SNN: A Supe ised Clus e ing Algo i hm
Jesús S. Aguila , Robe o Ruiz, José C. Riquelme, and Raúl Gi áldez
Depa men o Compu e Science. Uni e si y o Se illa
A da. Reina Me cedes s/n. 41011 Se illa. Spain.
[email p o ec ed]
Abs ac . In his pape , we p esen a new algo i hm based on he nea es
neighbou s me hod, o disco e ing g oups and iden i ying in e es ing
dis ibu ions in he unde lying da a in he labelled da abases. We in oduces he
heo y o nea es neighbou s se s in o de o base he algo i hm S-NN (Simila
Nea es Neighbou s). T adi ional clus e ing algo i hms a e e y sensi i e o he
use -de ined pa ame e s and an expe knowledge is equi ed o choose he
alues. F equen ly, hese algo i hms a e agile in he p esence o ou lie s and
any adjus well o sphe ical shapes. Expe imen s ha e shown ha S-NN is
accu a e disco e ing a bi a y shapes and densi y clus e s, since i akes in o
accoun he in e nal ea u es o each clus e , and i does no depend on a use -
supplied s a ic model. S-NN achie e his by collec ing he nea es neighbou s
wi h he same label un il he enemy is ound (i has no he same label). The
de e minism and he esul s o e ed o he esea che u n i in o a aluable ool
o he ep esen a ion o he inhe en knowledge o he labelled da abases.
Keywo ds: clus e ing, supe ised lea ning, nea es neighbou s.
1. In oduc ion
In he a ea o he supe ised lea ning he e a e se e al echniques o classi y a new
example om he labelled da abase om which he inhe en knowledge has been
ob ained. The o m in which i es a es he knowledge is dependen on he echnique
(decision ules, decision ees, associa ion ules, e c.); howe e , some me hods do no
p o ide ha knowledge limi ing hemsel es o ca y ou he classi ica ion (neu onal
ne wo ks, Bayesian model, nea es neighbou s, e c.).
F om he wo ks o [3], [5], [9], [6], [7], [4], [10], [11], o mo e ecen ly [12], [8],
and [1] he esea ch has been mainly ocused on he con e gence o he me hod, he
sea ch o p o o ypes o su aces o sepa a ion, he echniques o edi ing and
condensing and in he accele a ion o algo i hm. Howe e , he e has no been any
in e es on p o iding o he echnique o he nea es neighbou s a o m o ep esen
he inhe en knowledge o he in o ma ion.
Clus e ing, in Da a Mining, is a use ul echnique o g ouping da a poin s such ha
poin s in a single clus e ha e simila cha ac e is ics (o a e close o each o he ).
T adi ional clus e ing algo i hms a e applied in he a ea o he lea ning non-
supe ised.
S-NN employs a no el hie a chical clus e ing algo i hm based on he nea es
neighbou echniques. S-NN s a s wi h each inpu as a sepa a e clus e and a each
successi e s ep me ges he clus e s wi h iden ical neighbou s. We collec all he
L. Monos o i, J. Váncza, and M. Ali (Eds.): IEA/AIE 2001, LNAI 2070, pp. 207-216, 2001.
neighbou s ha hei dis ances a e sho e han he i s enemy, ha is o say, wi h no
he same label.
The emainde o he pape in o ganised as ollows. In sec ion 2 and 3, we su ey
basis con en s o he heo y o he nea es neighbou s’ se s. These ha d de ini ions
allow us apply he supe ised clus e ing algo i hm. The s ep in ol ed en clus e ing
using S-NN a e desc ibed in Sec ion 4. In Sec ion 5, we p esen he esul s o ou
expe imen s. Sec ion 6 concludes and p esen s ou ideas o u u e wo k.
2. Basic Concep s
Be o e beginning o desc ibe he nea se heo y, we ha e o men ion he concep s o
he classic heo y o se s ha a e necessa y o he de elopmen o ha heo y. We
will use he ope a ions known on se s: ³, , ¬ and # (ca dinal o a se ). Also we will
use he logic ope a ions on he se {F, T} ( alse and ue): ¾, ¿, and ½; and he
ollowing gene alisa ions: "
(uni e sal quan i ie ) and $
(exis en ial quan i ie ),
whe e
)(...)()().(: 21 n
xExExEExDiz ∧∧≡∀= )
)()()()·(: 21 n
xExExEExDiz ∨∨∨≡∃= K
(1)
and z is T i all he exp essions (i some o he exp essions) E(xi) a e T in he domain
D o ),...,( 1n
xxx =, i we conside he uni e sal quan i ie (exis en ial quan i ie ).
