Faster Algorithms for Constructing a Galois Lattice, Enumerating All Maximal Bipartite Cliques and Closed Frequent Sets

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1 Fastr Algorithms for Construting a Galois Latti, Enumrating All Maximal Bipartit Cliqus and Closd Frqunt Sts Viky Choi Yang Huang Fbruary 17, 2006 Abstrat In this papr, w giv a fast algorithm for onstruting a Galois latti of a binary rlation. Whn th binary rlation is rprsntd as a bipartit graph, ah vrtx of th latti (alld a onpt) orrsponds to a maximal bipartit liqu of th bipartit graph. Thus, our algorithm also numrats all maximal bipartit liqus. Furthr, our algorithm an b naturally modifid to omput only larg onpts that ar known as losd frqunt sts in data mining. Th running tim of our algorithm dpnds on th latti strutur and is fastr than all othr xisting algorithms for ths problms. Lt B dnot th st of all onpts, and L =< B, > b th orrsponding latti. For a onpt C B, a dsndant D = (xt(d), int(d)) of C is alld an uppr dsndant of C if thr xists i int(d) suh that for any dsndant E C with i int(e), xt(e) xt(d). Dnot th st of uppr dsndants of C by U C. For most of onpts, U C onsists of all sussors of C only. Th running tim of our algorithm is O( C B a xt(c) {D U G(C) : a xt(d)} ) whr G(C) is a prdssor of C. 1 Introdution Givn a bipartit graph G = (O M, E), a pair (A, B) with A O and B M of G is alld a bipartit liqu if for all a A and all b B, (a, b) E. A bipartit liqu (A, B) is maximal if thr dos not xist a bipartit liqu (C, D) suh that (1)A C, B D or (2) A C, B D. In Formal Conpt Analysis (FCA) [12], th bipartit graph G = (O M, E) is rprsntd by a binary rlation I O M. Th tripl (O, M, I) is alld a ontxt. Eah maximal bipartit liqu of th bipartit graph G = (O M, E) is alld a onpt of th ontxt (O, M, I). Conpts (ordrd by st-inlusion) onstitut a ni strutur, known as a latti. Th latti of all onpts is alld onpt or Galois latti. A onpt is also alld a losd frqunt st [32] in data mining. Th quivaln of all ths trminologis maximal bipartit liqus in thortial omputr sin (TCS), onpts for applid mathmatiians in FCA, and losd frqunt sts for rsarhrs in data mining (DM) was known,.g. [2, 32]. Thr is xtnsiv work of th rlatd problms in ths thr ommunitis,.g. [1] [7] in TCS, [9] [21] in FCA, and [23] [34] in DM. In gnral, in TCS, th rsarh fouss on ffiintly numrating all maximal bipartit liqus (of a bipartit graph); in FCA, on is intrstd in th latti strutur of all onpts; in DM, on is oftn intrstd in omputing only larg losd frqunt sts. Tim omplxity. Givn a bipartit graph, it is not diffiult to s that thr an b xponntially many maximal bipartit liqus. For problms with potntial xponntial (in th siz of th input) siz output, in thir sminal papr [5], Johnson t al introdud svral notions of polynomial tim for algorithms for ths problms: polynomial total tim, inrmntal polynomial tim, polynomial dlay tim. An algorithm runs in polynomial total tim if th tim is boundd by a polynomial in th siz of th input and th siz of th output. An algorithm runs in inrmntal polynomial tim if th tim rquird to gnrat a sussiv output is boundd by th siz of input and th siz of output gnratd thus far. An algorithm runs in Dpartmnt of Computr Sin, Virginia Th, USA. vhoi@s.vt.du. Dpartmnt of Computr Sin, Rutgrs Univrsity, USA. yahuang@s.rutgrs.du 1

2 polynomial dlay tim if th gnration of ah output is only polynomial in th siz of input. It is not diffiult to s that polynomial dlay is strongr than inrmntal polynomial (namly an algorithm with polynomial dlay tim is also running in inrmntal polynomial), whih is strongr than polynomial total tim. On says suh a problm is NP-hard if it has no polynomial total tim algorithm unlss P=NP. For polynomial dlay algorithm, w an furthr distinguish if th spa usd is polynomial or xponntial in th input siz. Prvious work. Obsrv that th maximal bipartit liqu (MBC) problm is a spial as of th maximal liqu problm in a gnral graph. Namly, givn a bipartit graph G = (V 1, V 2, E), a maximal bipartit liqu orrsponds to a maximal liqu in Ğ = (V 1 V 2, Ĕ) whr Ĕ = E (V 1 V 1 ) (V 2 V 2 ). Consquntly, any algorithm for numrating all maximal liqus in a gnral graph,.g., [7, 5], also solvs th MBC problm. In fat, th bst known algorithm in numrating all maximal bipartit liqus, whih was proposd by Makino and Uno [6] that taks O( 2 ) polynomial dlay tim whr is th maximum dgr of G, was basd on this approah. Th fat that th st of maximal bipartit liqus onstituts a latti was not obsrvd