data
m
ining
al
gorithm
s.
All
i
tem
set
s
wh
ic
h
has
the
val
ue
that
is
le
sser
than
m
ini
m
u
m
s
upport
is
cal
le
d
infr
e
qu
e
nt
it
e
m
set
s
.
The
idea
of
m
ining
associ
at
ion
r
ule
or
ig
inate
s
from
the
analy
sis
of
m
ark
et
bas
ket
data
[
2].
E
xam
ple
of
a
sim
ple
ru
le
is
a
cust
om
er
who
bu
ys
brea
d
a
nd
butt
er
will
al
so
te
nd
t
o
bu
y
m
il
k
with p
r
ob
a
bili
ty
s%
an
d
c%
.
The
a
pp
li
cabil
it
y
of
s
uc
h
r
ule to
busine
ss p
r
oble
m
s
m
akes
the
ass
ociat
ion ru
le
to
beco
m
e a popu
la
r
m
ining
m
eth
od.
P
re
viou
s
effort
s
on
ARM
ha
ve
m
anipu
la
te
d
the
tradit
io
nal
horizo
ntal
database
for
m
at
[
2
]
,
[
3].
Be
cause
of
the
per
sist
ent
iss
ue
s
in
stora
ge
a
nd
m
e
m
or
y,
la
t
er
effo
rts
tur
n
to
util
iz
e
on
t
he
ver
ti
cal
associ
a
ti
on
ru
le
s
m
ining
al
gorithm
s
[4
]
-
[
7].
T
he
th
ree
ba
sic
m
od
el
s
in
fr
e
qu
e
nt
it
e
m
s
et
m
ining
are A
pri
ori
[7
]
that
li
es
on
horizo
ntal f
orm
at
w
her
eas
Ec
la
t and
FP
-
G
rowth [
9
],
[
11]
unde
rly
ing data
base s
t
ru
ct
ur
e i
s on verti
cal
fo
rm
at
.
Seve
ral
w
orks
hav
e
bee
n
c
ondu
ct
e
d
on
ver
t
ic
al
data
assoc
ia
ti
on
r
ules
m
i
ning
[3
]
-
[
6],
[
8],
[
1
0]
-
[
12
]
.
Am
on
g
t
hose
effor
ts,
Ecl
at
a
lgorit
hm
is
known
f
or
it
s
‘f
a
st’
interse
ct
ion
of
it
s
ti
dlist
wh
e
re
by
the
re
su
lt
ing
nu
m
ber
of
ti
ds
is
act
ually
the
sup
port
(
fr
e
qu
ency)
of
eac
h
i
tem
set
s
[4
]
,
[
8]
.
T
hat
is,
we
s
hould
brea
k
off
eac
h
intersect
io
n
as
so
on
as
the
re
su
lt
ing
nu
m
ber
of
ti
ds
is
bel
ow
m
ini
m
u
m
s
upport
th
resho
ld
that
we
ha
ve
set
.
Stud
ie
s
on
Ecl
at
al
gorithm
has
at
tract
ed
m
any
de
vel
op
m
enteffo
rts
incl
ud
i
ng
[
5
]
,
[7
]
,
[
13
]
.
M
otivat
ed
to
it
s
‘f
ast
inter
sect
ion’,
this
pa
per
pr
ese
nts
a
cri
ti
cal
rev
ie
w
in
Ecl
a
t
as
well
as
to
it
s
var
i
ants.
O
ur
pro
pose
d
so
luti
on,
po
st
di
ff
set
al
gorith
m
per
form
s
mo
de
ratel
y
in
sel
ect
ed
den
se
da
ta
set
and
good
in
sel
ect
ed
sp
arse
dataset
s.
2.
RELATE
D
W
ORKS
The
Ecl
at
sta
nd
s
f
or
Eq
uiva
le
nce
Cl
ass
Transf
or
m
at
ion
[9
]
,
[
12]
ta
kes
a
de
pth
-
fi
rst
searc
h
a
nd
represe
nts
database
i
n
ver
ti
ca
l
la
yout
s
uc
h
t
hat
eac
h
it
em
i
s
re
pr
ese
nte
d
by
a
set
of
tran
s
act
ion
IDs
(c
al
le
d
a
ti
ds
et
)
w
hose
t
ran
sact
io
ns
co
ntain
t
he
it
em
.
Tidset
of
a
n
i
tem
set
is
gen
e
rated
by
inte
rs
ect
ing
ti
ds
et
s
of
it
s
ite
m
s.
Be
cause o
f
the
de
pth
-
fi
rst searc
h, it
is d
iffic
ult t
o uti
lize the
dow
nw
a
rd
cl
osure p
rop
erty
li
ke
in
Apr
iori
.
Howe
ver,
us
in
g
ti
ds
et
s
has
an
ad
van
ta
ge
th
at
there
is
no
ne
ed
f
or
co
unti
ng
suppo
rt,
the
su
pp
or
t
of
an
it
e
m
set
is
the
siz
e
of
t
he
ti
ds
et
rep
re
se
ntin
g
it
.
The
m
ai
n
op
erati
on
of
Ecl
at
is
i
ntersecti
ng
ti
ds
et
s,
thu
s
the
s
iz
e
of
ti
ds
et
s
is
on
e
of
m
ai
n
factor
s
aff
ect
in
g
the
r
unning
ti
m
e
an
d
m
e
m
or
y
us
age
of
Ecl
at
.
T
he
bi
gg
e
r
ti
ds
e
ts
are,
the m
or
e tim
e and m
e
m
or
y are n
ee
ded.
Ba
sed
upon
disco
ver
y
in
[
4],
a
ne
w
ver
ti
cal
data
re
present
at
ion
,
cal
le
d
D
iffset
is
propos
ed
[
5]
.
T
he
so
-
cal
le
d
dEcl
at
,
a
dif
fset
of
Ecl
at
al
gorith
m
.
In
ste
ad
of
usi
ng
ti
ds
et
s,
th
ey
us
e
t
he
dif
f
eren
ce
of
ti
dse
ts
(call
ed
diffset
s).
Using
di
ff
set
s
has
reduce
d
the
se
t
siz
e
rep
re
sen
ti
ng
it
em
s
e
ts
dr
am
atical
l
y
and
th
us
op
e
rati
ons
on
set
s
are
m
uch
faster.
T
he
dE
cl
at
has
show
n
to
achieve
si
gnific
ant
im
pr
ovem
ents
in
perform
ance
as
well
as
m
e
m
or
y
us
age
over
Ecl
at
,
e
s
pecial
ly
on
de
ns
e
databases
.
Howe
ver,
w
he
n
the
dataset
is
sp
a
rse,
dif
fset
loses
it
s
adv
a
ntage
ov
e
r
ti
ds
et
.
T
her
e
fore,
t
he
r
esearche
rs
s
ug
gested
us
in
g
t
idset
form
at
a
t
the
sta
rt
for
sp
ars
e
databases
and t
hen sw
it
c
hing t
o diffse
t f
or
m
at
lat
er when a s
witc
hing c
ondi
ti
on
is m
et
.
As
a
co
ntinu
it
y
in
[4
]
,
[5
]
,
a
novel
ap
proac
h
f
or
ve
rtic
al
rep
re
se
ntati
on
wh
e
re
in
the
a
uthors
us
e
d
the
com
bin
at
ion
of
ti
ds
et
an
d
dif
fset
an
d
s
or
te
d
the
di
ff
set
i
n
desce
nd
i
ng
or
der
t
o
re
prese
nt
data
bases
[
7].
