Anne Berry
Jean-PaulBordat y
AlainSigayret
Abstra t
Generating on epts de nedby a binary relationbetween a setP of properties
and a set O of obje ts is one of the important urrent problems en ountered in
Data Mining.
We presenta new algorithm whi h generates ea h on ept exa tly on e, using
graph-theoreti results. Thispro esshasatime omplexityofO(jPjm)permaximal
hain of the on ept latti e, where m denotes the number of non- rosses of the
relation,andusesadatastru turewhi hisofsmallpolynomialsize. Thisimproves
the urrent best-time algorithms for this problem, whi h require either O(2 jPj
)
spa e and O(jPj 2
) time per on ept, or, alternately, polynomial spa e but O(jPj 3
)
timeper on ept.
Our algorithm an be used, with no extra ost, to ompute the edges of the
latti e,and an, just as eÆ iently,generate onlyfrequent sets.
1 Introdu tion
Inthe ontext ofData Base Management andData Miningproblems, databases are oftenrepre-
sentedbya binaryrelationbetweena setP ofpropertiesand aset Oof obje ts. Oneof thetools
foranalyzing the data ontainedin the base is to ompute all possible ombinations of elements
of the relationinto maximal re tangles. These re tangles, alled on epts, are organized into a
hierar hi al stru ture alled a on ept latti e. This theory,thoughstudied bymathemati iansas
farba kastheNineteenth Century(see [1 ℄,[10 ℄), was madepopularand developed byWille and
his team ([10 ℄), and remains one of theimportant urrent trends ofresear h inData Mining and
Arti ialIntelligen e: on ept latti es areused in elds asvaried asthe dis overy of asso iation
rulesinData Bases([24℄), thegenerationoffrequent itemsets([23 ℄), ma hinelearning([15 ℄,[17 ℄)
andthe reorganizationof obje thierar hies([13℄, [5 ℄).
The main drawba k to this approa h is that a on ept latti e is, in general, of exponential
size. Asa onsequen e,itisofprimaryimportan etobeabletogenerate on eptseÆ iently,even
whenonlypart ofthe latti e isexplored.
Con ept generation has given rise to a steady ow of resear h for thepast thirty years. One
of the rst algorithms to be publishedin this eld is dueto Chein ([8 ℄); this algorithmgenerates
su essivelayersof thelatti e,byde ningpossible andidatesby ombinationsof on epts ofthe
previouslayer; ithas anexponentialworst-time omplexitypergenerated on ept (see [11 ℄).
Bordat'salgorithm ([7 ℄)wasan improvement,asit runsinO(n 3
)per on ept, wheren=jPj,
inaBreadth-Firstfashion;oneoftheinterestingfeaturesofthisalgorithmisthatitalso omputes
alltheedges ofthelatti e. Thistime omplexitywasre entlyimprovedbyNourineandRaynaud
([18℄)to O(n 2
) per on ept.
Allthese algorithmsrequireexponentialspa eand storethe omputed on epts.
LIMOS,bat. ISIMA,63173Aubiere edex,Fran e. Mail: berryisima.fr,sigayretisima.fr
y
LIRMM,161RueAda,34392 Montpellier,Fran e. Mail: bordatlirmm.fr.
lembe omes easier,though therunning timeper on ept is higher: thebest su h algorithm, due
toGanter([9 ℄), runsinO(n 3
)timeper on ept, usingtheinteresting notionof le ti order, whi h
avoids s anningallthepossiblesubsetsofproperties,without,however,avoidingre- omputingthe
same on ept O(n) times.
Inthispaper,we addresstheissueofeÆ iently omputingallthe on epts, en ounteringea h
exa tlyon e, usingonlypolynomialspa e.
Our ontribution isan algorithm whi h runs inO(nm) timepermaximal hain,where m de-
notesthenumberofnon- rossesoftherelation,andrequiresonlysmallpolynomialspa e(O(nm)).
This greatly improves [18 ℄, as we have a omparable omplexity without requiring exponential
spa e; it also signi antly improves Ganter's O(n 3
) time per on ept. Furthermore, the number
of on epts tends to be ome exponentialwhen therelationis dense; inthis ase, m is of order n,
andour omplexitybe omes O(n 2
) permaximal hain.
The algorithm we introdu e here presents similarities with [7 ℄. It uses ea h on ept AB
to generate the on epts whi h are just above AB in the latti e ( alled the over of AB),
working atea h step on thesub-relationR ((P A);B).
