SPIN: Mining Maximal Frequent Subgraphs from Graph Databases

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1 Reserch Trck Poster : Mining Miml Frequent Sugrphs from Grph Dtses Jun Hun, Wei Wng, Jn Prins Deprtment of Computer Science, Universit of North Crolin t Chpel Hill Chpel Hill,NC 27599, USA {hun, weiwng, prins}@cs.unc.edu Jiong Yng Deprtment of Computer Science, Universit of Illinois Urn-Chmpign, IL 61801, USA jiong@cs.uiuc.edu ABSTRACT One fundmentl chllenge for mining recurring sugrphs from semi-structured dt sets is the overwhelming undnce of such ptterns. In lrge grph dtses, the totl numer of frequent sugrphs cn ecome too lrge to llow full enumertion using resonle computtionl resources. In this pper, we propose new lgorithm tht mines onl miml frequent sugrphs, i.e. sugrphs tht re not prt of n other frequent sugrphs. This m eponentill decrese the size of the output set in the est cse; in our eperiments on prcticl dt sets, mining miml frequent sugrphs reduces the totl numer of mined ptterns two to three orders of mgnitude. Our method first mines ll frequent trees from generl grph dtse nd then reconstructs ll miml sugrphs from the mined trees. Using two chemicl structure enchmrks nd set of snthetic grph dt sets, we demonstrte tht, in ddition to decresing the output size, our lgorithm cn chieve five-fold speed up over the current stte-of-the-rt sugrph mining lgorithms. Ctegories nd Suject Descriptors: H.2.8 [Dtse Applictions]: Dt Mining Generl Terms: Algorithms Kewords: Sugrph Mining, Spnning Tree 1. INTRODUCTION In this pper, we focus on the prolem of finding recurring sugrphs from grph dtses, which is ver ctive topic in current dt mining reserch. Grphs provide generl w to model vriet of reltions mong dt, hence finding recurring sugrphs hs mn pplictions in interdisciplinr reserch such s chemicl informtics [2] nd ioinformtics [11]. There re lso mn pplictions in dt mngement reserch such s efficient storge of semi-structured dtses [5], efficient indeing [21], nd we informtion mngement [16]. One performnce issue (mong mn others) in mining lrge grph dtses is the huge numer of recurring ptterns. The phe- Permission to mke digitl or hrd copies of ll or prt of this work for personl or clssroom use is grnted without fee provided tht copies re not mde or distriuted for profit or commercil dvntge nd tht copies er this notice nd the full cittion on the first pge. To cop otherwise, to repulish, to post on servers or to redistriute to lists, requires prior specific permission nd/or fee. KDD 04, August 22 25, 2004, Settle, Wshington, USA. Copright 2004 ACM /04/ $5.00. nomenon is well understood in mining long frequent itemsets. Given frequent itemset I, n suset of I is lso frequent hence the numer of such frequent itemsets grows eponentill with I. In this pper, we propose new grph mining lgorithm tht mines onl miml frequent sugrphs. Given set of grphs G (referred to s grph dtse), the support of grph G is defined s the frction of grphs in G in which G occurs [9, 20]. G is frequent if its support is t lest user specified threshold; frequent sugrph is miml if none of its super grphs re frequent [10]. Mining onl miml frequent sugrph offers the following dvntges in processing lrge grph dtses. (1) It significntl reduces the totl numer of mined sugrphs. In eperiments we performed on some relistic dt sets, the totl numer of frequent sugrphs is up to one thousnd times greter thn the numer of miml sugrphs. We cn sve oth spce nd susequent nlsis effort if the numer of mined sugrphs is significntl reduced. (2) Severl pruning techniques, which re detiled in this pper, cn e efficientl integrted into the mining process nd drmticll reduce the totl mining time. (3) The non-miml frequent sugrphs cn e reconstructed from the miml sugrphs reported. To get the ctul frequenc (support) of non-miml sugrphs requires emintion of the originl dtse, ut it is certin to e t lest s high s the frequenc of the miml sugrph. In ddition, the techniques used in [15] cn e esil dpted to pproimte the support of ll frequent sugrphs within some error ound. (4) In some pplictions such s discovering structure motifs in group of homolog proteins [7, 11], miml frequent sugrphs re the sugrphs of most interest since the encode the miml structure commonlities within the group. Our mining method is sed on novel grph mining frmework in which we first mine ll frequent tree ptterns from grph dtse nd then construct miml frequent sugrphs from trees. This pproch offers smptotic dvntges compred to using sugrphs s uilding locks, since tree normliztion is simpler prolem thn grph normliztion. The proposed method enles us to integrte well-developed techniques from mining miml itemsets nd knowledge gined in grph mining into new lgorithm. According to our eperimentl stud, such comintion cn offer significnt performnce speedup in oth snthetic nd rel dt sets. The frmework of our method is verstile. Deping on the prticulr tree mining lgorithm, the serch cn e either redth-first or depth-first (preferred due to its etter memor utiliztion). It cn lso e designed to mine ll frequent sugrphs without mjor modifictions. Technicll, we mke three contriutions: (1) we propose novel lgorithm (SPnning tree sed miml grph mining) to mine onl miml frequent sugrphs of lrge grph dtses, 581