T
he
te
chn
iq
ue
is cl
aim
ed
to elim
i
nate the n
ee
d
of chec
king the s
witc
hing con
di
ti
on
an
d
c
onve
rting
ti
ds
et
to dif
f
set
form
at
reg
ar
dless
of
data
base
conditi
on
ei
th
er
sp
ar
se
or
de
ns
e.
Be
sides
,
the
com
bin
at
ion
can
f
ully
expl
oit
the
adv
a
ntage
s
of
bo
t
h
ti
ds
et
an
d
dif
fset
form
at
wh
ere
the
prel
i
m
resu
lt
s
hav
e
show
n
a
r
edu
ct
io
n
in
av
erag
e
diffset
size an
d spee
d of data
ba
se p
ro
ce
ssin
g.
3.
AS
S
OCIATI
ON R
ULE THE
ORETI
CAL
BACKG
ROUN
D
Fo
ll
owin
g
is
the
f
or
m
al
def
init
ion
of
t
he
pr
ob
le
m
def
ine
d
in
[
3].
Let
I
=
{i
1
,
i
2
,
…
i
m
}
fo
r
|m
|
>
0
be
the
set
of
it
e
m
s
.
D
is
a
database
of
transacti
ons
w
her
e
each tran
sact
io
n
has
a
un
iqu
e
ide
nt
ifie
r
cal
le
d
ti
d
.
Each
transacti
on
T
i
s
a
set
of
it
em
s
s
uch
that
⊆
.
An
associat
ion
ru
l
e
is
a
n
im
plication
of
t
he
f
orm
⊆
w
he
re
X
r
ep
rese
nt
the
antece
de
nt p
ar
t
of
the r
ule
an
d
Y
represe
nts the
co
ns
e
quent
pa
rt
of
th
e
r
ul
e
w
her
e
⊆
,
⊆
an
d
∩
=
∅
.
A
set
⊆
is
cal
le
d
an
it
emset
.
T
he
it
em
se
t
that
sat
is
fies
m
ini
m
u
m
su
pp
or
t
is
cal
le
d
fr
e
qu
e
nt it
em
s
et
.
The
sup
port
of rule
⇒
is t
he f
racti
on of tra
nsa
ct
ion
s i
n D c
onta
inin
g bo
t
h X a
nd Y.
(
⇒
)
=
∪
|
|
Evaluation Warning : The document was created with Spire.PDF for Python.
In
t J
Elec
& C
om
p
Eng
IS
S
N: 20
88
-
8708
Postd
if
fse
t Al
gorit
hm in
Rare
Patte
rn: A
n Imple
me
nta
ti
on v
ia Be
nchm
ar
k
Case
…
(
Wa
n Aezw
ani W.
A
.
B
.
)
4479
whe
re
|
D
|
is
th
e
tot
a
l
num
be
r
of
re
co
rds
in
d
a
taba
se
.
The
c
onfide
nc
e of rule
⇒
is t
he
f
racti
on
of tra
ns
act
io
ns
in
D
con
ta
ini
ng X t
hat also
contai
n Y.
(
⇒
)
=
(
∪
)
(
)
A
r
ule
is
fre
quent
if
it
s
s
upport
is
great
er
than
m
ini
m
um
su
pport
(m
i
n_
s
upp)
th
res
hold.
The
ru
le
s
wh
ic
h
sat
isfy
m
ini
m
u
m
con
fi
den
ce
(
m
in_
co
nf)
th
re
sh
ol
d
is
cal
le
d
strong
r
ule
an
d
both
min_s
upp
a
nd
mi
n_
c
onf
are
us
er
spec
ifie
d values
[4].
4.
REPRESE
NT
ATIO
N
O
F
D
ATA
Data
re
pr
es
e
nt
at
ion
is
c
riti
cal
in
ass
ociat
ion
ru
le
m
ining
.
How
data
is
st
or
e
d
in
data
ba
se,
data
base
la
yout and t
he se
arch
i
ng strat
egy in
vo
l
ved a
re all
contri
bu
t
e to the
p
e
rfo
r
m
ance of m
ini
ng each
it
em
se
ts.
4.1.
Searc
h Sp
ace
an
d
Da
ta
b
as
e Is
sues
Eit
her
with
horizo
ntal
data
fo
rm
at
or
ver
ti
cal
data
fo
rm
at
,
on
e
m
us
t
ta
ke
into
acco
un
t
on
the
sear
c
h
sp
ace
strat
e
gy
e
m
plo
y
m
ent
reg
a
rd
le
ss
t
he
database
c
onditi
on
of
wh
et
her
it
is
sp
ars
e
database
or
den
s
e
database
.
Th
e
Aprio
ri
-
ins
pire
d
al
gorithm
s
[5
]
perf
or
m
well
with
s
parse
dataset
s
su
c
h
a
s
m
ark
et
bask
e
t
data
wh
e
n
the
fr
e
quent
patte
rn
s
a
re
sho
rt.
But,
wh
e
n
the
fr
e
quent
patte
rn
s
a
re
lo
ng
with
de
ns
e
dataset
s
su
ch
a
s
bio
in
f
or
m
at
ic
s
and tel
ecom
m
u
nicat
ion
,
the
p
e
rfor
m
ance d
e
grades
dr
a
sti
cal
l
y.
The
degra
datio
n
is
cau
sed
by
m
any
passes
over
t
he
da
ta
bas
e
that
autom
at
i
cal
ly
incur
s
I/O
ove
rh
ea
ds
and
it
is
com
p
utati
on
al
ly
expensive
in
c
hec
king
la
rg
e
set
s
of
can
did
at
es
by
patte
rn
m
atch
in
g.
F
or
m
i
tem
s,
there
co
uld
im
ply
2
m
–
2
add
it
ion
al
fr
e
quent
patte
rn
s
that
w
il
l
exp
li
ci
tly
e
xam
ine
d
by
each
al
gorithm
s.
It
i
s
i
m
po
rtant
to
ge
ner
at
e
as
few
cand
i
dates
as
possible
sin
ce
c
om
pu
ti
ng
the
s
upports
is
ti
m
e
con
s
um
ing
[
14]
.
As
the
best
case,
on
ly
f
reque
nt
it
e
m
set
s
are
gen
erate
d
an
d
c
ounted
,
un
fortun
at
el
y,
the
i
de
a
is
i
m
po
ssib
le
in
gen
e
ral.
4.2.
Ho
ri
z
ontal
V
erses
Ver
tical La
youts
In
the
horiz
on
t
al
la
yout,
e
ach
transacti
on
is
represe
nted
a
s
:
(
,
)
w
her
e
is
the
transacti
on
identifie
r
a
nd
is
an
it
em
se
t
con
ta
ini
ng
it
e
m
s
occ
urrin
g
in
the
tran
sact
ion.
The
i
niti
al
transacti
on
co
ns
is
ts
of
al
l
transacti
on
s
.In
the
ver
ti
cal
la
yout,
eac
h
it
e
m
in
the
it
e
m
base
is
represente
d
as
:
{
,
(
)
}
a
nd
the
init
ia
l
trans
act
ion
datab
as
e
co
ns
ist
s
of
al
l
it
e
m
s
in
the
i
t
e
m
base.
Fo
r
both
la
youts
,
it
i
s
po
ssi
ble
to
use
the
bit
form
at
to
e
ncode
ti
ds
a
nd
al
so
a
c
om
bi
nation
of
both
la
youts
can
be
us
e
d
[7
]
,
[
8].