in th papr and thus th proprty was not utilizd for th numration algorithm. In FCA, muh of rsarh has bn dvotd to study th proprtis of th latti strutur. Thr ar svral algorithms,.g. [17, 21, 16], that onstrut th latti, i.. omputing all onpts togthr with its latti ordr. Thr ar also som algorithms that omput only onpts,.g. [19, 12]. S [14] for a omparison studis of ths algorithms. Th bst polynomial total tim algorithm was by Nourin and Raynaud [17] with O(nm B ) tim and O(n B ) spa, whr n = O and m = M and B dnot th st of all onpts. This algorithm an b asily modifid to run in O(mn) inrmntal tim [18]. Obsrv that th spa of total siz of all onpts is ndd if on is to kp th ntir strutur xpliitly. Thr wr svral othr algorithms,.g.[12, 16], all run in O(n 2 m) polynomial dlay. Thr is anothr algorithm [21] that is basd on divid-and-onqur approah, but th analytial running tim of th algorithm is unknown as it is diffiult to analyz. Thr ar svral huristis in data mining for omputing losd frqunt sts, suh as CHARM [33, 34], and CLOSET [27, 30]. To our bst knowldg, th algorithm with thortial analysis running tim was givn in [2] with O(m 2 n) inrmntal polynomial running tim, whr n = O and m = M. Our Rsults. In this papr, making us of th latti strutur of onpts, w prsnt a simpl and fast algorithm for omputing all onpts togthr with its latti ordr. Th main ida of th algorithm is that givn a onpt, whn all of its sussors ar onsidrd togthr (i.. in a bath mannr), thy an b ffiintly omputd. W omput onpts in th Bradth First Sarh (BFS) ordr th ordring givn by BFS travrsal of th latti. Whn omputing th onpts in this way, not only do w omput all onpts but also w idntify all sussors of ah onpt. Th ffiiny is furthr ahivd by using th sam implmntation thniqu that on mploys to implmnt th Lxiographi Bradth First Sarh (LBFS) travrsal of a graph in linar tim of numbr of vrtis and dgs. Anothr ida of th algorithm is that w mak us of th onpts gnratd to dynamially updat th adjany rlations. Th running tim of our algorithm dpnds on th latti strutur. Lt B dnot th st of all onpts, and L =< B, > th orrsponding latti. Eah onpt C B onsists of two omponnts and is writtn as (xt(c), int(c)), whr xt(c) and int(c) ar th xtnt and intnt of C rsptivly (s Stion 2 for rlatd bakground and trminology). For a onpt C B, a dsndant D = (xt(d), int(d)) of C is alld an uppr dsndant of C if thr xists i int(d) suh that for any dsndant E C with i int(e), xt(e) xt(d). Dnot th st of uppr dsndants of C by U C. In partiular, th st of uppr dsndants of C onsists of all sussors of C. For most of onpts, U C onsists of only sussors of C. Th running tim of our algorithm is O( C B a xt(c) {D U G(C) : a xt(d)} ), whr G(C) is a prdssor of C. Our algorithm is fastr than th bst known algorithms for omputing all onpts or onstruting a latti baus (1) th algorithm is improvd upon an asy algorithm (dsribd in Stion 4.1) and (2) th asy algorithm is as fast as th urrnt bst algorithms for th problm. 2

3 Outlin. Th papr is organizd as follows. In Stion 2, w rviw som bakground and notation on FCA. In Stion 3, w dsrib som basi proprtis of onpts that w us in our latti-onstrution algorithm. In Stion 4, w first dsrib an asy algorithm that is as fast as th bst known algorithms for omputing all onpts or onstruting a latti. Thn w dsrib how to improv th algorithm basd on th ida of omprssing attributs. Finally, w furthr improv th algorithm by liminating xpliit sussor-hking. W onlud with disussion in Stion 5. 2 Bakground and Trminology on FCA In FCA, a tripl (O, M, I) is alld a ontxt whr O and M ar sts and I O M. Th lmnts of O and M ar alld objts and attributs rsptivly. Th ontxt is oftn rprsntd by a ross-tabl as shown in Figur 1. For a M, dfin nbr(a) = {g O : (g, a) I}. Similarly, for g O, dfin nbr(g) = {a M : (g, a) I}. For a subst A of O (or M), A = a A nbr(a). By dfinition, it is asy to hk that A A, and if A B, thn B A. Th funtion indus a Galois onntion btwn O and M (th undrlying sts ar ordrd by st-inlusion rlation). Radrs ar rfrrd to [12] for proprtis of th Galois onntion. A pair C = (A, B), with A O and B M, is a onpt if and only if A = B and B = A. A and B ar alld th xtnt and intnt of C rsptivly, writtn as A = xt(c). and B = int(c). Th st of all onpts of th ontxt (O, M, I) is dnotd by B(O, M, I) or simply B whn th ontxt is undrstood. Lt (A 1, B 1 ) and (A 2, B 2 ) b two onpts in B. Obsrv that if A 1 A 2, thn B 2 B 1. W ordr th onpts in B by th following rlation : (A 1, B 1 ) (A 2, B 2 ) A 1 A 2 (B 2 B 