De ini ion 1 (Sequence): a sequence is a ini e o in ini e collec ion o elemen s wi h
an inhe en o de o access (sequen ial). I is always begun by i s and o accede o
any elemen i, i will be necessa y o pass h ough i-1 p e ious. Since i s de ini ion is
inhe i ed o se s, i also inhe i s he ope a ions associa ed o hese ³, , ¬, - and #.
Likewise, we de ined he ollowing ope a ions o sequence S o elemen s o T
ype: <>:S (emp y sequence); _+_:ST S (inse ion o an elemen in he end o
he sequence); [_]:S
N T (access o i h elemen o he sequence, wi h
i ³{1... #S}); and E·D:i+(gene alised conca ena ion o sequences), whe e
s(k) ...s(2)s(1))(}·..1{: +++=+iski (2)
wi h s(i) sequences. By con enience, i will be w i en as )(
1is
k
i=
+.
De ini ion 2 (O de ed Sequence): a sequence s, o size #s is o de ed i i sa is ies:
[] [ ]
1}·1)..(#1{: +
≤
−∀ isissi T(3)
whe e T is an es ablished ela ion o o al o de be ween he elemen s o T ype o he
sequence.
3. De ini ions
De ini ion 3 (A ibu e): a ibu e A is de ined by a se o alues. The a ibu e can be
con inuous o disc e e. I he a ibu e is con inuous, he se o alues will be limi ed
by he ex eme alues o an in e al, o ming he e o e he ank o alues o he
a ibu e. I he a ibu e is disc e e, he se o alues o his one will appea like an
enume a ion o he possible alues o he a ibu e. We will name C he se o alues
ha can adop he label.
De ini ion 4 (Example): an example E is one ow o med by he Ca esian p oduc o
he a ibu es o condi ion and decision. Likewise, we de ined he ollowing
ope a ions o ge o he a ibu es o condi ion o hei label.
CE:e iqANE:a →→× (4)
De ini ion 5 (Uni e se): he uni e se U is a sequence o examples. We will say ha a
da abase wi h n examples, each one o hem wi h m a ibu es ( he las one is
denomina ed label), will o m he pa icula uni e se om his momen . Then
U=<u[1]...,u[n]>.
Since we will model he da abase wi h a sequence, he access o an example o he
da abase will be made by means o he access o he sequence, ha is o say, he
sequence is s, hen s[i] ep esen s he example i h o he da abase. To accede o j h
a ibu e o he example, since we ha e modelled o his one wi h one ow, one will
become a (s[i],j), and o know i s label, e iq(s[i ]).
De ini ion 6 (Dis ance): he dis ance be ween wo examples is a unc ion ha ul ils
he p ope ies o a me ic space, ha is o say,
{}
0: ∪ℜ→× +
EEd (5)
wi h he ollowing p ope ies: e lec i e, de ined nonnega i e, symme ical and
ansi i e.
Since he examples belong o a sequence, we can ede ine he dis ance basing on
he posi ion ha hese examples occupy in he sequence, he e o e, compu e he ange
be ween wo examples ei and ej, we will do d(i,j).
De ini ion 7 (Sequence o Dis ances): sequence SD(i) o med by he dis ances o an
example i o all he o he s, and is de ined by
)),(,()(
#
1jidjiSD
s
j=
+=(6)
In he uni e se U, he sequence associa ed o he i s example will be:
SD(1)=<(1,d(1,1)),(2,d(1,2)),(3,d(1,3))> and each elemen o he sequence a e a pai
o med by he posi ion o he example o which i is wan ed o compu e he ange
and he alue o he dis ance. Hence, in he exp ession (j,d(i,j)) he i s coo dina e is
he index o an example and he second coo dina e is he dis ance o an example i o
he example whose index is indica ed in he i s coo dina e. In o de o access o
each one o he wo coo dina es easily we de ined wo ope a ions on he pai :
ℜ→ℜ×→ℜ× N:dis NN:ind (7)
De ini ion 8 (O de ed Sequence o Dis ances): as o a sequence P o pai s (index,
dis ance), we can ob ain a sequence Q o de ed by he dis ance i his one ul ils he
ollowing p ope y:
[]
()
[]
()
[]
()
[]
()
jQindkPind}·P..#1{:k}·Q..#1{:j1jQdis jQdis }·1Q..#1{:j =∃∀∧+≤−∀ (8)
ha we can be esumed like o de ed(Q,P). A o de ed sequence OSD(i) o dis ances
ul ils o de ed(OSD(i), SD(i)).