One of the new features whi h is addedis that ea h on ept inherits informationon the pre-
viously pro essed on epts, in order to avoid generating the same on ept more than on e, a
breakthroughin on ept generation.
Another di eren eis that we usea re ursive Depth-First Sear h of the latti e, whi h enables
to storeonlya polynomialnumberof bytes, asa on ept latti e, though itmaybeexponentially
large, is of small height (O(n)). Moreover, the fashion in whi h we ompute the over of ea h
on ept isdi erent.
Our approa h is based on ourexperien e on graphs. In [7℄, Bordat useda bipartite graphto
handlethe relation. In [4 ℄, Berry and Sigayret proposed a di erent en oding into a o-bipartite
graphforwhi hthey establisheda one-to-one orresponden ebetween the on eptsof thelatti e
andthe minimalseparators ofthe graph.
This is algorithmi ally interesting be ause, in the past de ade, mu h resear h has beendone
onusingminimalseparatorstoeÆ ientlysolvevariousgraphproblemssu has hordalembedding
([19℄,[2 ℄),andinparti ularseveralpapersdealwiththeeÆ ientenumerationofminimalseparators
([14℄, [21 ℄,[20 ℄,[3℄).
[4℄ pointed out that, using the underlying o-bipartite graph and these re ent results on the
emergingtheoryofminimalseparation, the urrent best algorithmsforgenerating on epts ould
easily be mat hed both in terms of time and spa e. In this paper, we use graph properties to
improve these. Thoughwewillnotexpli itlyuse resultsonminimalseparation, they underlyour
approa h (the readerisreferred to [4℄ forafullexplanation onthisrelationship).
In order to ompute the atoms of a on ept latti e, we use the graph notion of domination
betweenverti es: avertexxissaidtodominateanothervertexyiftheneighborhoodofxin ludes
the neighborhood of y. We use a propertyfrom [4℄: if all the verti es of a set A share thesame
neighborhood,andare nottogether dominatedbyanothervertexof theen odinggraphG
R ,then
Ade nes an atom AB ofthe on ept latti e.
Ourmain omplexityimprovement followsfromtheremarkthatmu hof theinformationne -
essarytodeterminethedominationrelationshipsbetweenverti es anbeinheritedasonemovesup
intothelatti e,alongapathfromthebottom tothetop( alleda maximal hain). Thisenables
usto avoid re omputing all the domination informationea h time a new on ept is en ountered
pro ess. Wewillalsopresentourdatastru ture,thedominationtable,andillustratethepro esses
involved witha smallexample.
2 Preliminaries
2.1 Latti es
Given a nitesetP of properties and a niteset O ofobje ts,abinary relationR isde ned asa
subsetoftheCartesian produ tPO. Given anelement ofPO,wewillrefer to itasa ross
of the relation if it is in R , as a non- ross if it is not. We will use n to denote jPj, and m to
denotethe numberof non- rosses.
A on ept, also alled a maximalre tangle or losed set of R , is a sub-produ t AB R
su h that 8x 2O B;9y 2Aj(y;x) 62 R , and 8x2P A;9y 2 Bj(x;y) 62R . A is alled the
intentof the on ept, B is alledtheextent.
The on epts, orderedbyin lusionontheintents, de nealatti e, alleda on eptlatti eor
Galoislatti e. A latti e is represented by its Hasse diagram: transitivityand re exivity ar s are
omitted. Con eptsareoftenreferred toaselementsof thislatti e. Su ha latti e,hasasmallest
element, alled the bottom element, and a greatest element, alled the top element. A path
from bottom to top is alled a maximal hain of the latti e. Ifthe relationR hasa fullline of
rosses,it iseasy to omputethe on epts of R from the on eptsof the sub-relationfrom whi h
thisline hasbeen removed ([4 ℄). In therest of this paper, we willdis ussonly relations with no
su h linesof rosses.
We will say that a on ept A 0
B 0
is a su essor of on ept AB if A A 0
and there is
no intermediate on ept A 00
B 00
su h that A A 0 0
A 0
. The set of su essors of an element
is alled the over of this element. The su essors of the bottom element are alled atoms. A
on ept A 0
B 0
is andes endant of on ept AB ifAA 0
. Thenotions of prede essor and
an estor arede neddually.