1 ). It is not diffiult to s that th rlation is a partial ordr on B. In fat, L =< B, > is a omplt latti and it is known as th onpt or Galois latti of th ontxt (O, M, I). For C, D B with C D, if for all E B suh that C E D implis that E = C or E = D, thn C is alld th sussor (or lowr nighbor) of D, and D is alld th prdssor (or uppr nighbor) of C. Th diagram rprsnting an ordrd st (whr only sussors/prdssors ar onntd by dgs) is alld a Hass diagram (or a lin diagram). S Figur 1 for an xampl of th lin diagram of a Galois latti. Not that ah onpt (A, B) is uniquly dtrmind by ithr its xtnt, A, or by its intnt, B. W dnot th onpts rstritd to th objts O by B O = {A O : A = A}. Whn thr is no dangr of onfusion, for A B O, w rfr to its orrsponding intnt by int(a) and th orrsponding onpt by (A, int(a)). Also, th ordr is ompltly dtrmind by th inlusion ordr on 2 O or quivalntly by th rvrs inlusion ordr on 2 M. That is, (A, int(a)) (P, int(p )) if and only if A P. Following th onvntion, w will simplify th standard st notation by dropping out all th sparators (.g., 123 will stand for th st of attributs {1, 2, 3} and abd for th st of objts {a, b,, d}). 3 Basi Proprtis In this stion, w dsrib som basi proprtis of th onpts on whih our latti onstrution algorithms ar basd. Proposition 3.1 Lt (C, int(c)) b a onpt in B(O, M, I). For i M \ int(c), if C i = C nbr(i) is not mpty, thn C i B O, that is, C i is an xtnt of som onpt. Convrsly, for any st C B O, C O, thr xists a onpt (P, int(p )) suh that C = P nbr(i) for som i M \ int(p ). Proof. For i M \ int(c), suppos that C i = C nbr(i) is not mpty. W will show that C i = C i. Sin C i C i, it rmains to show that C i C i. By dfinition, (int(c) {i}) = ( j int(c) nbr(j)) nbr(i) = Som authors alld this as immdiat sussor. 3

4 C nbr(i) = C i. Sin (int(c) {i}) (int(c) {i}), (int(c) {i}) Ci. By th proprty of, C i (int(c) {i}) = C i. Convrsly, for any st C B O, C O, lt (P, int(p )) b a prdssor of (C, int(c)), thn C = P nbr(i) whr i int(c) \ int(p ). (a) a b d (ab,1) (b) (a,13) (abd, ø) (bd,24) (b,124) () (ab,1) (abd, ø) (bd,24) (ø, 1234) Figur 1: (a) a ontxt (O, M, I) with O = {a, b,, d} and M = {1, 2, 3, 4}. Th ross indiats a pair in th rlation I. (b) th orrsponding Galois/onpt latti. () Child(abd, ) = {(ab, 1), (bd, 24), (a, 3)}. Exampl. Considr th onpt (C, int(c)) = (abd, ) of ontxt in Figur 1, w hav C 1 = ab, C 2 = bd, C 3 = a, C 4 = bd. For a onpt (C, int(c)) in B(O, M, I), dnot th st of rmaining attributs {i M \ int(c) : C nbr(i) } by R(C). Considr th following quivaln rlation on R(C): i j C nbr(i) = C nbr(j), for i j R(C). Lt I 1,..., I t b th quivaln lasss indud by, i.. R(C) = I 1... I t, and, C i = C j for any i j I k, 1 k t. For 1 k t, w dfin a nw pair (E k, pr int C (E k )) whr E k = C j for j I k and pr int C (E k ) = I k. Call th st of ths pairs {(E k, pr int C (E k )) : 1 k t} th hildrn of (C, int(c)), and dnot th st by Child(C, int(c)). For xampl, in Figur 1, Child(abd, ) = {(ab, 1), (bd, 24), (a, 3)}. From Proposition 3.1, th objt st E in ah pair (E, pr int C (E)) Child(C, int(c)) is an xtnt of som onpt. W dnot th xtnt st {E k : 1 k t} by ExtChild(C). Thus, Child(C, int(c)) = {(E, pr int C (E)) : E ExtChild(C)}. Rall that L =< B, > and L O =< B O, > ar ordr-isomorphi. W hav th proprty that (E, int(e)) is a sussor of (C, int(c)) in L if and only if E is a sussor of C in L O. For E ExtChild(C), w all E a hild (or uppr dsndant) of C and C a parnt of E. It is not diffiult to s that if E is a sussor of C, thn E is a hild of C. Howvr, not vry hild of C is a sussor of C. For th xampl in Figur 1, ExtChild(abd) = {ab, bd, a}, whr ab and b ar sussors of abd but a is not. Similarly, if P is a prdssor of C, thn P is a parnt of C but it is not nssarily tru that vry parnt of C is a prdssor of C. Lmma 3.2 If E ExtChild(C) is a sussor of C, thn int(e) = int(c) pr int C (E). If E ExtChild(C) is not a sussor of C, thn thr xists F ExtChild(C) suh that E F. Proof. Lt E ExtChild(C). Suppos that int(c) pr int C (E) int(e). Lt s int(e) \ (int(c) pr int C (E)), thn w hav E (C nbr(s)). By Proposition 3.1, thr xists F ExtChild(C) suh that F = C nbr(s). By dfinition of pr int C (E), E F, i.. E F. Thus, if E is a sussor of C, it would imply int(e) \ (int(c) pr int C (E)) =, that is, int(e) = int(c) pr int C (E). Corollary 3.3 Lt E ExtChild(C). Suppos that E is not ontaind in any of its siblings in ExtChild(C), i.. E F for any F ExtChild(C), thn E is a sussor of C. For E ExtChild(C), w all pr int C (E) th attribut st of E indud by C. Whn C is undrstood, w simply rfr pr int C (E) th indud attribut st of E. W all th st int(c) pr int C (E) th ombind attribut st of E. W say that pr int C (E) is omplt if its ombind attribut st quals to (a,3) 4

