Journal of Universal Computer Science, vol. 6, no. 1 (2000), submitted: 20/10/99, accepted: 6/11/99, appeared: 28/1/00 Springer Pub. Co.

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1 Journal of Univrsal Computr Sin, vol. 6, no. 1 (000), submittd: 0/10/99, aptd: 6/11/99, appard: 8/1/00 Springr Pub. Co. Galois Conntions and Data Mining 1 Dana Cristofor (Univrsity of Massahustts at Boston Dpartmnt of Mathmatis and Computr Sin Boston, Massahustts 015, USA dana@s.umb.du.) Laurntiu Cristofor (Univrsity of Massahustts at Boston Dpartmnt of Mathmatis and Computr Sin Boston, Massahustts 015, USA laur@s.umb.du.) Dan A. Simovii (Univrsity of Massahustts at Boston Dpartmnt of Mathmatis and Computr Sin Boston, Massahustts 015, USA dsim@s.umb.du.) Abstrat: W invstigat th appliation of Galois onntions to th idntiation of frqunt itm sts, a ntral problm in data mining. Starting from th notion of losur gnratd by a Galois onntion, w dn th notion of xtndd losur, and w us ths notions to improv th lassial Apriori algorithm. Our xprimntal study shows that in rtain situations, th algorithms that w dsrib outprform th Apriori algorithm. Also, ths algorithms sal up linarly. Ky Words: Galois onntion, losur, xtndd losur, support, frqunt st of itms Catgory: H..0, E.5 1 Intrution Galois onntions ar algbrai onstrutions whih play an important rol in latti thory, univrsal algbras and, mor rntly, in omputr sin (s []). W dmonstrat thir usfulnss as an algbrai and onptual tool in dsigning int algorithms for th idntiation of frqunt sts of itms, as dnd in data mining. Lt (P; ) and (Q; ) b two partially ordrd sts. A Galois onntion btwn P and Q is a pair of mappings (8; 9) suh that 8 : P 0! Q, 9 : Q 0! P and: for x; x 0 P and y; y 0 Q. x x 0 implis 8(x) 8(x 0 ); y y 0 implis 9(y) 9(y 0 ); x 9(8(x)) and y 8(9(y)); 1 C. S. Calud and G. Stfansu (ds.). Automata, Logi, and Computability. Spial issu ddiatd to Profssor Srgiu Rudanu Fstshrift.

2 Cristofor D., Cristofor L., Simovii D.A.: Galois Conntions It is asy to vrify (s []) that 8(9(8(x))) = x and 9(8(9(y))) = y for x X and y Y. Lt (P; ) b a omplt latti, that is, a post suh that for any K P thr xist both sup K and inf K. Alosur is a mapping l : P 0! P suh that th following onditions ar satisd: 1. x l(x),. l(x) =l(l(x)) and 3. if x x 0 thn l(x) l(x 0 ), for all x; x 0 P. An lmnt x P is l-losd if l(x) =x. Th st of l-losd lmnts will b dnotd by C l. For any Galois onntion =(8; 9) btwn th omplt lattis (P; ) and (Q; ) th mapping l = 98 is a losur on P whil th mapping l 0 = 89 is a losur on Q. It is asy to s that in this as th mapping = 8C l is a bijtion btwn C l and C 0 l. A standard mth for gnrating Galois onntions is through th notion of polarity. Lt X; Y b two sts and lt R X Y b a rlation. Dn th mappings 8 : P(X) 0! P(Y ) and 9 : P(Y ) 0! P(X) by for K X and 8(K) =fy j y Y;(x; y) R for all x Kg; 9(H) =fx j x X; (x; y) R for all y Hg; for H Y. Th pair =(8; 9) intrud abov is a Galois onntion and is usually rfrrd to as th polarity on X and Y dtrmind by th rlation R. Dnition 1.1 Lt =(8; 9) b apolarity on th sts X and Y. Th smidistans gnratd by ar th mappings d : P(X) P(X) 0! N and d 0 : P(Y ) P(Y ) 0! N dnd byd (U 0 ;U 1 ) = j8(u 0 ) 8 8(U 1 )j for U 0 ;U 1 P(X), whr 8 is th symmtri dirn opration, and d 0 (V 0 ;V 1 )= j9(v 0 ) 8 9(V 1 )j for V 0 ;V 1 Y. Th proximitis gnratd by ar th mappings p : P(X) P(X) 0! N and p 0 : P(Y )P(Y ) 0! N dnd byp (U 0 ;U 1 )=j8(u 0 )\8(U 1 )j for U 0 ;U 1 P(X), and p 0 (V 0 ;V 1 )=j9(v 0 ) \ 9(V 1 )j for V 0 ;V 1 Y. Th wight funtions ar th mappings w : P(X) 0! N and w 0 : P(Y ) 0! N givn by w (U) = j8(u)j and w (V ) = j9(v )j for vry U P(X) and V P(Y ). If th Galois onntion is lar from ontxt thn th subsript may b omittd. Also, if a st onsists of only on lmnt ` w may writ simply ` instad of f`g. Proposition 1. Lt b apolarity on th sts X and Y. W hav 1. d (U 0 ;U 0 [ U 1 )+d (U 1 ;U 0 [ U 1 )=d (U 0 ;U 1 ),. p (U 0 ;U 0 [ U 1 )=p (U 1 ;U 0 [ U 1 )=p (U 0 ;U 1 ), 3. d (U 0 ; l (U 0 ))=0, for vry U 0 ;U 1 P(X).