Fig
ur
e
1
il
lu
strat
e
s
horizo
ntal
an
d
ver
ti
cal
la
yout
of
data
repre
sentat
ion
by
[
7].
The
i
te
m
s
i
n
B
co
ns
ist
of
{a,b,
c
,d,e}
an
d
each
it
e
m
set
s
are
a
ll
ocated
with
un
i
qu
e
i
den
ti
fi
ers
(tids
)
f
or
each
tra
ns
act
ion
s
.
T
his
is
cl
early
visu
al
iz
ed
i
n
horizo
ntal
for
m
at
.
To
switc
h
to
ver
ti
cal
fo
r
m
at
,
ever
y
it
e
m
s
{a,b,
c,
d,
e} are
then
organ
i
zed
w
her
e
al
l
it
e
m
s
are
al
locat
ed
with
their
c
orres
pondin
g
ti
ds.
Whe
n
t
his
is
done
,
it
is
cl
early
vis
ualiz
ed
t
he
s
uppo
rt
of
eac
h
it
e
m
s
thr
ough the
coun
ti
ng
num
ber
of eve
ry it
em
’s
ti
ds
.
Figure
1
.
H
or
iz
on
ta
l a
nd
ve
rtic
al
lay
ou
t
Evaluation Warning : The document was created with Spire.PDF for Python.
IS
S
N
:
2088
-
87
08
In
t J
Elec
&
C
om
p
En
g,
V
ol.
8
, N
o.
6
,
Dece
m
ber
2
01
8
:
4477
-
4485
4480
5.
DESIG
N OF
ECLAT
AND E
CLAT
-
LI
K
E ALGO
RIT
HMS
Ther
e
are
t
wo
m
ai
n
ste
ps
:
cand
i
date
ge
nera
ti
on
a
nd
pru
ni
ng.
I
n
ca
nd
i
dat
e
ge
ner
at
io
n,
e
ach
k
-
it
em
set
cand
i
date
is
ge
ner
at
e
d
f
ro
m
two
fr
e
quent
(k
-
1)
-
it
em
set
s
and
it
s
sup
port
is
counte
d,
if
it
s
su
pp
or
t
is
lo
we
r
tha
n
the
thres
ho
l
d,
t
hen
it
will
be
discar
ded,
o
th
e
rw
ise
it
is
fr
eq
uen
t
it
em
se
ts
a
nd
us
e
d
to
ge
ne
rate
(k
+
1)
-
it
em
se
ts.
Since
Ecl
at
us
es
the
ver
ti
cal
layou
t,
co
unti
ng
su
pp
or
t
is
trivia
l.
Dep
th
-
fir
st
searchi
ng
strat
egy
is
do
ne
w
he
re
it
sta
rts
with
fr
e
qu
e
nt
it
e
m
s
in
the
it
e
m
base
and
the
n
2
-
it
e
m
se
ts
fr
om
1
-
it
e
m
set
s,
3
-
it
em
se
ts
fr
om
2
-
it
e
m
sets
and s
o on.
5.1.
Tr
ad
iti
onal Ec
lat
(
Ti
ds
et
)
A
k
-
it
em
set
is
gen
e
rated
by
t
akin
g
un
i
on
of
two
(k
-
1)
-
it
em
se
ts
wh
ic
h
ha
ve
(k
-
2)
it
e
ms
in
c
omm
on
,
the two
(
k
-
1
)
-
it
e
m
set
s ar
e cal
l
ed
par
e
nt it
e
m
s
et
s o
f
the k
-
it
e
m
se
t. Fo
x
exa
m
pl
e, {
, {ab
} a
nd
{ac} are p
a
r
ent o
f
{abc}.
T
o
a
void
ge
ner
at
in
g
duplica
te
it
e
m
set
s,
(
k
-
1
)
-
it
em
se
ts
are
so
rt
ed
in
s
om
e
order.
To
ge
nerat
e
al
l
po
s
sible
k
-
it
em
se
tsfro
m
a
set
of
(k
-
1)
-
it
em
se
ts
sh
arin
g
(k
-
2)
-
it
e
m
s,
un
i
on
operati
on
i
s
co
nducted
of
a
(k
-
1)
-
it
e
m
set
swith
t
he
it
em
se
ts
that
sta
nd
beh
i
nd
it
in
the
sorte
d
order,
a
nd
t
his
process
ta
kes
place
f
or
al
l
(
k
-
1)
-
it
e
m
set
s
excep
t
the
la
st
on
e.
Fo
r
e
xam
ple,
f
ro
m
a
set
of
{
,
,
,
,
}
,
wh
ic
h
sha
re
0
it
e
m
,
then
this
cou
l
d
be
so
rte
d
i
nto
al
phabet
order.
T
o
ge
ner
at
e
al
l
2
-
it
em
set
s,
the
unio
n
of
{
}
w
it
h
{
b,
c
,d,e}
w
il
l
resu
lt
i
nto
2
-
it
e
m
set
s
{a
b,a
c,ad,
ae}
,
the
n
for
the
unio
n
of
{b}
with
{c
,
d,
e}
will
res
ult
in
{
bc,b
d,be}
,
si
m
il
arly
fo
r
{
c}
an
d
{d
}
.
Finall
y,
al
l
po
ssi
b
le
2
-
it
em
se
ts
{a
b,ac
,ad,ae,
bc,bd,be
,c
d,
ce,
de}
is
ge
ne
rated
to
get
al
l
po
ssi
ble
3
-
it
em
se
ts
un
ti
l t
he rest
of the
nu
m
ber
of
po
s
sible i
te
m
se
ts.
Ecl
at
sta
rts
with
pr
e
fix
{}
an
d
the
searc
h
t
r
ee
is
act
ually
the
init
ia
l
searc
h
tree
.
T
o
div
i
de
the
init
ia
l
search
t
ree
,
it
picks
the
pr
e
fix
{a},
ge
ner
at
e
the
cor
re
spo
nd
i
ng
e
qu
i
valence
cl
ass
and
do
e
s
fr
e
quent
it
e
m
set
m
ining
i
n
the
su
b
tree
of
al
l
it
e
m
set
s
con
ta
inin
g
{a},
in
th
is
sub
tree
it
di
vid
es
furthe
r
i
nto
t
wo
s
ub
tr
ees
by
picking
t
he
prefix
{a
b}:
the
first
s
ub
t
re
e
c
on
sist
s
of
al
l
it
e
m
set
con
ta
in
ing
{a
b},
t
he
oth
e
r
co
ns
ist
s
of
al
l
it
e
m
set
s
co
ntainin
g
{a} bu
t n
ot {b
}, and th
is
p
ro
ce
ss is r
ecur
si
ve
unti
l al
l
it
e
m
set
s
in
the
init
ia
l search
tree are
visit
ed.
T
he
se
arch t
ree
of an
it
e
m
b
ase {a,b
,
c,d
,e}
is re
pres
ented by
the
tr
ee as s
how
n
in
Figure
2.
Figure
2. Searc
h
tree
for {a
,
b,c,d,
e}
w
it
h n
ul
l set
Figure
3
il
lustr
at
es
of
de
ta
il
ste
ps
ta
ken
in
Ec
la
t
al
go
rithm
wh
e
n
a
ssu
m
ing
that
the
i
niti
al
transacti
on
database
is
in
ver
ti
cal
la
yout
and
represe
nted
by
an
e
quiv
al
ence
cl
ass
E
with
prefi
x
{}.