Example2.1 Binaryrelation R ; the asso iated on ept latti eL(R ) isshown in Figure1.
Set of properties:
P =fa;b; ;d;e;f;g;hg;
Set of obje ts:
O=f1;2;3;4;5;6g.
a b d e f g h
1
2
3
4
5
6
2.2 Graphs
Given a relation R , [4 ℄ de ned an underlying en oding graph G
R
as the graph with vertex set
P[O;P and O are liques,and fora vertex x of P and a vertex y of O, there is an xy edgein
G
R
i (x;y) is not in R . For X P;Y O, we willdenote byG
R
(X[Y) thesubgraph of G
R
indu edbyvertex set X[Y.
We willuseonlyexternalneighborhoods,whi hwe willdenotebyN +
: ifx2P;N +
(x)=fy2
Oj(x;y)2Gg,and ifx2O;N +
(x)=fy2Pj(y;x)2Gg,whereGisthe urrentsubgraphofG
R .
In our example, N (a) = f1;4;5g, N (b) = f4;5;6g, N ( ) = f3;4;6g, N (d) = f2;3;6g,
N +
(e)=f2;3;5;6g, N +
(f)=f1;2;4;5;6g, N +
(g)=f1;4;5;6g, N +
(h)=N +
(a).
Inorder to omputethe over ofa on ept, we needseveral graphnotions.
De nition 2.2 A vertex x is said to dominate a vertex y i N +
(y) N +
(x); we will say that
this domination isstri t when x dominates y and N +
(y)6=N +
(x).
Anotherimportantnotionforthispaperisthat of maximal liquemodules.
De nition 2.3 A set X of properties is said to form a maximal lique module of G
R
if every
vertex x of X dominates every other vertex of X; we will all X a maxmod.
ThemaxmodsofG
R
de nea partitionofP;essentially,amaxmodbehavesasasinglevertex,
asall theverti esofamaxmodX sharethesameexternalneighborhood,denotedbyN +
(X). We
thusextendDe nition2.2tomaxmods: wewillsaythatamaxmodXdominatesanothermaxmod
Y ifN +
(Y)N +
(X).
Theorem 2.4 ([4 ℄)A on ept (A+X)B 0
overs a on ept AB i X, in G
R
((P A)[B),
isa non-dominating maxmod.
In our example, in G
R
, the partition of P into maxmods is: fa;hg, fbg, f g , fdg, feg, ffg,
fgg, fhg. feg dominates fdg , ffg dominates fgg, fgg dominates fa;hg and fbg; f g is neither
dominated nor dominating. The non-dominating maxmods of G
R
are: fa;hg, fbg, f g and fdg.
3 Algorithmi Pro ess
Theorem2.4 an be used to re ursively ompute the over of an element, starting with thebot-
tom element. This pro ess will generate ea h on ept exa tly as many times as the number of
prede essorsithas.
Ourstatedgoalistogenerateea h on eptexa tlyon e. Be ausetheorderingonthe on epts
isde nedbyin lusion,any on eptA 0
B 0
whi hisades endantof AB veri esAA 0
. Sin e
ouralgorithmworksupina depth- rstfashion,when amaxmod X isused to generatea on ept
(A+X)B,then all on epts ontaining X intheir intent will be de ned bythe re ursive all
upon (A+X)B. Ifabrother on ept of(A+X)B usesa maxmod ontaining somevertex x
ofX,this on ept hasalreadybeengeneratedpreviously. Ifweare arefultostoreinformationon
themaxmodswhi h have already beenused bya brother on ept or by a brother of an an estor
on ept,we an avoid omputingthe same on ept more thanon e.
General algorithmi pro ess on on ept AB:
Computeset ND of non-dominating maxmods of G
R
((P A)[B);
//Ea h maxmod X of ND de nes a on ept overing AB.
Computeset New of maxmodsof ND ontaining no already pro essed vertex;
//Ea h maxmod X of New de nes a new on ept overing AB.
For ea h maxmod X in New:
Re ursively apply pro ess to new on ept withintent A+X;
ADD all verti es of X to set of already pro essed verti es.