3 6 Cristofor D., Cristofor L., Simovii D.A.: Galois Conntions... Proof. Th dnition of d implis that whih imply d (U 0 ;U 0 [ U 1 )=j8(u 0 ) 8 8(U 0 [ U 1 )j = j8(u 0 ) 8 (8(U 0 ) \ 8(U 1 ))j = j8(u 0 ) 0 8(U 1 )j d (U 1 ;U 0 [ U 1 )=j8(u 1 ) 8 8(U 0 [ U 1 )j = j8(u 1 ) 8 (8(U 0 ) \ 8(U 1 ))j = j8(u 1 ) 0 8(U 0 )j; d (U 0 ;U 1 )=j8(u 0 ) 8 8(U 1 )j = j8(u 0 ) 0 8(U 1 )j + j8(u 1 ) 0 8(U 0 )j = d (U 0 ;U 0 [ U 1 )+d (U 1 ;U 0 [ U 1 ): For th sond part of th proposition w not that p (U 0 ;U 0 [ U 1 )=j8(u 0 ) \ 8(U 0 [ U 1 )j = j8(u 0 ) \ (8(U 0 ) \ 8(U 1 ))j = j8(u 0 ) \ 8(U 1 )j = p (U 0 ;U 1 ) for vry U 0 ;U 1 P(X). Th proof of p (U 1 ;U 0 [ U 1 )=p (U 0 ;U 1 )isntirly similar. Finally, not that d (U 0 ; l (U 0 )) = j8(u 0 ) 8 8(9(8(U 0 )))j = j8(u 0 ) 8 8(U 0 )j =0. Proposition 1.3 Th wight funtion w X and Y has th following proprtis: gnratd byapolarity =(8; 9) on 1. maxfw (U 0 );w (U 1 )gd (U 0 ;U 1 )+p (U 0 ;U 1 ),. p (U 0 ;U 1 )=w (U 0 [ U 1 ) minfw (U 0 );w (U 1 )g, 3. d (U 0 ;U 1 )+p (U 0 ;U 1 ) w (U 0 \ U 1 ),. w (U 0 )+w (U 1 )=d (U 0 ;U 1 )+p (U 0 ;U 1 ), 5. w (U 0 )=w (l (U 0 )), for vry U 0 ;U 1 P(X). Proof. W giv th argumnt for th third part of th proposition. Not that d (U 0 ;U 1 )+p (U 0 ;U 1 )=j8(u 0 ) 8 8(U 1 )j + j8(u 0 ) \ 8(U 1 )j = j8(u 0 ) [ 8(U 1 )j j8(u 0 \ U 1 )j = w (U 0 \ U 1 ): Th rst of th proof is lmntary and is lft to th radr. Proposition 1. Th proximity funtion p gnratd byapolarity =(8; 9) on X and Y has th following proprtis:

4 Cristofor D., Cristofor L., Simovii D.A.: Galois Conntions p (U; U) =w (U). p (U 0 ;U 1 )=p (U 1 ;U 0 ) 3. p (U 0 ;U 1 )=p (l (U 0 ); l (U 1 )). p (U 0 ;U ) p (U 0 ;U 1 )+p (U 1 ;U ) 0 w (U 1 ), for vry U; U 0 ;U 1 ;U P(X). Proof. Th argumnt is lmntary and is omittd. Proposition 1.5 Lt U 0 ;U 1 X and lt b apolarity on th sts X and Y. For vry sts U 0 ;U 00 suh that U 0 U 0 l (U 0 ) and U 1 U 00 l 0 (U 1 ) w hav d (U 0 ;U 00 )=d (l(u 0 ); l(u 1 )). Proof. Not that if U 0 U 0 l (U 0 ), thn 8(U 0 ) 8(U 0 ) 8(9(8(U 0 ))) = 8(U 0 ), so 8(U 0 )=8(U 0 ). This implis d (U 0 ;V 0 )=j8(u 0 ) 8 8(V 0 )j = j8(u 0 ) 8 8(U 1 )j = d (U 0 ;V 0 ). Frqunt Sts of Itms and Closurs of Sts of Itms Th notion of frqunt sts of itms is formulatd starting from two nit, nonmpty sts, a st of transations T and a st of itms I, and from a rlation R T I. Th mappings of th polarity dtrmind by th rlation R ar dnotd by ti R : P(T ) 0! P(I) and it R : P(I) 0! P(T ). If R is lar from ontxt, th subsript will b omittd. Th tabl assoiatd to this Galois onntion is a tabl whos hading is th st of itms I. Eah itm, rgardd as an attribut has th binary domain f0; 1g. Ift is a transation in T, thn T is rprsntd in th tabl by a tupl (dnotd by th sam lttr) suh that t[i] = 1 if and only if (t; i) R. Dnition.1 Lt b apolarity on th st of transations T and th st of itms I. Th support of a st of itms K, K I, is th numbr supp (K) = w (K)=jT j. A st K is -frqunt if supp (K) and is -maximal if it is -frqunt and thr isno-frqunt st L suh that K L. Th wight of a st of itms K is givn by w (K) =jti(k)j = jt T j t[k] = 1 for vry k Kj: In othr words, th wight of th st of itms K quals th numbr of transations that ar assoiatd with vry itm k K. In viw of Proposition 1.3 w hav Thorm. If isapolarity on th st of transations T and th st of itms I, thn for vry two sts of itms K 0 ;K 1 I w hav: 1. maxfsupp (K 0 ); supp (K 1 )g(d (K 0 ;K 1 )+p (K 0 ;K 1 ))=jt j;. supp (K 0 [ K 1 ) minfsupp (K 0 ); supp (K 1 )g; 3. d (K 0 ;K 1 )+p (K 0 ;K 1 ) supp (K 0 \ K 1 ) 1jT j;. supp (K 0 )+supp (K 1 )=(d (K 0 ;K 1 )+p (K 0 ;K 1 )) 1jT j; 5. supp (K 0 )=supp (l (K 0 ));