It
us
es
prefi
x
-
base
d
equ
i
valence
cl
ass
al
ong
with
bo
tt
om
-
up
sea
rch.
Fr
e
quent
i
tem
set
s
are
gen
erate
d
by
inte
rsecti
ng
ti
dlist
of
al
l
disti
nct
pairs
of
at
om
s
(i.e.
i
1..
i
n
)
and
c
hec
king
the
ca
rd
i
nalit
y
of
the
ti
dlist
.
This
pr
ocess
i
s
rep
eat
e
d
unti
l
all
fr
e
qu
e
nt it
em
s
et
s ar
e e
nu
m
erated.
Evaluation Warning : The document was created with Spire.PDF for Python.
In
t J
Elec
& C
om
p
Eng
IS
S
N: 20
88
-
8708
Postd
if
fse
t Al
gorit
hm in
Rare
Patte
rn: A
n Imple
me
nta
ti
on v
ia Be
nchm
ar
k
Case
…
(
Wa
n Aezw
ani W.
A
.
B
.
)
4481
:
(
(
1
,
1
)
,
…
(
,
)
)
|
)
,
:
(
,
)
1
:
2:
≔
∪
// add
to
cre
a
te
a
new
pre
fix
3:
(
′
)
//
ini
t
ial
ize
a
new
equ
iv
al
ence
cl
ass
wi
th
th
e
n
ew
pref
ix P
4
:
ℎ
ℎ
>
5
:
=
∩
6
:
|
|
≥
ℎ
7
:
′
≔
∪
(
,
)
8
:
=
∪
(
∪
)
9
:
10:
11:
′
≠
{
}
the
n
12:
(
′
,
)
13:
14:
Figure
3. Pse
udoc
ode
for
Ecl
at
al
gorithm
5.2.
dEcl
at (
Di
ff
se
t
)
The
dEclat
(
dif
fer
e
nt
set
or
diff
set
)
is
pro
pos
ed
by
[5
]
wh
e
r
e
the
aut
hors
r
epr
ese
nt
a
n
it
em
se
t
by
ti
ds
that
app
e
ar
in
t
he
ti
ds
et
of
it
s
pr
e
fix
bu
t
do
not
ap
pear
i
n
it
s
ti
ds
et
s.
I
n
ab
breviat
io
n,
diffs
et
is
the
diff
e
r
ence
betwee
n
t
wo
(
2)
ti
ds
et
s
(i.e.
ti
ds
et
of
t
he
i
tem
set
s
and
it
s
prefi
x)
.
T
hro
ugh
dif
fset,
t
he
car
din
al
it
y
of
set
s
represe
nting
it
e
m
set
s
is
redu
ced
rig
orously
and
that
c
on
t
ribu
te
s
i
n
faster
intersect
io
n
a
nd
le
ss
m
e
m
or
y
u
sa
ge
.
C
on
si
der
an
e
quivale
nc
e
cl
ass
with
prefi
x
P
con
ta
in
s
the
it
em
se
ts
X
a
nd
Y
[
7]
.
Let
t(
X
)
de
no
te
s
the
ti
dse
t
of
X
an
d
d(X
)
de
no
te
s
t
he
di
ff
s
et
of
X.
When
us
in
g
ti
ds
et
for
m
at
,
we
will
ha
ve
t(P
X)
an
d
t(PY)
avail
a
ble
in
th
e
equ
i
valence
class an
d
t
o ob
ta
in
t(P
XY)
we
c
heck the c
ar
dina
li
ty
o
f
(
)
∩
(
)
=
(
)
.
Wh
e
n
us
i
ng
di
ff
set
form
at
,
we
will
ha
ve
d(PX)
instea
d
of
t(
PX
)
and
(
)
=
(
)
−
(
)
,
t
he
set
of
ti
ds
in
t(P
)
bu
t
not
in
t(X
).
Si
m
i
la
rly
,
we
hav
e
d(PY
)
=
t(P)
–
t(Y
).
S
o
the
sup
port
of
PX
is
not
the
siz
e
of
it
s
diffset
.
By
the
de
finiti
on
of
d(PX)
,
it
can
be
seen
that
|
(
)
|
=
|
(
)
|
−
|
(
)
−
(
)
|
=
|
(
)
|
−
|
(
)
|
. In othe
r wor
d,
sup
(
)
=
(
)
−
|
(
)
|
.
Re
fer t
o
t
he i
ll
us
trat
ion
i
n Fi
g
ure
4
.
Figure
4
.
D
if
fe
ren
ce
of ite
m
se
t A a
nd B
To
us
e
dif
fset
form
at
,
the
initial
transacti
on
database
in
vert
ic
al
layou
t
is
firstly
con
ve
rted
to
dif
fset
form
at
in w
hic
h
dif
fset of it
em
s
are
set
s o
f
ti
ds
who
se tra
nsa
ct
ion
s
do
no
t
co
ntain it
em
s.
This is d
e
du
ce
d
from
the
de
finiti
on
of
dif
fset,
the
init
ia
l
transacti
on
databa
se
in
ver
ti
cal
la
yout
is
an
eq
uiv
al
ence
with
t
he
pr
e
fix
P={}
,
s
o
the
t
idset
of
P
incl
ud
e
s
al
l
ti
ds
,
a
ll
transacti
ons
con
ta
in
P,
a
nd
the
dif
fset
of
an
it
e
m
i
is
(
)
=
(
)
−
(
)
,
this
is
a
set
of
ti
ds
w
hose
tr
ansacti
ons
do
no
t
c
onta
in
i
.
Fr
om
this
init
ia
l
equ
i
valence
cl
ass,
we
cou
l
d
ge
ner
at
e
al
l
i
tem
set
s
with
their
di
ffset
s
and
s
upports.
T
he
dE
cl
at
is
diff
ere
nt
from
Ecl
a
t
in
ste
p
5,
instea
d
of
ge
ne
rati
ng
a
new
t
idset,
a
ne
w
di
ff
set
is
ge
ner
at
ed.
T
he
pe
rfo
r
m
ance
and
m
em
or
y
us
age
of
dEclat
h
as
sho
wn
to
achieve
sig
nifi
cant
i
m
pr
ovem
ents
over
tra
diti
on
al
Ecl
at
(ti
ds
et
)
es
pecial
ly
in
den
se
dat
abase
.
But
wh
e
n
data
base
is
sp
ar
se,
it
loses
it
s
adv
anta
ges
ove
r
ti
ds
et
s.
The
n
in
[
5]
the
auth
or
s
s
uggeste
d
to
use
ti
ds
et
form
at
a
t
sta
rting
f
or
s
par
se
data
ba
se
and
la
te
r
s
wit
ch
to
diffset
f
or
m
at
wh
en
s
witc
hing
co
ndi
ti
on
is
m
et
. F
ro
m
this starti
ng
point,
po
st
dif
fset is
pro
posed
.
Evaluation Warning : The document was created with Spire.PDF for Python.
IS
S
N
:
2088
-
87
08
In
t J
Elec
&
C
om
p
En
g,
V
ol.
8
, N
o.
6
,
Dece
m
ber
2
01
8
:
4477
-
4485
4482
5.3.
C
om
-
Ec
l
at
(
So
r
td
if
fse
t
)
The
c
om
-
Ecl
at
(c
om
bin
at
ion
of
ti
ds
et
s
+
di
ff
set
s
a
nd
s
or
t
)
is
i
ntrod
uced
by
[7
]
t
o
e
nhance
dEclat
duri
ng
switc
hi
ng
c
onditi
on.