5 its intnt, i.. int(e) = int(c) pr int C (E). If E ExtChild(C) is a sussor of C, w all th orrsponding pair (E, pr int C (E)) a sussor pair of (C, int(c)), othrwis, th non-sussor pair. From Lmma 3.2, w know that all indud attribut sts in sussor pairs ar omplt. Rall that to omput th latti strutur of all onpts, it suffis to find all sussor/prdssor rlations in addition to omputing all onpts. Dnot th sussors of C by Su(C). Thn by Lmma 3.2, Su(C) = ExtChild(C) \ {E : F ExtChild(C) suh that E F }. In gnral, an xtnt an hav svral parnts. For xampl, b is in ExtChild(ab) and ExtChild(bd), with pr int ab (b) = 24 and pr int bd (b) = 1 rsptivly. In th following, w giv anothr haratrization of Su(C) in trms of th siz of th attribut sts. Lmma 3.4 Suppos that E is not a sussor of C, thn E ExtChild(F ), for all F ExtChild(C) with E F. Furthrmor, int(c) + pr int C (E) < int(f ) + pr int F (E). Proof. Suppos that E is not a sussor of C and E F with F ExtChild(C). For i pr int C (E), w hav E = C nbr(i). Sin E F, intrsting both sids by nbr(i), w hav E nbr(i) F nbr(i). On th othr hand, F C, F nbr(i) C nbr(i) = E. Thus, w hav E = E nbr(i) F nbr(i) E, implying that E = F nbr(i). By Proposition 3.1, w hav E ExtChild(F ). Also, w hav i pr int F (E) (as i int(f )) and hn pr int C (E) pr int F (E). Sin F ExtChild(C), int(c) < int(f ). Consquntly, int(c) + pr int C (E) < int(f ) + pr int F (E). Data Struturs. Throughout this papr, th binary rlation (or th bipartit graph) is rprsntd in adjany lists of all objts and attributs. Th adjany list of ah objt (attribut rsptivly) ontains a list of pointrs to th adjant attributs (objts rsptivly). WLOG, w assum that no objt is adjant to all attributs, and no attribut is adjant to all objts. W us a tri ovr th objt st to stor th xtnts. Eah xtnt in th tri is assoiatd with a st of attribut st. Thus, ah onpt is rprsntd by its xtnt in th tri and th orrsponding intnt is assoiatd with th xtnt. To ffiintly math th xtnts, w us hash kys ovr ah branh in th tri. It will tak tim O( C ) to sarh or insrt an objt st C in th tri. 3.1 Computing Child(C, int(c)) by Produr SPROUT W will dsrib how to omput Child(C, int(c)) in O( a C nbr(a) ) tim, using a produr alld SPROUT. Lmma 3.5 For (C, int(c)) B, it taks O( a C nbr(a) ) to omput Child(C, int(c)). Proof. Lt R(C) = a C nbr(a) \ int(c). For ah i R(C), w assoiat it with a st C i (whih is initializd as an mpty st). For ah objt a C, w san through ah attribut i in its nighbor list nbr(a), appnd a to th st C i that is assoiatd with i. This stp taks O( a C nbr(a) ). Nxt w ollt all th sts {C i : i R(C)}. W us a tri to group th sam objt st: sarh C i in th tri; if not found, insrt C i into th tri with {i} as its attribut st, othrwis w appnd i to C i s xisting attribut st. This stp taks O( i R(C) C i ) = O( a C nbr(a) ). Thus, this produr, alld SPROUT(C, int(c)), taks O( a C nbr(a) ) tim to omput Child(C, int(c)). In ordr to ompar th running tim with th algorithm by Makino and Uno [6], w xprss th running tim in trms of th maximum dgr of vrtis (objts/attributs). Th siz of a xtnt st is O( ). Thus, O( a C nbr(a) ) = O( 2 ). 3.2 Effiint Intnt Compltion by Produr COMPLETEINTENT Rall that not all sts in ExtChild(C) ar th sussors of C. From Lmma 3.2, if E is not a sussor of C, thn its indud attribut st is not omplt, i.. int(c) pr int C (E) int(e). Thus, by omparing th ombind attribut st, int(c) pr int C (E), with int(e) = a E nbr(a), w an dtrmin if E is a sussor or not. This an b implmntd in Lxiographi Bradth First Sarh-lik mannr in 5

6 O( a E nbr(a) ). W all this produr COMPLETEINTENT. Th produr rturns Ys if th indud attribut st is omplt, othrwis No and st th attribut st pr int C (E) to int(e) \ int(c). That is, upon rturn, w always hav int(c) pr int C (E) = int(e). W summariz this in th following lmma. Lmma 3.6 For (E, pr int C (E)) Child(C, int(c)), w an dtrmin whthr pr int C (E) is omplt, i.. pr int C (E) = int(e) \ int(c), in O( a E nbr(a) ) tim. 4 Algorithms for Construting A Galois Latti 4.1 An Easy Algorithm In this stion, w dsrib an asy algorithm to onstrut th latti strutur, inluding gnrating all onpts and a list of its sussors for ah onpt, in O( C B a C nbr(a) ) tim. Our algorithm starts with prossing th top onpt (C, int(c)) = (O, ). First, w gnrat all th hildrn Child(C, int(c)) by produr SPROUT. Thn w hk ah pair (E, pr int C (E)) Child(C, int(c)) by th produr COMPLETEINTENT. If th indud attribut st pr int C (E) is omplt, w add E to th sussor list of C. To systmatially pross onpts so that ah onpt is prossd (and thus sproutd) on and only on, w pross onpts in a Bradth First Sarh (BFS) ordr. Namly, w nquu an xtnt into a quu whn th xtnt is gnratd (or sproutd) by its first parnt. Eah xtnt an hav svral parnts and thus an b gnratd svral tims. Rall that th produr COMPLETEINTENT also omplts th attribut st pr int C (E) and an xtnt is a sussor if its indud attribut st is omplt. Thus, aftr th first tim an xtnt was gnratd, w stor E togthr with th intnt st int(e) in th tri (rgardlss if E is a sussor of C or not). Whn an xtnt is gnratd again, w an avoid alling COMPLETEINTENT to dtrmin if it is a sussor of th urrnt parnt. Instad w ompar its ombind attribut st with th intnt st in th tri. If th ombind attribut st is th sam as th intnt st, thn th xtnt is a sussor of its (urrnt) parnt, othrwis it is not. S Figur 2 for an illustration of th algorithm and Algorithm 2 in th appndix for th psudo-od. Tim omplxity analysis. Eah onpt alls on th produr COMPLETEINTENT and on th produr SPROUT. Thus, th total running tim is O( C B a C nbr(a) ). In trms of polynomial dlay, it taks O( 3 ) to output a sussiv onpt togthr with its sussor list. If w nd only onpts, th algorithm an b modifid to run in O( a C nbr(a) ) = O( 2 ) polynomial dlay. Thus, this asy algorithm is as fast as th algorithm by Makino & Uno [6] that omputs only onpts. It is also as fast as th bst latti-onstrution algorithm givn by by Nourin and Raynaud [17], in total polynomial tim of O(mn B ) (or O(mn) inrmntal polynomial), and svral othr algorithms (.g. s [14]) of O(n 2 m) polynomial dlay. Rmark: Th ordr of th onpts bing prossd is not important as long as th pair that is bing prossd is a onpt (i.. th indud attribut st is omplt). If w us stak instad of quu, thn th onpts ar gnratd in Dpth First Sarh (DFS) ordr. Also, in th DFS vrsion, if w nd only onpts, w an avoid th intnt ompltion produr. Howvr, w will nd to ompar th attribut sts for th latti strutur. 4.2 An Improvd Algorithm Th running tim of th abov algorithm is dominatd by produrs COMPLETEINTENT and SPROUT. Eah produr taks O( a C nbr(a) ) for prossing a onpt (C, int(c)). Thus, if w an rdu th siz of th adjany lists (i.. nbr(a) ), w an rdu th running tim of th algorithm. Obsrvation. Considr a onpt (C, int(c)) L, th xtnts of all dsndants of (C, int(c)) ar all substs of C. To omput th dsndants of (C, int(c)), it suffis to onsidr th adjany lists of attributs with rstrition to C. That is, for i R(C), w rstrit th adjant list to C, nbr(i) C = {a C : (a, i) I}, dnotd by nbr C (i). Comprssing attributs. For (E, pr int C (E)) Child(C, int(c)), nbr C (i) = nbr C (j), for any i, j pr int C (E) with i j. That is, all attributs in pr int C (E) hav th sam adjany list whn th objt 6

7 (abdf, ) a b d f (abd,15) (a,23) (bf,4) (a,1235) (b,145) (,12356) (,123456) (a) (b) ø (abdf, ) b a b f (abd,15) (a,23) (bf,4) (,6) 23 d 15 () (d) ø (abdf, ) a b (abd,15) (a,23) (bf,4) (,6) b 1235 f 4 (a,23) (b,4) (,6) d () Figur 2: (a) A ontxt; (b) Th orrsponding latti; () Child(abdf, ), a solid lin indiats a sussor whil a dash lin indiats a non-sussor; (d) Th tri aftr insrting Child(abdf, ), ah xtnt is assoiatd with its orrsponding intnt. At this stp, th quu Q = (abd, a, bf). () Sam as () for Child(abd, 15); (f) Th orrsponding tri aftr insrting Child(abd, 15). A filld box indiats th nwly gnratd xtnts. At this stp, Q = (a, bf, a, b). (f) 7

8 st is rstritd to C. To rdu th sizs of adjant lists, w thus an rprsnt th attributs in pr int C (E) by a singl lmnt, alld a omprssd attribut. For xampl, using a singl lmnt 15 to rprsnt th two attributs 1 and 5. W all th rdud adjany lists th omprssd adjany lists. Th st of omprssd adjany lists orrsponds to a rdud ross-tabl. For xampl, th rdud ross tabl of Child(abdf, ) of th abov xampl is shown in Figur a b d f Figur 3: Rdud ross-tabl of Child(abdf, ) of th ontxt in Figur 2. To mak us of th omprssd adjant lists, w prod th algorithm in a two-lvl mannr. For a onpt (C, int(c)), w first gnrat all its hildrn Child(C, int(c)) by th produr SPROUT. Thn w omprss th attributs in ah hild of (C, int(c)) to gt a st of omprssd adjany lists. W thn us ths omprssd adjany lists to pross ah hild (E, pr int C (E)) of (C, int(c)), by produrs SPROUT and