5 6 Cristofor D., Cristofor L., Simovii D.A.: Galois Conntions K 0 K 1 implis supp (K 1 ) supp (K 0 ). Proof. Th argumnts for th rst v parts follow immdiatly from Dnition.1 and th orrsponding parts of Proposition 1.3. Th last part follows from Part (ii). Thus, if K 1 is an -frqunt st of itms and K 0 K 1, thn K 0 is also -frqunt. Corollary.3 For vry sts of itms K; H suh that K H l (K) w hav supp (K) =supp (H). Dnition. Lt b apolarity on th st of transations T and th st of itms I and lt L I. Th family of (`; )-xtndd losurs of L rlativ to is th olltion of sts (L) =fl (L) [ H jjhj = `; H \ l (L) =;; and supp(l [ H) g: Not that ; if supp(l) CL 0; <; (L) = fl (L)g; othrwis. CL`; Furthr, CL 1; (L) =fl (L) [fig ji 6 l (L) and supp(l [fig) g: W rfr to any mmbr of CL`; (L) asan(`; )-xtndd losur. Unlss statd othrwis w always work with (1;)-xtndd losurs; so, w will just rfr to thm as xtndd losurs. W will oftn rfr to a st of itms using th trm itmst. Dnition.5 An itmst I annot b xtndd by losur if it is l-losd and its family of xtndd losurs is ;. To omput th family of (1;)-xtndd losurs for a st of itms w nd to dn th k-matrix M [k] of th Galois onntion. Evry row of this matrix orrsponds to a st of itms of ardinality k. W assum that th st of itms is I = fi 0 ;:::;i n01 g and that th sts of itms of lngth k ar arrangd in lxiographial ordr. Th olumns of th matrix M [k] orrspond to th itms of I. IfC is a st of itms suh that jcj = k and i p I, thn M [k] C;i p, th lmnt loatd in th C-row and p-th olumn is M [k] C;i p = p (C; fi p g)=jft T j t[c; i p ]=(1;:::;1)gj: Not that th matrix M 1 is symmtri. Also, th largst valu of th ntris of th C-lin of th matrix M [k] will b found in th olumns that orrspond to th mmbrs of C and possibly in othr olumns. Th losur and family of xtndd losurs of a st of itms K an b omputd starting from th tabl assoiatd to th Galois onntion, as shown in th nxt stion.

6 Cristofor D., Cristofor L., Simovii D.A.: Galois Conntions Thorm.6 Lt =(ti R; it R) b th polarity assoiatd to th rlation R T I. W hav i l (K) if and only if M [k] K;i k = jkj. Proof. Not that maxfm [k] K;j M [k] = maxfm [k] K;j j j Ig, whr j j Ig = supp (K). For any i I suh that = K;i supp (K) and i 6 K w hav to prov that i l (K). This follows from th dnition of th valu M [k] K;i whih tlls us that i appars in all transations in whih K appars so i blongs to th losur of th itmst K. Finding all -frqunt sts of itms for a spid is an important data mining problm as it rprsnts th most omputationally xpnsiv stp in nding assoiation ruls in a databas [1]. In litratur ([1], []) th valu of is alld minimum support and -frqunt sts of itms ar alld larg itmsts. In th following stions w prsnt on algorithm (Closur) for nding all - frqunt itmsts and two algorithms (MaxClosur, AltMaxClosur) for nding all maximal -frqunt itmsts. 3 Th Closur Algorithm W us th notions of losur and xtndd losur to improv on th numbr of databas sans don by th Apriori algorithm. Our algorithm is basd on th following rsult: Thorm 3.1 Lt F ;k b th olltion [ of -frqunt k-itmsts. Dn 1; CF ;k = fl (F )g[cl (F ): F F ;k Thn, for vry G F ;k+1 thr is a st C CF ;k suh that G C. Proof. Suppos G F ;k+1 ; thn, all substs of G that hav ardinality k ar inludd in F ;k.w an writ G = F S fig whr F F ;k and i is an itm. W now hav two ass: 1. If th itm i appars in all transations in whih F appars, thn i l (F) and sin l (F ) is on of th lmnts of CF ;k it follows that G must b inludd in on of th lmnts of CF ;k.. If i dos not appar in all transations in whih F appars, but it appars in a fration of transations gratr than, thn this will nsur that on (1;)- xtndd losur of F will ontain i whih mans that G must b inludd in on of th lmnts of CF ;k. Th algorithm basd on this ida uss th matrix M prsntd bfor and works as follows: Lt Maximal b th olltion of all -maximal itmsts.