Wh
e
n
switc
hi
ng
process
ta
ke
s
place,
the
re
e
xist
ti
ds
et
s
w
hich
do
no
t
sat
isfy
th
e
switc
hing
c
ondi
ti
on
,
t
hu
s
thes
e
ti
ds
et
s
rem
ai
n
as
ti
ds
et
s
i
ns
te
ad
of
di
ff
set
f
or
m
at
.
The
sit
ua
ti
on
res
ults
in
both
ti
ds
et
s
an
d
dif
fsets
f
or
m
at
of
it
e
m
s
et
s
in
pa
rtic
ular
e
qu
i
va
le
nce
cl
ass
a
nd
the
ne
xt
inte
rsecti
on
pr
oces
s
will
involve
both
f
orm
at
s.
6.
POSTD
IFFS
ET
A
LGO
RI
THM
Po
stdi
ff
set
is
desig
ne
d
pri
or
to
sugg
e
sti
on
that
is
m
ade
in
[
5]
to
us
e
t
idset
form
at
at
sta
rting
for
sp
ars
e
data
bas
e
an
d
la
te
r
s
witc
h
to
diffset
f
or
m
at
wh
en
s
witc
hing
c
ondi
ti
on
is
m
et
.
C
on
ce
ptu
al
ly
,
b
y
giv
e
n
equ
i
valence
cl
ass
with
prefi
x
P
consi
sti
ng
of
it
e
m
set
s
in
so
m
e
or
der,
int
ersecti
on
of
w
it
h
al
l
with
j>i
is
to
be
perform
ed
in
orde
r
to
obta
in
a
ne
w
e
qu
i
valence
cl
ass
with
pref
ix
and
f
re
qu
e
nt
it
e
m
set
s
.
an
d
co
uld
be
in
ei
ther
ti
ds
et
or
dif
fset
form
at
.
If
is
in
dif
f
set
form
at
and
is
in
ti
dse
t
f
orm
at
,
(
)
∩
(
)
=
(
)
wh
ic
h
bel
ongs
to
the
e
qu
i
va
le
nce
cl
ass
of
pr
e
fix
,
not
as
exp
ect
e
d.
In
oth
e
r
w
ords,
i
n
orde
r
to
do
intersect
io
n
bet
ween
it
em
set
s
in
diffset
form
at
and
it
e
m
sets
in
ti
ds
et
fo
r
m
at
to
pro
du
ce
ne
w
equ
i
valence
cl
asses
prop
e
rly
,
it
e
m
s
et
s
in
ti
ds
et
form
at
m
us
t
sta
nd
be
f
ore
it
em
set
s
in
diffset
form
at
in
the
or
de
r
of
their
eq
uiv
al
enc
e
cl
ass.
That
can
be
a
chieve
d
by
sw
app
i
ng
(sorti
ng
)
it
e
m
set
s
in
diff
se
t
and
ti
dse
t
for
m
at
,
a
pr
oces
s
wh
ic
h
ha
s
th
e
com
plexity
O(n)
wh
ere
n
is
the
num
ber
of
it
e
m
set
s
of
t
he
equ
i
valence
class.
In
po
st
diffset
al
gorithm
,
the
fi
rst
le
vel
of
lo
opin
g
is
based
on
ti
ds
et
s
proce
ss,
fo
ll
ow
s
by
the
seco
nd
le
vel
onwards
of
lo
op
i
ng
are
getti
ng
the
res
ult
of
dif
fset
(
diff
e
re
nce
intersect
ion
set
)
be
tween
i
th
colu
m
n
and
i+
1
th
col
um
n
and
sa
ve
to
db.
Re
fer
ri
ng
to
F
i
gure
5
,
the
m
in_
sup
port
th
res
ho
l
d
value
is
de
te
rm
ined
in
te
rm
s
of
per
ce
ntage
w
he
re
the
us
e
r
-
spe
ci
fied
m
in_
suppo
rt
value
will
be
div
ide
d
by
100
an
d
m
ul
ti
ply
with
total
rows
(r
ec
ords
)
of
ea
ch
dataset
.
T
he
n
in
each
lo
op,
sta
rting
with
t
he
first
lo
op,
if
the
su
pp
or
t
is
gr
eat
er
t
han
or
equ
al
(>=)
to
m
in_
suppo
rt,
then
,
i
n
po
st
dif
fset,
the
first
le
vel
of
l
ooping
is
base
d
on
ti
ds
et
s
pr
ocess,
fo
ll
ows
by
the
seco
nd
le
vel
onwa
r
ds
of
loopin
g
are
getti
ng
the
res
ult
of
diff
set
(
dif
f
e
ren
ce
inte
rse
ct
ion
set
)
bet
ween
i
th
colum
n
an
d
i+
1
th
col
um
n
an
d save
to d
at
a
b
a
se.
:
(
(
1
,
1
)
,
…
(
,
)
)
|
)
,
:
(
,
)
1.
s
tart
2.
/
/
get mi
n_
s
upp
ort
3.
mi
n_
s
upp
=numbe
r_o
f_rows
*pe
rce
nt
age
_m
in
_s
upp
ort
;
4.
run t
ids
et for
firs
t l
oo
p;
5.
if
(s
upp
ort
<
=min
_supp
ort
){
6.
ad
d d
ata to
th
e nex
t proces
s
;
7.
ad
d d
ata in
to
db
8.
}
9.
end tidse
t
10.
/
/
for nex
t l
o
op
11.
s
tart l
oo
ping ;
12.
run diff
s
et;
13.
if
(s
upp
ort
<
=min
_supp
ort
){
14.
ad
d d
ata to
th
e nex
t proces
s
;
15.
ad
d d
ata in
to
db
16.
}
17.
end lo
opin
g.
18.
end dif
fs
et;
19.
flush valu
e
for curre
nt
/
last tran
s
actio
n data;
20.
end
Figure
5
.
P
os
td
iffset
ps
eu
doco
de
Evaluation Warning : The document was created with Spire.PDF for Python.
In
t J
Elec
& C
om
p
Eng
IS
S
N: 20
88
-
8708
Postd
if
fse
t Al
gorit
hm in
Rare
Patte
rn: A
n Imple
me
nta
ti
on v
ia Be
nchm
ar
k
Case
…
(
Wa
n Aezw
ani W.
A
.
B
.
)
4483
In
Fig
ur
e
5,
t
he
m
in_
sup
port
is
m
easur
ed
base
don
the
m
ul
ti
plica
ti
on
of
t
he
num
ber
of
r
ows
of
it
e
m
set
s
in
database
with
the u
ser
s
pecified p
erce
ntage
of
m
in_
sup
port.
The
n,
each
it
em
se
t
is
intersect
ed
wit
h
it
s
transacti
on
id
(tid)
f
or
t
he
first
lo
opin
g.
If
the
sup
port
of
each
it
e
m
set
is
le
ss
than
or
eq
ual
to
the
m
in_
sup
port
(item
set
in
this
conditi
on
is
very
rar
e
as
to
in
di
cat
e
the
it
e
m
s
et
of
ab
norm
al
and
pecu
li
ar
c
ases)
,
then
that
it
em
s
et
is
passed
to
the
seco
nd
le
ve
l
of
lo
op
i
ng.
Starti
ng
from
s
econdlo
opin
g
onwa
rd
s
,
each
ti
ds
is
intersect
ed
with
it
s
diff
e
re
nc
e
set
(d
if
fset)
un
ti
l
finis
h.