COMPLETEINTENT. That is, instad of using th global adjany lists, whn prossing (E, pr int C (E)), w us th omprssd adjany lists with rstrition to C. Rall that for a r-gnratd xtnt E, w nd to ompar th nw ombind attribut st with its (alrady omputd) intnt. Howvr, instad of xpanding th omprssd attributs to mak th omparison, w an dtrmin if an xtnt is a sussor or not by th siz of attribut sts. This is baus an xtnt is a sussor (of its parnt) if and only if its ombind attribut st is th sam as th intnt st. This is quivalnt to th siz of th ombind attribut st is th sam siz as its intnt. If an xtnt is not a sussor, its ombind attribut st will b a subst of its intnt and thus th orrsponding siz will b smallr. W summariz this haratrization in th following lmma. Lmma 4.1 If E ExtChild(C) is a sussor of C, thn int(c) + pr int C (E) = int(e). If E ExtChild(C) is not a sussor of C, thn int(c) + pr int C (E) < int(e). By assoiating ah omprssd attribut with th numbr of th original attributs that it ontains, w an asily kp trak of th siz of attribut sts. Aftr omputation, w an xpand th omprssd attributs to rstor th original attributs. It is not diffiult to s that this will not tak mor asymptoti tim. Radrs ar rfrrd to th full vrsion of this papr for th implmntation dtails. Tim omplxity analysis. Now th running tim of ah produr (SPROUT or COMPLETEINTENT) will dpnd on th siz of omprssd adjany lists it uss. It will tak O( a C nbr G(C)(a) ) whr nbr G(C) (a) is th omprssd adjany lists with rstrition to G(C), whn prossing an xtnt C. For th produr SPROUT, G(C) is a prdssor of C; howvr, for th COMPLETEINTENT, G(C) might b an anstor of C that is a fw lvls highr than C. Sin th siz of a adjany list rstritd to a prdssor is smallr than th siz of a adjany list rstritd to a highr anstor, th running tim of this algorithm is dominatd by COMPLETEINTENT. 4.3 A Furthr Improvd Algorithm In this stion, w furthr improv th abov algorithm by liminating th COMPLETEINTENT produr. That is, th running tim will b dominatd by th produr SPROUT on ah onpt. W mak two modifiations to th abov algorithm: First, w pross th hildrn aording to th siz of thir xtnts in drasing ordr. Basd on this prossing ordr, w laim that w an avoid alling COMPLETEINTENT to omput th siz of intnt. Instad, w dtrmin if an xtnt is a sussor or not by 8

9 omparing th sizs of th xtnt s attribut sts gnratd so far. Namly, whn w pross an xtnt, th xtnt is a sussor of its urrnt parnt if th siz of its ombind attribut st is not smallr than th siz of attribut st gnratd up to this point. Th orrtnss follows from th following lmma. Lmma 4.2 If E ExtChild(C) is not a sussor of C, thn on of its siblings F in ExtChild(C) will b prossd arlir than E, and int(c) + pr int C (E) < int(f ) + pr int F (E). Proof. By Lmma 3.2, if E is not a sussor of C, thr xists P ExtChild(C) suh that E P. Lt F b th largst suh st. That is, F is not ontaind by any othr sibling in ExtChild(C). By Corollary 3.3, F is a sussor of C. Thus, by th ordring, F will b prossd (not nssarily as a hild of C) arlir than E. Thn by Lmma 3.4, E ExtChild(F ) and thus int(c) + pr int C (E) < int(f ) + pr int F (E) as laimd. S Algorithm 1 for th psudo-od and Figur 4 for an illustration of th algorithm. Algorithm 1 Construting a Galois Latti Input: (O, M) and th adjany list of ah lmnt; Output: All onpts (C, int(c)) and a sussor list for ah onpt. 1: Initializ a tri T for th objt st O; 2: Start with a quu Q ontaining O; Eah lmnt in Q onsists of a pointr pointing to an xtnt in T. 3: int(o) = ; (O, ) is th top onpt. 4: Gnrat th hildrn of O by produr SPROUT; 5: whil Q is not mpty do 6: C = dquu(q); 7: Comprss attributs in ah indud attribut st in th hildrn of C and gnrat th orrsponding omprssd adjany lists aordingly; 8: Sort th hildrn of C aording to th xtnt siz in drasing ordr. Suppos th hildrn ar (E k, pr int C (E k )), 1 k t and E i E j for i < j. 9: for k = 1... t do 10: if th siz of E k s ombind attribut st is not smallr than th largst attribut st gnratd thn 11: Add E k to th sussor list of C; 12: if E k is not sproutd thn 13: Gnrat th hildrn of E k by produr SPROUT; 14: For ah hild of E k, if its xtnt xists in T, ompar and kp th siz of th largr ombind attribut st for th xtnt, othrwis insrt th xtnt into T with th siz of its ombind attribut st. 15: Enquu E k into Q; 16: nd if 17: nd if 18: nd for 19: Expand th omprssd attributs in int(c); output th onpt (C, int(c)) and its sussor list. 20: nd whil Tim omplxity analysis. Th total running tim of th algorithm is O( C B a C nbr G(C)(a)) ), whr O( a C nbr G(C)(a)) ) is th tim for sprouting th xtnt C, and G(C) is th lftmost prdssor of C and nbr G(C) (a) is th siz of th omprssd adjany list of a with rstrition to G(C). Obsrv that nbr G(C) (a)) = {E ExtChild(G(C)) : a E}. Rall that U C is th st of uppr dsndants of C dfind by {D B : D C and thr xists i int(d) suh that for all E C with i int(e), 9