7 66 Cristofor D., Cristofor L., Simovii D.A.: Galois Conntions... Candidat = all 1 itmsts KFrqunt = mpty Maximal = mpty whil (tru) do initializ th lmnts of matrix M with 0; ount th support for all lmnts of Candidat and updat thir orrsponding rows in M; for all itmsts C in Candidat do if (C is not frqunt) ontinu to nxt itmst; add C to KFrqunt; omput losur of C in l(c); omput all xtndd losurs of C and add thm to Maximal; if no xtndd losurs wr found add l(c) to Maximal; mpty Candidat; gnrat nw andidat itmsts in Candidat basd on KFrqunt; if Candidat is mpty brak; mpty KFrqunt; for all itmsts L in Maximal mark as frqunt all itmsts from Candidat that ar ontaind in L; for all itmsts C in Candidat if C is markd as frqunt add C to KFrqunt; mpty Candidat; gnrat nw andidat itmsts in Candidat basd on KFrqunt; if Candidat is mpty brak; mpty KFrqunt; rturn Maximal Not that th Closur algorithm rquirs (b n + 1) sans of th databas whr n is th dimnsion of th longst frqunt itmst. Th MaxClosur Algorithm W wrot th MaxClosur algorithm in ordr to provid a fast algorithm for nding th maximal frqunt sts. As mntiond in [3], nding all frqunt sts is an xtrmly xpnsiv omputation for som databass, so it maks sns to hav instad a fast algorithm that would allow on to know what th maximal itmsts ar (whih also mans what th frqunt itmsts ar) without nding th support for all frqunt andidats. Basd on this knowldg, on ould guid th mining pross to xtrat only th ruls on is intrstd in.

8 Cristofor D., Cristofor L., Simovii D.A.: Galois Conntions Th MaxClosur algorithm is a spialization of th Closur algorithm. It uss a similar matrix strutur for omputing th losurs and xtndd losurs and it maks us of th fat that if at any point a frqunt itmst has an mpty xtndd losur, thn that itmst is a maximal frqunt itmst (it blongs to th positiv bordr [6]). Th matrix usd in this algorithm dirs from th on dsribd bfor in th fat that its rows an orrspond to itmsts of dirnt ardinality. Thorm.1 If a frqunt itmst F annot b xtndd by losur thn it is a maximal frqunt itmst. Proof. Suppos that F annot b xtndd by losur but is not a maximal frqunt itmst. Thn, thr xists a frqunt itmst G suh that F G and G 0 F 6= ;. Lt i b an itm suh that i G and i 6 F. Sin G is frqunt it folows that i is frqunt and appars with F in a fration of transations gratr than. Thn i should ithr blong to th losur of F or to an xtndd losur of F and w hav a ontradition sin F ould not b xtndd by losur. Th algorithm's psudo is prsntd blow. It uss thr olltions: Candidat to kp th andidat maximal frqunt itmsts, Frqunt whih is usd for intrmdiat storag, and Maximal whih ontains th maximal frqunt itmsts found at any point. Candidat = all 1-itmsts; Frqunt = mpty; Maximal = mpty; whil (tru) do initializ th lmnts of matrix M with 0; ount th support for all lmnts of Candidat and updat thir orrsponding rows in M; for all itmsts C in Candidat do if C is not frqunt ontinu to nxt itmst; omput losur of C in l(c); for all xtndd losurs xl(c) of C if xl(c) has not bn alrady addd to Frqunt add xl(c) to Frqunt; if no xtndd losurs wr found if l(c) has not bn alrady addd to Maximal add l(c) to Maximal; if Frqunt is mpty brak; mpty Candidat; mov into Candidat all lmnts from Frqunt; rturn Maximal; Not that th MaxClosur algorithm rquirs in th worst as n 0 1 sans of th databas whr n is th dimnsion of th longst frqunt itmst. Th