The
e
xp
e
rim
e
ntati
on
of
pos
tdiffset
al
gorit
hm
i
s
pr
ese
nted
in
t
he
n
e
xt secti
on.
7.
E
X
PERI
MEN
TATIO
N
All
exp
e
rim
ent
s
are
pe
rfor
m
ed
on
a
Dell
N
5050,
I
ntel
®
Pentium
®
CPU
B9
60
@
2.2
0
G
Hz
wit
h
8G
B
R
AM
in
a
W
in
7
64
-
bi
t
platfor
m
.
Th
e
so
ft
war
e
s
pe
ci
ficat
ion
f
or
a
lgorit
hm
dev
el
op
m
ent
is
dep
l
oyed
us
in
g
open
sou
rce
s
of
t
war
e
i.
e.
My
SQ
L
ve
r
sion
5.6
.20
–
My
SQ
L
c
o
m
mu
nity
ser
ve
r
(
GP
L
)
f
or
our
da
ta
bas
e
serv
e
r,
A
pache/
2.
4.1
0
(
W
i
n32)
Op
e
nSSL/
1.0.1i
PHP/
5.5
.15
f
or
ou
r
w
eb
se
rv
e
r,
php
as
a
pr
ogram
m
ing
la
nguag
e
an
d
phpMyA
dm
in
with
ver
si
on
4.2.7.
1,
the
la
te
st
sta
ble
ve
rsion
as
to
ha
nd
l
e
the
adm
inist
rati
on
of
My
SQ
L
ove
r
the
W
e
b
.
T
he
phpMyA
dm
in
[
91
]
is
a
fr
ee
s
of
t
war
e
to
ol
w
ritt
en
in
P
HP
,
t
hat
suppo
rts
a
wide
range
of ope
rati
on
s
on MyS
Q
L,
Ma
ria
DB and
Dr
iz
zl
e.
T
he
database
ch
a
ra
ct
erist
ic
s
is sh
o
wn in
Table
1
.
Table
1.
Datab
ase Cha
racteri
s
ti
cs
Datasets
Nu
m
.
o
f
Tr
an
sactio
n
s
Leng
th
(Attr
ib
u
te)
Size
(KB)
Categ
o
ry
Ch
ess
3196
37
335
Den
se
Mus
h
roo
m
8125
43
558
Den
se
Retail
8
8
1
6
2
68
5143
Sp
arse
T40
I10
D1
0
0
K
1
0
0
0
0
1
32
1
5
1
1
6
Sp
arse
7.1.
Em
piri
cal R
esul
ts
Fo
r
the
e
ase
a
nd
fa
st
ex
pe
rim
entat
ion
pur
po
ses,
we
hav
e
m
od
i
fied
dataset
s
to
be
on
ly
th
ou
s
an
d
rows
of
it
e
m
set
s
th
at
are
r
an
dom
l
y
processe
d
f
or
m
ining
pur
poses
.
Our
e
xperi
m
entat
ion
is
w
it
h
re
gards
to
dEclat
(d
if
fset)
,
com
-
Ecl
at
(so
rt
diffset
)
an
d
post
diffset
al
gorithm
becau
se
f
ro
m
ou
r
past
exp
e
rim
entat
i
on
on
po
st
dif
fset
i
m
ple
m
entat
ion
in
fr
e
qu
e
nt
it
e
m
s
et
m
ining
,
the
resu
lt
s
of
tradi
ti
on
al
-
Ecl
at
(tidset)
will
al
ways
be
the
la
st
in
pe
rfor
m
ance
an
d
m
e
m
or
y
us
ag
e
am
on
g
t
ho
se
three
(3)
al
gor
it
h
m
s.
Fig
ur
e
6
s
hows
the
gr
aph
of
perform
ance
evaluati
onwit
h
reg
a
rds
to
ex
ecuti
on
ti
m
e
(in
sec
ond)
w
it
hin
f
our
(
4)
dataset
s
i.e
.
chess
,
m
us
hr
oom
, r
et
ai
l and
T1
0I4
D
100K.
Re
fer
ri
ng
t
o
Figure
6,
in
de
nse
dataset
,
p
ost
diffset
lose
it
s
perform
ance
by
63
%
to
di
ff
s
et
and
44
%
to
s
or
tdi
ff
set
in
c
hess.
I
n
m
us
hro
om
,
po
st
diff
set
outpe
rform
with
23
%
in
dif
fset
a
nd
84%
in
sortdif
fse
t.
F
or
sp
ars
e
dataet
cat
egory,
po
st
diffset
trem
end
ously
outpe
rfor
m
with
94
%
an
d
95
%
to
diff
set
i
n
re
ta
il
an
d
T1
0I4D1
00K.
The
al
gorithm
con
ti
nues
to
outpe
rfor
m
dr
a
m
at
ic
ally
in
sortdif
fset
with
99%
both
i
n
ret
ai
l
and
T1
0I4D1
00K d
at
aset
.
Figure
6
.
Per
f
orm
ance o
n dif
f
set
, s
ort
dif
fset
and post
diff
set
in
c
hess, m
us
hro
om
, r
et
ai
l an
d
T
10I
4D1
00K
Evaluation Warning : The document was created with Spire.PDF for Python.
IS
S
N
:
2088
-
87
08
In
t J
Elec
&
C
om
p
En
g,
V
ol.
8
, N
o.
6
,
Dece
m
ber
2
01
8
:
4477
-
4485
4484
8.
CONCL
US
I
O
N AND F
UT
U
RE DIRE
CTI
ON
The
pe
rfor
m
ance
of
postdif
fs
et
var
ie
s
de
pe
ndin
g
upon
data
set
s.
It
is
be
st
execu
te
d
i
n
s
pa
rse
dataset
s
i.e.
retai
l
and
T1
0I4D1
00K
wh
il
e
in
de
nse
dataset
s
i.e.
chess,
it
loses
it
s
rep
utati
on
towa
r
ds
dif
f
set
and
so
rt
diffset
.
B
u
t
in
m
us
hro
om
,
po
st
dif
fset
did
well
am
on
g
t
he
ot
her
tw
o
(
2)
al
gorithm
s.
T
he
sim
ple
co
nc
lusio
n
can b
e
m
ade
w
her
e
the n
at
ure
of
dataset
s
in
t
erm
s
of
ho
w
m
any
tim
e
the
oc
currence
of
it
e
m
set
s
cou
ld
m
e
on
e
of
the
co
ntri
buti
ng
facto
r
to
the
overall
perf
or
m
ance
o
f
cer
ta
in
associat
ion
ru
le
in
fr
e
qu
e
nt
m
ining
al
gorithm
s.
Our
nex
t
fo
c
us
co
uld
be
the
enforc
em
ent
of
c
onfi
den
ce
le
vel
or
ot
her
interen
sti
ngne
ss
m
easur
e
to
ward
s
it
e
m
set
s r
at
her
than j
us
t f
oc
us
i
ng on m
ini
m
um
su
ppor
t
valu
e.
ACKN
OWLE
DGE
MENTS
We
wish t
o
t
ha
nk
al
l fac
ulty
m
e
m
ber
s for
s
upportin
g o
ur
work in
r
e
view
ing
f
or
sp
el
li
ng
erro
rs
a
nd
synch
ronizat
io
n
c
onsist
encies
and als
o
f
or th
e m
eaning
f
ul c
omm
ents and s
uggestio
ns
.