10 (abd,) a b d (ab,4) (abd,357) (a,6) (b,12) (ab,3457) (a,34567) (b,123457) (,126) (a) (b) (abd,ø) a ø :2 b :3 (ab,4) (abd,3 ) (a,6) (b,1 ) b :1 :2 (ab,3 ) (a,6) (b,1 ) :1 d :3 () (abd,ø) (d) ø (ab,4) (abd,3 ) (a,6) (b,1 ) a :5 b :6 :3 (ab,3 ) b :1 :2 (a,6) (b,1 ) :1 d :3 () Figur 4: (a) A ontxt; (b) Th orrsponding latti. A solid lin indiats th hild is gnratd by th prdssor, whil a dottd lin indiats th hild is gnratd by anothr prdssor. () Comprss attributs in Child(abd, ): 3 = 357, 1 = 12 and gnrat Child(ab, 4). (d) th tri T aftr insrting Child(ab, 4). Eah xtnt is assoiatd with th siz of its largst attribut st gnratd so far. For illustration purpos, w skip stps of sprouting (ab, 3 ), (a, 6), (b, 1 ). () Gnrat Child(ab, 3 ). (f) Not that th attribut st siz of (a, 6) as a hild of (ab, 4) is only 2, lss than its siz of 5 (aftr a is sproutd by (a, 6)) in th tri, thrfor (a, 6) is not a sussor of (ab, 4). Similarly, (b, 1 ) is not a sussor of (ab, 4). (f) 10

11 xt(e) xt(d)}. It is not diffiult to s that ExtChild(C) = {E : (E, int(e)) U C }. Hn th total running tim of th algorithm is O( C B a xt(c) {D U G(C) : a xt(d)} ), and it taks O( a xt(c) {D U G(C) : a xt(d)} ) polynomial dlay for prossing onpt (C, int(c)). 5 Disussion In this papr, by making us of th latti strutur of onpts, w prsnt a simpl and fast algorithm for onstruting a Galois latti of a ontxt. W gnrat all onpts, togthr with a list of sussors of ah onpt, in th BFS ordr. Our algorithm is fastr than th xisting algorithms for onstruting lattis [17, 21, 16, 14], and algorithms for omputing only all onpts [6]. Th algorithm by Makino and Uno [6] gnrats th onpts aording to th (rvrs sarh) numration tr ordr. As it dos not sm to hav a lar orrspondn btwn th latti and th numration tr, it is not lar that on an modify thir algorithm to output th onpts togthr with its latti ordr. In FCA, muh of rsarh has bn dvotd to th study of th strutur proprtis of onpt lattis. Howvr, th latti-onstrution algorithms in th ara do not sm to tak advantag of ths proprtis. In data mining, on is in gnral intrstd in onpts with larg xtnt siz, and th latti strutur. Sin w gnrat our onpts in drasing ordr of xtnt siz, w an naturally modify our algorithm to output ths larg onpts and its latti ordr, namly by kping and sprouting only onpts with larg nough xtnts. Thortially, our algorithm is muh fastr than th urrnt stat-of-art algorithm CHARM [34] in data mining. W ar tsting and omparing our algorithm with CHARM on th bnhmark datast. Th rsults will b rportd lswhr. As FCA finds its wid appliations in aras ranging from data mining to bioinformatis, ffiint algorithms for onstruting Galois lattis ar muh ndd. Our algorithm is fastr than prvious algorithms for this problm, nvrthlss, it sms to hav muh room to improv. Aknowldgmnt W would lik to thank Rinhard Laubnbahr for introduing us FCA. Rfrns [1] J. Abllo, A. Pogl, L. Millr. Bradth first sarh graph partitions and onpt lattis. J. of Univrsal Computr Sin, 10(8), p , [2] E. Boros, V. Gurvih, L. Khahiyan, K. Makino. On maximal frqunt and minimal infrqunt sts in binary matris. Annals of Mathmatis and Artifiial Intllign, 39, p , (STACS 2002) [3] D. Eppstin. Arboriity and bipartit subgraph listing algorithm. Information Prossing Lttrs, 51(4), p , [4] T. Kashiwabara, S. Masuda, K. Nakajima, T. Fujisawa. Gnration of maximal indpndnt sts of a bipartit graph and maximum liqus of a irular-ar graph. J. Algorithms, 13, p , [5] D.S. Johnson, M. Yannakakis, C.H. Papadimitriou. On gnrating all maximal indpndnt sts. Information Prossing Lttrs, 27, p , [6] K. Makino, T. Uno. Nw algorithms for umrating all maximal liqus. Pro. 9th Sand. Workshop on Algorithm Thory (SWAT 2004), p Springr Vrlag, Ltur Nots in Computr Sin 3111, [7] S. Tsukiyama, M. Id, H. Ariyoshi, and I. Shirakawa. A nw algorithm for gnrating all th maximal indpndnt sts SIAM Journal on Computing, 6(3), p , [8] M. Barbut and B. Montjardt. Ordr t Classifiations: Algbr t ombinatoir. Hahtt, [9] F. Baklouti, R.E. Grarvy. A fast and gnral algorithm for Galois lattis building. J. of Symboli Data Analysis, 3(1), p19-31, [10] A Formal Conpt Analysis Hompag. [11] B. Gantr, G. Stumm, R. Will (ds.). Formal Conpt Analysis: Foundations and Appliations. Ltur Nots in Computr Sin, vol 3626, Springr, [12] B. Gantr, R. Will. Formal Conpt Analysis: Mathmatial Foundations. Springr Vrlag, 1996 (Grmany 11