9 68 Cristofor D., Cristofor L., Simovii D.A.: Galois Conntions... rason is that in th worst as, a andidat itmst an b xtndd with only on itm during on san of th databas so it will tak n 0 1 sans of th databas to gnrat an n-itmst. 5 An Extnsion of th MaxClosur Algorithm Th algorithm prsntd in this stion is dsignd to guarant fwr passs ovr th databas, as ompard to MaxClosur. AltMaxClosur rquirs (b n +1) sans of th databas whr n is th dimnsion of th longst frqunt itmst, th sam numbr that Closur dos. AltMaxClosur is an xtnsion of MaxClosur. Th ida is that instad of starting with a andidat C, omputing th xtndd losur of C and thn using this xtndd losur as a andidat, w an try to xtnd vn furthr th xtndd losur and us this xtnsion as a andidat. This nsurs that w will do half th numbr of passs rquird by MaxClosur. Th psudo for th algorithm is prsntd blow. It uss on additional olltion: XPrvCandidat to kp th xtndd losurs gnratd from th prvious andidat itmsts. Candidat = all 1-itmsts; Frqunt = mpty; XPrvCandidat = mpty; Maximal = mpty; whil (tru) do initializ th lmnts of matrix M with 0; ount th support for all lmnts of Candidat and updat thir orrsponding rows in M; if XPrvCandidat is not mpty do for all itmsts C in Candidat mark as not bing maximal all itmsts from XPrvCandidat that ar ontaind in C; for all itmsts L in Maximal mark as not bing maximal all itmsts from XPrvCandidat that ar ontaind in L; for all itmsts P in XPrvCandidat if P is markd as bing maximal add P to Maximal; mpty XPrvCandidat; for all itmsts C in Candidat do if C is not frqunt ontinu to nxt itmst; omput losur of C in l(c); for all xtndd losurs xl(c) of C if xl(c) has not bn alrady addd to XPrvCandidat do add xl(c) to XPrvCandidat;

10 Cristofor D., Cristofor L., Simovii D.A.: Galois Conntions for all itms i not in xl(c) suh that M(C,i) >= psilon do rat itmst C' from th union of xl(c) and i; if C' has not bn alrady addd to Frqunt add C' to Frqunt; if no xtndd losurs wr found if l(c) has not bn alrady addd to Maximal add l(c) to Maximal; if Frqunt is mpty brak; mpty Candidat; mov into Candidat all lmnts from Frqunt; rturn Maximal; Givn an xtndd losur of a andidat C, w try to xtnd it by looking at all itms that appar with C in a numbr of transations gratr or qual to. W also stor xl(c) in XPrvCandidat in th vntuality that non of ths xtnsions prov to b frqunt whih would man that xl(c) an not b furthr xtndd so it is maximal. A nw stp has bn intrud for vrifying whthr any of th lmnts of XPrvCandidat ar maximal. 6 Exprimntal rsults W implmntd Apriori, Closur, MaxClosur, and AltMaxClosur in C++. W hav usd th hash-tr strutur intrud in [] in ordr to intly tst for itmst inlusions. Th data was gnratd using th IBM Qust synthti data gnrator and was stord in binary ls. As obsrvd by othr rsarhrs [7], whn working with ls th bottlnk is in th CPU omputations, not in assing th disk. For this rason, th dirn btwn Closur and Apriori is not as big as on would xpt. In a prvious xprimnt in whih w ran a prliminary vrsion of th Apriori and Closur algorithms using Oral databass, th rsults showd a 50% improvmnt in tim on bhalf of th Closur algorithm. Our rst xprimnts usd databass with an avrag numbr of itms pr transation qual to 10, and a total of 100 itms. W hav gnratd a 50 thousand, a 100 thousand and a 1 million rows databas in ordr to hk for th salability of ah algorithm. Th rsults of th xprimnts, prsntd in Figur 1, show larly that all algorithms sal linarly with th siz of th databas. Th graphs in Figur show that at rasonabl lvls of support, our algorithms outprform th Apriori algorithm. Similar rsults hold for AltMaxClosur whn ompard to Apriori, as shown in Figur 3. On th othr hand, undr our urrnt implmntations, whih ar basd on raw ls rathr than on rlational databass, AltMaxClosur is always slowr than MaxClosur, dspit th fat that it dos fwr passs ovr th data. Th

11 70 Cristofor D., Cristofor L., Simovii D.A.: Galois Conntions... Support % Tim in sonds Apriori Closur MaxClosur AltMaxClosur 50 thousand rords thousand rords million rords Figur 1: Prforman of Algorithms thousand rows Apriori Closur MaxClosur Tim (s) Minimum support (%) Figur : Prforman Graphs for Apriori, Closur, and MaxClosur