The bottlene k omplexity forthispro ess is omputingthe setof non-dominatingmaxmods,
amaxmodX of minimumdegree, omputethemaxmodswhi hdominate it,remove these andX
fromthevertexset,beforereiterating. AlgorithmMCS([22℄)wouldyielda ompatibleorderingof
themaxmodsinlinearO(m)time,and omputingthemaxmodsdominatingXwould ostthesame
time; globally, with this pro ess, we would obtain the set of non-dominating maxmods in O(m)
timepernon-dominating maxmod generated, whi h would add up to a worst-time omplexity of
O(nm)per on ept. Experimentally,thisalgorithm,implementedasexplainedabove,runsrapidly,
be ause oftheextrainformationonthealreadypro essedverti es,espe iallyifat ea hstepwhere
a maxmod X isadded to the setof already pro essed verti es, the maxmodswhi h dominate X
arealso added.
However, in order to improve the worse-timebehaviour,we propose a more sophisti ated ap-
proa h to omputingthenon-dominated maxmods,asdes ribedbelow.
Updating the domination information
InordertoeÆ ientlyanswerrequestsonthesetofnon-dominatingmaxmods,weuseadomination
table ontaining information on the urrent graph. As thisinformation an be inherited along a
maximal hain,maintaining thistable inthe ourseofthe Depth-Firsttraversalalong a maximal
hain avoids re omputing theentiredomination informationat ea h step ofthealgorithm.
The inheritan e me hanism involved is the following: when moving up into the latti e, say
froma on ept AB representedbytheunderlyinggraphG
R
((P A)[B)to a se ond on ept
(A+X)B 0
, overingthe rst,andrepresentedbytheunderlyinggraphG
R
((P (A+X)[(B
N +
(X)), two thingshappen:
1. SetX ofpropertiesdisappear.
2. SetN +
(X) ofobje tsdisappear.
In our example, when moving up from the bottom element ;O to element ah236, the new
subgraph will be de ned on (P fa;hg)[f2;3;6g, so that properties a and h will disappear, as
well as obje ts1, 4 and 5.
A vertexx isde nedasdominating anothervertexy ingraphG
R
ifwhenthere isan yiedge,
there also is an xi edge. Equivalently, if(y;i) is a non- ross of R , then (x;i) is also a non- ross
of R . Our idea, used to maintain Galois sub-hierar hies in [5 ℄, is to list into a table L, for ea h
pair ofproperties (x;y), theobje ts whi hprevent x from dominating y. Thismeans thatiffor
obje ti, (x;i)2R and (y;i)62R , iwillappear inthelistL[x;y℄.
The orresponding lists in our example are givenin table Lbelow:
a b d e f g h
a ; f1g f1;5g f1;4;5g f1;4g ; ; ;
b f6g ; f5g f4;5g f4g ; ; f6g
f3;6g f3g ; f4g f4g f3g f3g f3;6g
d f2;3;6g f2;3g f2g ; ; f3g f2;3g f2;3;6g
e f2;3;6g f2;3g f2;5g f5g ; f3g f2;3g f2;3;6g
f f2;6g f1;2g f1;2;5g f1;4;5g f1;4g ; f2g f2;6g
g f6g f1g f1;5g f1;4;5g f1;4g ; ; f6g
h ; f1g f1;5g f1;4;5g f1;4g ; ; ;
From this table L, we an seethat will dominate a whenobje ts1 and 5 have disappeared.
orrespond to distin tneighborsof y inG
R
,and ea h rowy ontainsn su hlists.
In our example, L[ ;a℄=f1;5g; (a;1) and (a;5) are edges of G
R .
UpdatingthetableLmeansforea h (x;y)-pairinP 2
,removingfromlistL[x;y℄theobje tswhi h
disappearfrom thegraphwhen movingup froma on ept to one of its su essors.
A tually, we are only on erned with the number of verti es whi h a vertex x dominates in
a given graph, so that ardinalities are suÆ ient for our data stru ture: a maxmod X will be
non-dominating when, for any x 2 X, the number of verti es whi h x dominates is exa tly jXj.
ThedominationtableT weusethus ontainsnumbersbetween0andjOj,T[x;y℄representingthe
sizeof list L[x;y℄, thusvertex x dominates vertex y i T[x;y℄=0. In order to have rapida ess
to thisinformation, we also keepa table D, s anningP,where D[x℄ givesthe numberof verti es
ysu hthatT[x;y℄=0,i.e. thenumberof verti eswhi hx dominates. A maxmodX willthusbe
non-dominatingif and onlyifforan arbitrary x 2X, D[x℄=jM(x)j, and the query: 'Whi h are
the non-dominating maxmods?' an beansweredinvery eÆ ient O(n)time usingtable D.