De ini ion 9 (Rela ion o Neighbou hood): wo examples whose posi ions in he
o de ed sequence a e i and j a e neighbou s o a hi d k i in he o de ed sequence o
dis ances o example k, OSD(k), any example be ween i and j does no exis (o
be ween j and i) whose label is di e en om which hey ha e i and j. The e o e, i
examples i and j do no ha e he same label, a e no neighbou s. The ela ion can be
de ined o he ollowing way:
Rk={(u[ind(OSD(k)[i])],u[ind(OSD(k)[j])])³U2|"h:{min(i,j)+1..max(i,j)-1})
·(e iq(u[ind(OSD(k)[i])])=e iq(u[ind (OSD(k)[j])])=e iq(u[ind(OSD(k)[h])])}
(9)
Fo example, i is OSD(1)=<(1,0),(4,1),(5,2),(34,3),(3,4)>, we will say ha
examples 4 and 3 a e neighbou s o 1 i examples 4 and 3 ha e he same label and all
he examples ha a e be ween hese (5 and 34) ha e he same label ha hose oo.
De ini ion 10 (Class o Neighbou s o an Example i espec o ano he Example
k): class o neighbou s o an example i is de ined espec o ano he k like all hose
ha in he o de ed sequence o dis ances o k can be g ouped a ound i in a egion o
examples o he same class. The class is de ined om he neighbou hood ela ion as i
ollows: [ i]k={j³N | u[i ] Rk u[j]}
This class o neighbou s can also be unde s ood like a sequence, because
in insically an o de ela i e o he dis ance exis s [2].
De ini ion 11 (O de ed Subsequence o O de k espec o an Example i): gi en
o he ela ion o neighbou hood and he de ini ion o class o neighbou s, we can
cons uc he o de ed sequence o dis ances o an example i om he conca ena ion o
o de ed subsequences, ha is o say,
OSD(i)=OSD(i)1+OSD(i)2+... +OSD(i)k+... +OSD(i)z(10)
whe e e e y subsequence OSD(i)k is cons uc ed om he classes o sepa a es
examples in ela ion o i. This way, each OSD(i)k is a class o examples whose
a ibu e o decision is he same one, bu ha di e s om he decision a ibu e o he
classes OSD(i)k-1 and OSD(i)k+1. The e o e, we could ep esen he da abase ( oge he
wi h he in o ma ion ela i e o he dis ances) o he ollowing way:
OSD(1)=OSD(1)1+OSD(1)2+... +OSD(1)k1
OSD(2)=OSD(2)1+OSD(2)2+... +OSD(2)k2
………………………….…………………
OSD(n)=OSD(n)1+OSD(n)2+... +OSD(n)kn
(11)
whe e Ki ³{1..n}; howe e , i some Ki we e 1, all he examples would belong o he
same class, and he mo e i app oaches n, in p inciple, he mo e homogeneously
dis ibu ed will be he examples o he same class in he da abase.
Since he sequence OSD has go associa ed wi h each elemen an example and he
dis ance o i , we a e going o do wi hou he dis ance now o associa e o each
example o he da abase a sequence o classes, whe e he conca ena ion o all o hem
will g oup he o al o examples o he da abase. Hence we will ha e associa ed o
each example (p eceding o he symbol ) an inde ini e numbe o classes in a speci ic
o de ha implici ly con ains he in o ma ion abou he "p oximi y" o hese o he
example o he beginning. Then,
[1] [1]1+[1]2+…+[1]k1
[2] [2]1+[2]2+…+[2]k2
……………………………………
[n] [n]1+[n]2+…+[n]kn
(12)
i indica es ha example k has a class o neighbou ing examples [k]1 whose labels a e
he same ones ha he one o k. A e wa ds, he e is ano he class o neighbou ing
examples [k]2 whose labels a e di e en ha hose o he p e ious class [k]1 and he
la e class [k]3. And so on. To hese classes, [i]j, we will denomina e classes j-
neighbou s o an example i.