12 Cristofor D., Cristofor L., Simovii D.A.: Galois Conntions Tim00 (s) thousand rows Apriori MaxClosur Minimum support (%) 3 Figur 3: Prforman Graphs for Apriori and AltMaxClosur rason is that AltMaxClosur has to do svral quality and inlusion tsts in ordr to nsur that no dupliats ar addd to its various olltions and thos oprations prov to b quit xpnsiv. In Figur w prsnt th rsults obtaind whn w ran th algorithms on a 100,000 thousand rows databas having an avrag of 5 itms pr transation. In this xprimnt Closur prforms slightly wors than Apriori baus it dos mor omputations whih ar not ompnsatd by th drasd numbr of passs ovr th databas. MaxClosur ontinus to prform bttr than th othr two although th improvmnt oms from ounting th support for fwr itmsts rathr than from doing fwr sans ovr th databas. For 5% minimum support th algorithms disovr 1699 maximal frqunt itmsts. W also did a tst on a databas with 100,000 transations and an avrag of 50 itms pr transation. W ran th algorithms for 50% minimum support and w obtaind 3 maximal frqunt itmsts: Apriori Closur MaxClosur AltMaxClosur Tim Databas passs 8 7 W onlud that as th avrag numbr of itms pr transation inrass, th prforman of th Closur algorithm drass. MaxClosur prforms its worst as numbr of passs but it oms prtty los to Closur baus it dos not omput all frqunt itmsts.

13 7 Cristofor D., Cristofor L., Simovii D.A.: Galois Conntions Tim (s) thousand rows Apriori Closur MaxClosur Minimum support (%) Tim (s) thousand rows Apriori MaxClosur Minimum support (%) Figur : Prforman Graphs at 5 Itms pr Transation 7 Conlusions W intrud thr nw algorithms: Closur, MaxClosur and AltMaxClosur that mak us of Galois onntions. Th xprimnts w hav don show that Closur and MaxClosur prform fastr than th standard Apriori algorithm whn running on databass with an avrag numbr of itms pr transation of lss than 5% of th total numbr of itms. For highr avrags (lik 50%), our algorithms ar outprformd by Apriori. Howvr, if databas ass is xpnsiv, Closur might still prform bttr than Apriori du to th fwr numbr of passs ovr th databas it rquirs. AltMaxClosur may b an altrnativ

14 Cristofor D., Cristofor L., Simovii D.A.: Galois Conntions for MaxClosur whn databas ass is xpnsiv; howvr, its prforman drass vry fast as th minimum support taks smallr valus. Rfrns [1] Agrawal R., Imilinski T., Swami A.:Mining Assoiation ruls btwn Sts of Itms in Larg Databass, SIGMOD 1993, pags [] Agrawal R., Srikant R.:Fast Algorithms for mining assoiation ruls, VLDB 199. [3] Bayardo R. J.:Eintly Mining Long Pattrns from Databass, SIGMOD 1998, pags [] Birkho G.:Latti Thory, Amrian Mathmatial Soity, Colloquium Publiations, 3rd dition, [5] Girz G.t al.:a Compndium of Continuous Lattis, Springr-Vrlag, Brlin, [6] Gunopulos D., Khardon R., Mannila H., Toivonn H.:Data mining, Hyprgraph Transvrsals, and Mahin Larning, PODS 1997, pags [7] Mullr A.:Fast Squntial and Paralll Algorithms for Assoiation Rul Mining: A Comparison, CS-TR-3515, Univrsity of Maryland-Collg Park, 1995.

Homework #3. 1 x. dx. It therefore follows that a sum of the

Homework #3. 1 x. dx. It therefore follows that a sum of the Danil Cannon CS 62 / Luan March 5, 2009 Homwork # 1. Th natural logarithm is dfind by ln n = n 1 dx. It thrfor follows that a sum of th 1 x sam addnd ovr th sam intrval should b both asymptotically uppr-