In our example, tablesT and D would be:
a b d e f g h
a 0 1 2 3 2 0 0 0
b 1 0 1 2 1 0 0 1
2 1 0 1 1 1 1 2
d 3 2 1 0 0 1 2 3
e 3 2 2 1 0 1 2 3
f 2 2 3 3 2 0 1 2
g 1 1 2 3 2 0 0 1
h 0 1 2 3 2 0 0 0
a b d e f g h
2 1 1 1 2 5 4 2
The pro ess for onstru ting theinitial dominationtable T from atable initializedto ontaining
zerovaluesisthe following:
For ea h x inP
For ea h y inP
For ea h z inO
If (x;z) 2R and (y;z)2= R thenadd 1 to T[x;y℄;
Thesetablesarepre-updatedatea h step todes ribethedominationrelationshipsinthenew
graph before a re ursive all, and then post-updated ba k to its original form. The global time
requiredfor pre-updating T along a maximal hain of the latti e doesnot ex eed the number of
bits ontainedinL,sothepre-updatingwill ostO(nm). Clearly,thepost-updating ostsexa tly
thesame. The updatingalgorithm isgiven inthenext se tion.
4 The algorithm
Thealgorithm isinitially alledbyCONCEPTS(;O;;) on anempty Markedset.
TablesT and D are onstru tedfrom G
R
asdes ribedinthe previousse tion.
Algorithm CONCEPTS
Input: A on ept AB,aset Markedof verti esof P
Initialization:
G G
R
((P A)[B);
Computethepartitionof P A inG into maxmods;
// Themaxmod whi h a vertex x2P belongs tois denoted by M(x).
For x inMarkeddo Marked Marked[M(x);
//1. Compute the setND of non-dominating maxmods of G.
ND ;;
For xin P A do
If D[x℄=jM(x)j thenND ND[M(x);
//2. If desirable,generate the over of AB.
For X inNDdo
A 0
A+X; B 0
O N
+
(X);
PRINT(A 0
B 0
);
//3. Generate the unpro essed des endants of AB.
For X inND Markeddo
A 0
A+X; B 0
O N
+
(X);
PRINT(A 0
B 0
);
//Whengenerating frequentsets, testsize of B 0
; iftoo small,take nextX in ND{Marked.
UPDATE(pre);
CONCEPTS(A 0
B 0
,Marked);
UPDATE(post);
Marked Marked[X;
End.
Algorithm UPDATE
Input: A variableV set to preorto post.
Output: TablesT and D aremodi ed using urrentvaluesof X and AB.
Begin
Choose arepresentativex inX;
//1. Updatetable D by simulating deletion of property set X.
For y in(P A) X do
If T[y;x℄=0then
If V =prethen D[y℄ D[y℄ jXj;
else D[y℄ D[y℄+jXj;
//2. Updatetables T andD by simulating deletion of obje ts in N +
(x).
For j inN +
(x) do
Z N
+
(j) X;
U (P A) Z X;
For (u;z) inUZ do
If V =prethen
T[u;z℄ T[u;z℄ 1;
If T[u;z℄=0 thenD[u℄ D[u℄+1;
else //V=post
T[u;z℄ T[u;z℄+1;
If T[u;z℄=1 thenD[u℄ D[u℄ 1;
End.
Ea hstepofAlgorithmCONCEPTSrequires omputingthemaxmodsofthegraph,whi h an
be doneinO(m)timeusingthealgorithmofHsuandMafrom [12 ℄;usingtableD, ndingtheset
ofnon-dominatingmaxmodsrequiresO(n)time; omparingthese withMarked ostsO(n) time,
thusa on ept is pro essedin globalO(m)time. Asdis ussed inSe tion3, Algorithm UPDATE
globally ostsO(nm)permaximal hain. Theglobaltime omplexityisthusinO(m)per on ept
plusO(nm) permaximal hain.
Spa e omplexity: the re ursive queue ontains at most O(n) on epts of size O(n) ea h,
Markedisof sizeO(n);T ontainsO(nm)bits; theglobalspa e omplexityisthusin O(nm).
Example4.1
Step1: Theexe utionstartswiththebottomelement;123456. InG=G
R
,thenon-dominating
maxmods are fa;hg, fbg, f g and fdg. The over of ;123456 is: ah236, b123, 125,
d145. ThesetMarkedofalreadypro essedverti esisempty. ah236is hosentobepro essed
next.