F om a ma hema ical poin o iew, we ha e ob ained he join quo ien acco ding
o he ela ion o neighbou hood R o each example o he uni e se:
{}
[]
()
R
iOSD
iUi =⋅∀ ..#1: (13)
Fo example, i OSD(1)=<(1,0),(4,1),(5,2),(34,3),(3,4)> + <(73,5),(2,6),(31,7)>+
... we a e indica ing ha examples 1, 4, 5, 34 and 3 ha e he same label ( alues 0, 1,
2, 3 and 4 would be he dis ance o each example o example 1), ha in addi ion
di e s om he one o examples 73, 2 and 31. F om his we can he e cons uc he
class [1] o he ollowing o m: [1][1,4, 5, 34,3]+[73, 2, 31]+.…The class 1-
neighbou o 1 is [1, 4, 5, 34,3]; he class 2-neighbou o 1 is [73, 2, 31]; and so on.
De ini ion 12 (Class o O de k j-Neighbou o an Example i): he class o o de 0
is de ined 1-neighbou o an example i,
[]
0
1
i, like he class o neighbou s o an example
i espec o i sel . Tha is o say, since in he o de ed sequence o dis ances o an
example i, OSD(i), he i s example always will be he own i, since he dis ance o
i sel is 0, he class o o de 0 1-neighbou o example i will be he se o
neighbou ing examples o his one whose label is he same one, o o ano he o m,
hey will be hose ha belong o OSD(i)1.
On he o he hand, he class o o de 0 j-neighbou o an example i will be
[]
0
j
i. We
a e specially in e es ed on he classes o o de k 1-neighbou s. We de ine hen he
o de class 1 1-neighbou like:
[] [] []
[]
U0
1
0
1
0
1
1
1
ik
kjjii
∈
∈∪= (14)
The in e p e a ion o his exp ession is he ollowing one: since i con ains all he
examples wi h he same label han i ha is nea e o i (including i ) un il inding
ano he example o di e en label, he class
[]
1
1
i has o hose con ained in
[]
0
1
i plus he
1-neighbou s o hem. Fo example, in Fig. 1 we ha e
[]
0
1
i=<i,1,2,3,4,5> ( he 6 does
no belong o i , i has ano he label).

i
5
3
2
1
4
6
Fig. 1.
In gene al, he class o o de k 1-neighbou o an example i is de ined as:
[] []
U
kj
jk ii
<≤
=
011 (15)
And, he e o e, he class o o de k j-neighbou o an example i is de ined as:
[] []
U
kh
h
j
k
jii
<≤
=
0
(16)
By con enience, we will speak o k-class ins ead o class o o de k, and he e o e,
k-class j-neighbou , ins ead o class o o de k j-neighbou . In pa icula , we a e
in e es ed on he k-classes 1-neighbou s, and when we will speak abou hem we omi
subsc ip 1, ha is o say, ins ead o
[]
k
i1we will w i e
[]
k
i. Only when he
neighbou hood o de is di e en om 1 we will exp ess his o de .
De ini ion 13 (Equali y o Classes): wo classes a e equal when bo h con ain exac ly
he same examples, al hough in di e en o de . Fo mally,
[] [ ] [] [ ] [ ] []
()
iejejeieji ∈⇒∈∀∧∈⇒∈∀⇔= (17)
De ini ion 14 (Se o k-Classes 1-Neighbou s): we de ine he se o k-classes 1-
neighbou like he se o med by he k-classes 1-neighbou o each example. Also,
by con enience, k-se o neighbou ing classes will be named, ins ead o se o k-
classes 1-neighbou , and i will w i en like SNk whe e
[] [ ] [ ]
{}
kkk
knSN ,...,2,1=(18)
De ini ion 15 (Reduced k-Se o Neighbou ing Classes): we de ine k-se educed o
neighbou ing classes as he se o k-classes 1-neighbou s whe e he e a e no wo
equal classes. Fo mally,
[] [] [] []
{}
kk
K
k
k
k
kjijiSNjSNiRSN ≠⇒≠⋅∈∀∈= (19)
We will iden i y he educed se s o neighbou ing classes wi h he examples ha
ha e been educed so ha we do no lose in o ma ion, ha is o say, i he class [i] and
he class [j] a e equal, since hei neighbou s a e he same [w1...,wk], hen [i, j] has as
neighbou s o [w1...,wk].