12 vrsion), 1999 (English vrsion). [13] S.O. Kuzntsov. Complxity of larning in ontxt lattis from positiv and ngativ xampls. Disrt Applid Mathmatis, 142, p , [14] S.O. Kuzntsov and S.A. Obdkov. Comparing prforman of algorithms for gnrating onpt lattis. Journal of Exprimntal and Thortial Artifiial Intllign, 14(23), p , [15] S.O. Kuzntsov. On omputing th siz of a latti and rlatd dision problms. Ordr, 18, p , [16] C. Lindig. Fast onpt analysis. Grhard Stumm d. Working with Conptual Struturs - Contributions to ICCS 2000, Shakr Vrlag, Aahn, Grmany, [17] L. Nourin, O. Raynaud. A fast algorithm for building lattis. Information Prossing Lttrs, 71, p , [18] L. Nourin, O. Raynaud. A fast inrmntal algorithm for building lattis. J. of Exprimntal and Thortial Artifiial Intllign, 14, p , [19] E.M. Norris. An algorithm for omputing th maximal rtangls in a binary rlation. Rvu Roumain d mathmatiqus Purs t Appliqus, 23(2), p , [20] P. Valthv, R. Missaoui, R. Godin, M. Mridji. Gnrating frqunt itmsts inrmntally: two novl approahs basd on Galois latti thory. Journal of Exprimntal and Thortial Artifiial Intllign, 14(2-3), p , [21] P. Valthv, R. Missaoui, P. Lbrun. A partition-basd approah towards onstruting Galois (onpt) lattis. Disrt Mathmatis, 256(3), p , 2002 [22] R. Will. Rstruturing th latti thory: An approah basd on hirarhis of onpts. In I. Rival, ditor, Ordrd sts, pags , Dordrht-Boston, 1982, Ridl. [23] F. Afrati, A. Gionis, H. Mannila. Approximating a olltion of frqunt sts. Pro. 10th ACM SIGKDD Intrnational Confrn on Knowldg Disovry and Data mining, p12 19, [24] G. Cong, K.L. Tan, A.K.H. Tung, F. Pan. Mining frqunt losd pattrns in miroarray data. Pro. 4th IEEE Intrnational Confrn on Data Mining, p , [25] T. Caldrs, C. Rigotti, J.F. Bouliaut. A survy on ondnsd rprsntations for frqunt sts. Constraindbasd Mining, Springr, 3848, [26] N. Pasquir, Y. Bastid, R. Taouil, L. Lakhal. Effiint mining of assoiation ruls using losd itmst lattis. Information Systms, 24(1), p25 46, [27] J. Pi, J. Han, R. Mao. CLOSET: An ffiint algorithm for mining frqunt losd itmsts. Pro. ACM SIGMOD Workshop on Rsarh Issus in Data Mining and Knowldg Disovry, p21 30, [28] F. Rioult, J.F. Bouliaut, B. Crmillux, J. Bsson. Using transposition for pattrn disovry from miroarray data. Pro. 8th ACM SIGMOD workshop on Rsarh issus in Data Mining and Knowldg Disovry, p73 79, [29] G. Stumm, R. Taouil, Y. Bastid, N. Pasquir, L. Lakhal. Fast Computation of Conpt Lattis using data mining thniqus. Pro. 7th Intrnational Workshop on Knowldg Rprsntation mts Databass, p , [30] J. Wang, J. Han, J. Pi. CLOSET+: Sarhing for th bst stragis for mining frqunt losd itmsts. Pro. 9th ACM SIGKDD intrnational onfrn on Knowldg Disovry and Data mining, p , [31] G. Yang. Th omplxity of mining maximal frqunt itmsts and maximal frqunt pattrns. Pro. 10th ACM SIGKDD Intrnational Confrn on Knowldg Disovry and Data mining, p , [32] M.J. Zaki, M. Ogihara. Thortial foundations of assoiation ruls. Pro. 3rd ACM SIGMOD Workshop on Rsarh Issus in Data Mining and Knowldg Disovry, p1 7, [33] M.J. Zaki, C.-J. Hsiao. CHARM: An ffiint algorithm for losd assoiation rul mining. Pro. 2nd SIAM Intrnational Confrn on Data Mining, [34] M.J. Zaki, C.-J. Hsiao. Effiint Algorithms for mining losd itmsts and thir latti strutur. IEEE Trans. Knowldg and Data Enginring, 17(4), p ,

13 Appndix Algorithm 2 Computing a Galois Latti Input: (O, M) and th adjany list of ah lmnt; Output: All onpts (C, int(c)) and a sussor list for ah onpt. 1: Initializ a tri T for th objt st O; 2: Start with a quu Q ontaining O; Eah lmnt in Q onsists of a pointr pointing to an xtnt in T. 3: int(o) = ; (O, ) is th top onpt. 4: whil Q is not mpty do 5: C = dquu(q); 6: Child(C, int(c)) = SPROUT(C, int(c)); 7: for ah pair (A, pr int C (A)) Child(C, int(c)) do 8: if A dos not xists in T thn 9: if COMPLETEINTENT(A, pr int C (A))=Ys thn 10: Add A to th sussor list of C; 11: Enquu A into Q; 12: nd if 13: int(a) = int(c) pr int C (A). 14: Insrt A into T ; 15: ls if (int(c) pr int C (A) = int(a)) thn 16: Add A to th sussor list of C; 17: if A is not in Q thn 18: Enquu A into Q; 19: nd if 20: nd if 21: nd for 22: Output th onpt (C, int(c)) and th sussor list of C; 23: nd whil 13