Step2: Con eptah 236is hosentobepro essednext;thetableisa ordinglypre-updated: sin e
obje ts1,4and5disappear,pairsfromtheCartesianprodu tsfb; ;d;egff;gg,fd;egfb; ;f;gg
andf ;dgfb;e;f;gg should ause the orrespondingnumbersfrom T do bede rementedby1.
Newtables T and Dobtained:
b d e f g
b 0 0 0 0 0 0
1 0 0 0 1 1
d 2 1 0 0 1 2
e 2 1 0 0 1 2
f 1 1 0 0 0 1
g 0 0 0 0 0 0
b d e f g
2 3 6 6 3 2
Graph G be omes G
R
(fb; ;d;e;f;g;2;3;6 g). Maxmods of G: fb;gg, f g, fd;eg, ffg ; non-
dominatingmaxmod: fb;gg. Con eptabgh23is generated.
Step3: abgh23ispro essed. Non-dominatingmaxmods: f gandffg. Con eptsab gh2 and
abfgh3 aregenerated; abfgh3 is hosen to be pro essednext.
Step 4: abfgh3 ispro essed. Non-dominatingmaxmod: f ;d;eg; topelement ab defgh; is
generated.
Step 5: ab defgh ; is pro essed; the graph G obtained is empty; no new on ept an be
generated.
Step6: step3re ursively allsab gh2,withMarked=ffg. Non-dominatingmaxmod: fd;e;fg;
sin ef isinMarked,no new on ept isgenerated.
Step 7: step 1 re ursively alls 125 with Marked=fa;hg. Non-dominating maxmods: fbg
andfdg. Con eptsb 12 and d15are generated. b 12is hosento be pro essednext.
Step 8: b 12 is pro essed, with Marked=fa;hg. Non-dominating maxmods: fd;eg and
fa;g;hg; sin e a and h are in Marked, only fd;eg will be used to generate a new on ept:
b de1.
Step9: b de1 ispro essed,withMarked=fa;hg. Non-dominatingmaxmod: fa;f;g;hg;sin e
aand hare inMarked,nonew on ept isgenerated.
Step10: step7re ursively alls d15,withMarked=fa;b;hg;fa;hgisinheritedfrom on ept
ah236, a brother of father 125, and fbg is inherited from brother on ept b 12. Non-
dominatingmaxmod: fb;eg. Sin eb isin Marked,no new on ept isgenerated.
Step 11: step 1 re ursively alls b123 with Marked=fa; ;hg. Non-dominating maxmods:
f g andfeg. Sin e is inMarked,only on ept de14 isgenerated.
Step13: de14ispro essed, withMarked=fa;b; ;hg. Non-dominatingmaxmod: fb; g. Sin e
b and are in Marked, no new on ept is generated. The re ursive queue is empty and the
algorithmterminates.
x 123456 φ
cd x 15
ah x 236
de x 14 abfgh x 3
abcdefgh x φ
{c} {d,e}
{d} {e}
{d}
{b}
{a,h} {c}
5
abcgh x 2 4 6
abgh x 23
3 8 bc x 12
bcde x 1 9
13
2 11 b x 123 7 c x 125 12 d x 145
{c,d,e}
{b,g}
[ah]
[abch]
[abh]
[ah]
[ach] [ah] [abch]
[f]
1
10 {f}
{b}
Figure 1: Con ept latti e L(R )of relation R ; the on epts are numbered in pre x order following
our sample re ursive exe ution; the inherited sets of already pro essed verti es appear between
bra kets. The edges of the Depth-First tree are labeled by the non-dominating maxmod used to
ompute ea h new on ept.
5 Con lusion
In this paper, we propose a new algorithmi approa h to on ept generation, whi h enables us
to improve all existing algorithms for this problem. Our omplexity analysis ould probably be
streamlined, as O(nm) per maximal hain is a very rough overestimation for the ost of the
updatingpro ess.
Another promisingapproa h to improve the omplexityfor thisproblem would be to usethe
propertythat thereis aone-to-one orresponden ebetweenthemaximal hainsof thelatti e and
the minimal triangulationsof the underlying o-bipartite graph ([4 ℄) and apply Meister's re ent
work (see [16 ℄) on thelinear-timetriangulationsof AT-free and Claw-free graphs,a super lassof
o-bipartitegraphs.
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