4. Algo i hm ‘‘Simila Nea es Neighbou " (S-NN)
Once seen all he necessa y de ini ions ha suppo he heo y ha we p esen in his
wo k, we desc ibe he de ails o ou algo i hm in igu e 2.
S-NN (U: Da abase) e (RSN: Se o Classes)
0←i
{}
←
i
SN1
Fo each example j de U
[]
{}
i
ii jSNSN 111 ∪←
{}
←
−1
1
i
RSN (by con enience RSN1-1)
While 1
11
−
≠ii RSNSN
()
ii SN educ ionRSN 11 ←
{}
←
+1
1
i
SN
Fo each
[]
i
iRSNj 11 ∈
Fo each
[]
i
jk 1
∈
[] [] []
iii kjj 1
1
1
1∪←
+
[]
{}
1
1
1
1
1
1
+
++ ∪← i
ii jSNSN
1+← ii
Fig. 2. Algo i hm
The Inpu pa ame e s a e he U da abase, con aining n examples wi h m a ibu es.
As we men ioned ea lie , s a ing wi h he indi idual poin s as indi idual clus e s, a
each successi e s ep he clus e s wi h iden ical neighbou s a e me ged. The p ocess is
epea ed un il we can no simpli y he se o clus e s.
S-NN ea s each inpu poin as a sepa a e clus e , in each i e a ion o he while-
loop, un il we can no simpli y he se o clus e s, we compu e he neighbou s o each
clus e membe .
educ ion (C:Se o Classes) e (RSN:Se o Classes)
CRSN ←
Fo each (x,y) con x, y ³ C
I [x]=[y] ([x]=[x][y]=[x]¬[y])
{}
yRSNRSN −← yxx ∪←
Fig. 3. Reduc ion
The exp ession )
i
1
(SN educ ion
i
1
RSN ← in okes o he ollowing algo i hm
shown in igu e 3, whose assignmen is o simpli y he se o classes by means o he
elimina ion o hose classes ha ha e exac ly he same neighbou s. I he e a e wo
classes x and y and hey ha e he same neighbou s, hen he examples o y a e added
o hose o x, which bo h, will ha e exac ly he same neighbou s.
5. Resul s
5.1. I is
We ha e used he da abase I is o illus a e he comple e esul s o he me hod
because hey a e possible o be included in he a icle. Howe e , we eg e no o be
able o include, by lack o space, he in e media e esul s (se SN and RSN o each
o de o i e a ions).
The nex able con ains wo ypes o ows:
 odd ows: [class, examples o he class, neighbou s o he examples o he class].
The i s alue e e s o he class o labels; he second indica es how many
examples belong o he class ha has he men ioned label; and he hi d alue
co esponds wi h he numbe o neighbou s ha ha e go ha class.
 e en ows: he example o he class a e placed on he le column, whose ca dinal
co esponds wi h he second numbe o he p e ious ow; and in he igh column
he neighbou s o he examples o he class a e placed, o he le column, and has
go as ca dinal he hi d alue o he p e ious ow.