REFERE
NCE
S
[1]
R.
Agrawal
and
R.
Srikant
,
“
Fast
al
gorit
hm
s
for
m
ini
ng
associa
ti
on
rule
s”
,
in
P
roce
edi
ngs
of
2
0th
Inte
rnationa
l
Confe
renc
e
on
V
ery
Lar
ge
Data
Bases
(
VLDB)
,
1215,
487
–
499
,
1
994.
[2]
R.
Agrawal
,
et
al
.
,
“
Mining
ass
oci
a
ti
on
rule
s
bet
wee
n
sets
of
it
ems
in
la
rge
dat
aba
ses”
,
ACM
SIGMO
D
Re
cord
,
22(2),
207
–
216
,
1993.
[3]
J.
Han,
et al.
,
“
Mining
fre
qu
ent
pat
t
er
ns wit
hout
ca
ndid
at
e
gen
erati
on
”
,
ACM
SIG
MOD
Re
cord
,
2
9(2),
1
–
12
,
2000
.
[4]
M.J.
Z
aki
,
e
t
al
.
,
“
New
al
gori
th
m
s
for
fast
disc
over
y
of
association
ru
le
s”
,
In
P
roce
edi
ngs
o
f
th
e
ACM
SIGKD
D
int
ernati
ona
l co
nfe
renc
e
on
Kno
wle
dge
Discov
ery
and
Data
Min
i
ng
(
KDD
’97
)
,
283
–
286,
1997
.
[5]
M.J.
Za
ki
and
K.
Gouda,
“
Fast
ver
tica
l
m
ini
n
g
using
diffse
ts
”
,
In
Proc
ee
d
in
gs
of
the
nint
h
ACM
SIGKD
D
int
ernati
ona
l co
nfe
renc
e
on
Kno
wle
dge
Discov
ery
and
Data
Min
i
ng
.
326
–
335
,
20
03.
[6]
P.
Sheno
y
,
et al.
,
“
Turbo
-
cha
rg
in
g
ver
t
ic
a
l
m
ini
n
g
of
l
arg
e
da
ta
ba
ses
”
,
ACM
SIG
MOD
Re
cord
,
2
9(2),
22
–
33
,
200
0.
[7]
T.
A,
Tri
eu
and
Y.
Kunieda,
“
An
improvem
ent
for
dec
l
at
al
gori
thm”
,
In
Proc
e
edi
ngs
of
th
e
6t
h
Inte
rnationa
l
Confe
renc
e
on
Ubiquit
ous Inf
or
mation
Manage
ment
and
Comm
unic
ati
on
(ICUI
MC’12),
54
,
1
–
6
,
2012
.
[8]
J.
Hipp,
et
al.,
“
Algorit
hm
s
for
association
r
ule
m
ini
ng
a
ge
ner
al
surve
y
an
d
compari
son
”
,
ACM
SIGKD
D
Ex
plorati
ons
Ne
wslet
te
r
,
2(1)
,
5
8
–
64,
2000
.
[9]
J.
Han,
et
al
.
,
“
Freque
nt
pat
t
er
n
m
ini
ng:
cur
re
nt
stat
us
and
f
uture
dir
ec
t
ions
”
,
Data
Mini
ng
and
K
nowle
dge
Discov
ery
,
15(1)
,
55
–
86
,
2007
.
[10]
C.
Borge
lt
,
“
Eff
ic
i
ent
imple
m
ent
at
ions
o
f
apr
ior
i
and
ecla
t
”
,
In
Proceedi
ngs
of
the
IEEE
ICD
M
Workshop
on
Freque
nt
Ite
ms
e
t
Mini
ng
Imple
m
ent
ati
ons
(FIM
I03),
2003
.
[11]
L.
Schm
idt
-
Thie
m
e,
“
Algorit
hm
ic
fea
tur
es
of
ec
l
at
”
,
In
Proce
ed
i
ngs
of
the
IEE
E
ICDM
Worksh
op
on
Freque
nt
Ite
ms
et
M
ini
ng
I
mpleme
ntations
(FIM
I04),
2004.
[12]
M.J.
Z
aki
,
“
Scalable
al
gor
it
hm
s
for
association
m
ini
ng”
,
I
EEE
Tr
ansacti
ons
on
Knowle
dge
and
Data
Engi
n
ee
rin
g
,
12(3),
372
–
390
,
2000.
[13]
X.Yu
and
H.
W
ang,
“
Im
pr
ovement
of
e
cl
a
t
al
go
rit
hm
base
d
on
support
in
fre
qu
ent
item
set
m
ini
ng”
,
Journal
o
f
Computers
,
9(9), 2116
–
2123,
201
4.
[14]
B.
Goethals,
“
Freque
nt
se
t
m
ini
n
g
”
,
In
Data
M
ini
ng
and
Knowle
dge
Disco
ve
ry
Handbook
,
Springer,
321
–
338
,
2010
[15]
Borgel
t
,
C.
;
and
Krus
e,
R
.
;
“
Ind
uct
ion
of
associ
at
ion
rule
s:
Apri
ori
implement
at
i
on”
,
In
Comps
ta
t,
Spring
er,
395
–
400,
2002
.
[16]
A.
Savase
r
e,
et
al
.
,
“
An
eff
ic
i
en
t
a
lgori
thm
for
m
ini
ng
associati
on
rule
s
in
la
rge
databa
ses”
,
In
Proce
ed
ing
of
t
he
21th
Int
ernati
on
al
Conf
ere
nce o
n
Ve
ry
Lar
ge
D
a
ta
Bases
(V
LDB
'
95
), 432
–
444,
1
995.
[17]
H.
Toi
vonen
,
“
Sam
pli
ng
la
rg
e
da
ta
base
s
for
associa
ti
on
rule
s
”
,
In
Proce
ed
ing
of
th
e
22nd
Inte
rnat
i
onal
Confe
ren
c
e
on
Ve
ry
Lar
ge
D
ata
Bases
(V
LDB
'
96)
,
134
–
145,
1996.
[18]
J.
Han,
et
al
.
,
“
Mining
fre
quen
t
pat
t
ern
s
withou
t
ca
nd
ida
t
e
g
enerat
ion:
A
fre
qu
e
nt
-
pat
t
ern
tre
e
a
pproa
ch
”
,
Data
Mini
ng
and
Kno
wle
dge
Discov
ery
,
8(1
), 53
–
87,
2
004.
[19]
T.
Slim
an
i
and
A.
Lazz
ez,
“
Eff
ic
i
ent
ana
l
y
sis
o
f
pat
t
ern
and
as
s
oci
at
ion
rul
e
m
ini
ng
appr
o
ac
h
e
s
”
,
Int
ernati
onal
Journal
of
Infor
mation
Techno
lo
gy
and
Comput
e
r Sc
ie
n
ce
,
6(3)
,
70
–
81,
2014
.
[20]
M.
Man,
e
t
a
l.,
“
Spati
al
infor
m
at
ion
databa
se
s
int
egr
a
ti
on
m
odel
”
,
In
A.A.
Mana
f
et
a
l.
(Ed
s.):
ICIEI
S
201
1,
Informatic
s E
ng
i
nee
ring a
nd
Info
rm
ati
on
Scienc
e
,
Springer, 77
–
90
,
2011
.
[21]
S.
Shrivasta
va
a
nd
P.K.