[A,50,50]
1, 6, 10, 18, 26, 31, 36, 37, 40, 42, 44, 47, 50, 51, 53, 54, 55,
5
8, 59, 60, 63, 64, 67, 68, 71, 72, 78, 79, 87, 88, 91, 95, 96,
100, 101, 106, 107, 112, 115, 124, 125, 134, 135, 138, 139,
143, 144, 145, 149, 136
1, 95, 106, 55, 36, 64, 125, 88, 107, 112,145, 134, 72, 67, 63, 37, 100,
31, 54, 135, 50, 47, 68, 6, 18, 101, 144, 78, 42, 143, 149, 139, 53, 51,
2
6, 115, 40, 10, 44, 96, 60, 124, 91, 58, 79, 138, 87, 59, 136, 71
[C, 47, 48]
2, 4, 17, 21, 23, 24, 39, 41, 45, 73, 80, 89, 102, 110, 126, 7,
13, 15, 20, 35, 27, 49, 104, 56, 57, 74, 148, 77, 81, 83, 111,
122, 123, 127, 131, 132, 146, 16, 75, 32, 34, 46, 52, 62, 82,
108, 137
2, 57, 122, 83, 131, 4, 15, 132, 13, 35, 81, 27, 41, 111, 123, 74, 17,
102, 23, 20, 127, 148, 34, 7, 80, 146, 46, 32, 75, 16, 5, 56, 49, 77, 126,
5
2, 104, 110, 73, 24, 62, 45, 108, 137, 82, 39, 21, 89
[B, 47, 49]
3, 28, 113, 8, 11, 14, 76, 85, 86, 116, 109, 121, 129, 19, 29,
30, 43, 66, 70, 99, 33, 98, 48, 133, 38, 61, 119, 65, 69, 84, 150,
9
3, 92, 94, 97, 103, 105, 114, 141, 118, 120, 140, 128, 130,
142, 22, 147
3, 92, 141, 113, 142, 119, 61, 29, 128, 11, 117, 103, 130, 28, 118, 147,
2
2, 69, 30, 105, 114, 76, 86, 66, 94, 19, 121, 43, 99, 116, 85, 65, 8, 109,
14, 33, 140, 98, 133, 70, 48, 129, 93, 38, 9, 150, 97, 84, 120
[C, 1, 1]
55
[B, 1, 1]
99
[B, 1, 1]
12 12
[C, 1, 1]
25 25
[C, 1, 1]
90 90
[B, 1, 1]
117 117
Fo he da abase I is 4 i e a ions ha e been needed, in each one o which he
ca dinal one o se RSN has been: 98, 62, 19 and las he 9 ha is in he able.
The me hod o e s a e y aluable in o ma ion because i p o ides:
 The numbe o egions: 9.
 I he examples o he class ag ee wi h he neighbou s ( ac ha happens o he
class A) hen he egion is clea ly sepa able o he es .
 Which a e he examples ha make di icul he classi ica ion (5, 9, 12, 25, 90 and
117), he e o e, we could ex ac hem om he da abase o a la e classi ica ion.
 An es ima ion o he e o a e on he aining ile (le us ake in o accoun ha i
we keep he h ee i s egions we would be a ound 96%, ha is app oxima ely
wha hey p o ide o he good so keys).
5.2. B eas Cance
Fo his da abase (wi h he 683 examples wi hou noise), and needing 5 i e a ions we
ha e ob ained 44 egions. The egions calcula ed in each i e a ion a e: 440, 308, 142,
45 and 44. The wo i s a e s ops A (427 examples) and o B (210 examples), which
means ha only wi h hese wo we would be a ound he 93,26% o he in o ma ion.
Rega ding he compu a ional cos o he algo i hm, he execu ions ha e been made
in a PC Pen ium 550 MHz and o he da abase I is i uses less han 1 second; o he
b eas cance da abase i uses 2 minu es.
6. Conclusions
The de ini ions p esen ed in his a icle base he heo y ha i suppo s on algo i hm
S-NN. The algo i hm, besides does no need pa ame e s is de e minis . As o he
in o ma ion ha i p o ides, in he example I is i is demons a ed ha i is able o
ob ain: a geome ic idea o he dis ibu ion o examples o he da abase; an es ima ion
o he numbe o egions (possible ules); an es ima ion o he di icul y o
classi ica ion o he da abase; and which a e he examples ha make di icul he
lea ning o he da abase, wi h a iew o elimina e hem in he phase o lea ning.
On he o he hand, algo i hm S-NN allows some in e es ing di ec ions, which we
a e s udying, as a as he educ ion c i e ion is conce ned (i see poin 2,1 o he
educ ion algo i hm). Th ee c i e ia o educ ion exis : a es ic i e c i e ion ( he one
ha a he momen is applied, ha is o say, hey will be educ ion i he wo classes
a e exac ly equal); a mode a e c i e ion ( he e will be educ ion i one o he classes
is included in he o he ); and, inally, a elaxed c i e ion ( he e will be educ ion i he
in e sec ion o he classes is no emp y). These c i e ia p o ide di e en solu ions, as
much mo e nume ous as o egions, as es ic i e is he educ ion c i e ion. The
cha ac e is ics o he con ibu ed solu ions as well as hei di e ences, bo h analy ical
and geome ically, will be objec o nex wo ks. In he same way, ano he in e es ing
line is he use o he se o classes o neighbou s like so key.
Acknowledgemen s. This wo k has been suppo ed by he Spanish Resea ch Agency
CICYT unde g an TIC99-0351.