Johari,
“
Anal
y
sis
on
high
uti
li
t
y
infre
q
uent
Ite
m
Sets
m
in
ing
over
tr
ansa
ctional
da
ta
b
ase
”
,
In
Re
c
ent
Tr
ends
in
El
ectronics,
Information
&
Com
municat
ion
Technol
og
y
(
RTEICT
)
,
IEE
E
Inte
rnat
iona
l
Confe
renc
e
on
p
p.
897
-
902
,
201
6.
[22]
M.A.
Thalor
an
d
S.
Pati
l
,
“
Inc
rement
al
Learni
ng
on
Non
-
stat
i
onar
y
Data
Stre
am
using
Ensemble
Approac
h
”
,
Inte
rnational
Jo
urnal
of El
e
ct
ri
c
al
and
Comput
er
Engi
n
ee
ring
,
A
ug
1;
6(4)
:
1811
,
2016.
[23]
G.
Bat
hla,
et
a
l.,
“
A
Novel
App
roa
ch
for
cl
ust
e
ring
Big
Data
b
ase
d
on
Map
Reduc
e
”
,
Int
ernat
ional
Journal
o
f
El
e
ct
rica
l
and
C
omputer
Engi
n
e
ering
(
IJE
CE)
,
J
un
1;
8(3)
,
2018
.
[24]
M.B.
Man,
et
a
l
.
,
“
Mining
As
so
ci
a
ti
on
Rule
s:
A
Case
Stud
y
on
Benc
hm
ark
Dense
Data
”
,
Indon
esian
Journal
of
El
e
ct
rica
l
Eng
in
ee
ring a
nd
Computer
Sc
ie
nc
e
on
pp.
546
-
553,
Se
p
1;
3(3)
,
2016
.
Evaluation Warning : The document was created with Spire.PDF for Python.
In
t J
Elec
& C
om
p
Eng
IS
S
N: 20
88
-
8708
Postd
if
fse
t Al
gorit
hm in
Rare
Patte
rn: A
n Imple
me
nta
ti
on v
ia Be
nchm
ar
k
Case
…
(
Wa
n Aezw
ani W.
A
.
B
.
)
4485
BIO
GRAP
H
I
ES
OF
A
UTH
ORS
W
an
Aez
wani
Bt
W
an
Abu
Baka
r
rec
eived
her
P
hD
in
Com
pute
r
Scie
nc
e
at
Univ
ersit
i
Mal
a
y
s
ia
Te
ren
gg
anu
(U
MT)
Te
r
engga
n
u
in
Nov,
2016.
Her
foc
us
area
i
s
in
associ
at
ion
rule
in
fr
eque
nt
it
ems
et
m
ini
ng.
She
rec
e
ive
d
h
er
m
aste
r’s
degr
ee
in
Master
of
Sc
i
enc
e
(Com
put
er
Scie
nc
e)
from
Univer
siti
T
ekn
ologi
Malay
sia
(UTM)
Skudai,
Johor
in
2000
prior
to
fini
shin
g
her
stud
y
in
Bac
he
lor’s
degr
ee
al
so
in
th
e
sam
e
strea
m
from
Univer
siti
Putra
Malay
si
a
(UP
M)
Serda
ng,
Sela
ngor
in
1998.
Her
m
aste
r’s
r
ese
a
rch
was
form
erly
on
Finger
print
Im
age
Segm
ent
at
ion
in
the
strea
m
of
Im
age
Proce
ss
ing.
Now
she’s
pursuing
her
rese
ar
ch
towar
ds
associa
ti
on
rel
a
ti
onship
in
infre
quen
t
i
te
m
set
m
ini
ng
which
is more
downs
tre
am t
o
educat
io
nal
d
at
a
set
ti
ngs.
Mus
ta
fa
Man
is
an
As
socia
te
Profess
or
in
Scho
ol
of
Inform
at
ic
s
and
Applie
d
Mathe
m
at
i
cs
and
al
so
as
a
Deput
y
Dire
ct
or
a
t
Rese
arc
h
Mana
g
ement
Innova
ti
on
Ce
ntre
(RMIC),
UM
T.
He
start
ed
his
PhD
studie
s
in
Jul
y
2009
an
d
fini
shed
h
is
st
udie
s
in
Com
put
er
Sci
enc
e
from
UTM
in
201
2.
He
has
recei
ved
Com
pute
r
Science
Diploma,
C
om
pute
r
Scie
nc
e
Degre
e
,
Master
s
Degre
e
from
UP
M.
In
2012,
he
has
bee
n
awa
rde
d
a
“
MIM
OS
Prestigi
ous
Awa
rds”
for
his
PhD
b
y
MIM
OS
Berha
d.
His
res
ea
rch
is
foc
us
ed
on
the
dev
el
op
m
ent
of
m
ult
iple
t
y
p
es
of
da
ta
b
ase
s
integra
t
ion
m
odel
and
al
so
in
Augm
ent
ed
Rea
lit
y
(AR),
a
ndroid
base
d
,
an
d
IT
r
el
a
te
d
int
o
ac
ross
dom
ai
n
pla
tform.
Masita
@
Masil
a
Abdul
Jalil
re
c
ei
ved
h
er
B.
Eng
(Hons
)
in
Comput
er
S
y
st
em
En
gine
er
ing
from
the
Univer
si
t
y
o
f
W
arwic
k,
UK
in
1997.
Aft
er
g
r
adua
t
ed,
she
joined
CEL
COM
(
M),
one
of
the
le
ad
ing
te
l
ec
om
m
unic
at
ion
prov
ide
rs
in
Malay
si
a
as
a
s
y
st
em
engi
nee
r
.
She
lat
er
pursued
her
Master
stud
y
i
n
Engi
nee
r
ing
Business
Mana
gement
at
th
e
sam
e
unive
rsit
y
bef
ore
joi
n
ing
Univer
siti
Ma
lay
sia
T
ere
ngganu
(UM
T
)
as
a
lec
ture
r
in
2001.
In
2012,
she
obtai
ned
her
PhD
in
Inform
at
ion
Te
c
hnolog
y
from
Univer
siti
Keba
ngsaa
n
Malay
si
a
(UK
M).
Her
cur
ren
t
rese
arc
h
int
er
ests
inc
lude
software
reu
se,
computer
science
educat
ion
a
nd
computer
appl
icati
ons
in
fore
nsics.
Julai
l
y
Aid
a
Jus
oh
recei
ved
h
er
B.
Eng
(Hons
)
in
Software
Eng
in
ee
ring
f
rom
the
Univer
siti
Putr
a
Malay
s
ia
(UP
M),
Sel
angor
in
20
04.
After
gra
dua
te
d,
she
furth
ered
her
Master
stu
d
y
in
Softwar
e
Engi
ne
eri
ng
in
Univer
siti
Mal
a
y
sia
Te
r
engga
nu
(UM
T)
in
2005.
In
200
9,
she
jo
ine
d
Univer
sit
i
Sulta
n
Za
in
al
Abidin
(UN
ISZA)
as
a
le
ct
ure
r
.
Now
,
she
furthe
re
d
her
PhD
studie
s
in
Univer
siti
Malay
s
ia
Teren
gganu
(UM
T)
si
nce
Sep
te
m
ber
2016.
She
cur
re
ntly
works
in
in
fre
quent
item
set
m
ini
ng
using
E
cl
a
t
Algorit
hm
for
her
PhD
re
sea
rch
.
Her
cur
ren
t
res
ea
rch
in
te
rests
in
cl
ud
e
software
engi
ne
e
ring,
form
a
l
m
ethods
and
pa
ttern
m
ini
ng.
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