2 Reserch Trck Poster (2) we integrte severl optimiztion techniques, some from eisting miml itemset mining reserch nd some developed ourselves, to speed up the mining process, (3) we perform n etensive nlsis of the proposed lgorithm nd nlze how its performnce on grph dt sets with different chrcteristics. The reminder of the pper is orgnized s follows. In Section 2, we present the dt structure nd the proposed lgorithm. Section 3 presents the results of our eperimentl stud using snthetic grph dtses nd two enchmrk chemicl dt sets. We conclude the pper with discussion, relted works, nd conclusion. 2. MAXIMAL SUBGRAPH MINING In the following discussion, we present novel frmework for mining miml frequent sugrphs from grph dtse. The frmework comines tree mining nd sugrph mining; we first find ll frequent trees from grph dtse nd then reconstruct the group of frequent sugrphs from the mined trees. There re two importnt components in the frmework. The first is grph prtitioning method through which we group ll frequent sugrphs into equivlence clsses sed on the spnning trees the contin. The second importnt component is set of pruning techniques which im to remove some prtitions entirel or prtill for the purpose of finding miml frequent ones onl. There re three resons we dvocte this two-step method for finding miml grph ptterns. First, tree relted opertions such s isomorphism, normliztion, nd testing whether tree is sutree of nother tree re smptoticll simpler thn the comprle opertions for grphs, which re NP-complete. Second, in certin pplictions, such s chemicl compound nlsis, most of the frequent sugrphs re rell trees. Lst ut not lest, this frmework dpts well to miml frequent sugrph mining, which is the focus of this pper. Using chemicl structure enchmrk, we show 99% of cclic grph ptterns nd 60% of tree ptterns cn e eliminted our optimiztion technique in serching for miml sugrphs; further detils out the efficienc of the optimiztion techniques cn e found in [10]. To the est of our knowledge, we re the first to comine the two distinct methodologies: mining frequent sugrphs in grph dtses nd mining frequent trees in forests ( set of trees) for the purpose of designing efficient sugrph mining lgorithm. 2.1 Tree-sed Equivlence Clsses We define sutree of n undirected grph G s n cclic connected sugrph of G. A sutree T is spnning tree of G if T contins ll nodes in G. Given grph G, there re mn spnning trees nd we define the miml one, ccording to totl order defined on trees [4, 10], nd cll it the cnonicl spnning tree of G. EXAMPLE 2.1. In Figure 1, we show n emple of leled grph P (upper-left) with ll four-node sutrees of P. Ech sutree is represented its cnonicl representtion nd sorted ccording to the totl order (s given in [10]). Ech such tree is spnning tree of the grph P nd the first one (T 1) is the cnonicl spnning tree of P. DEFINITION 2.1. Tree-sed Equivlence Clsses: Given two grphs P nd Q, we defined inr reltion = such tht P = Q if nd onl if their cnonicl spnning trees re isomorphic to ech other. The reltion = is refleive, smmetric, trnsitive, nd hence n equivlence reltion. p 2 p 1 p 3 (P) p 4 (T 1 ) (T 2 ) (T 3 ) (T 4 ) (T 5 ) (T 6 ) (T 7 ) Figure 1: Emple of leled grph P (upper-left), P s sutrees, spnning trees, nd its cnonicl spnning tree (T 1 ). EXAMPLE 2.2. In Figure 2, we show sugrphs of the grph P in Figure 1 which re not necessril trees. Sugrphs re grouped together if the shre the sme cnonicl spnning tree. The five non-singleton groups re shown here nd the remining twelve groups re ll singletons 1 Clss I Clss II Clss IV Clss III Clss V Figure 2: Emple of tree sed equivlence clsses for sugrphs in grph P, presented in Figure 1. We cn use simple greed serch lgorithm to find the cnonicl spnning tree of grph G, the detils of which re given in [10]. The frequent sugrph mining cn conceptull e roken into two steps: (1) mine ll the frequent trees from grph dtse nd (2) for ech such frequent tree T, find ll frequent sugrphs whose cnonicl spnning trees re isomorphic to T. Miml frequent sugrphs cn e found mong frequent ones. We skip the first step in the following discussion for two resons. First, s pointed out in [20], the current sugrph mining lgorithms cn e esil tilored to find onl trees from grph dtse limiting the topolog of the ptterns. This is true for Closegrph [20] s well s for FFSM [9], which is our recentl developed depth-first sugrph mining lgorithm. Second, most of the techniques developed for mining sutrees from forest cn lso e esil dpted 1 Throughout the pper, we re interested onl in sugrphs with t lest n edge (i.e. ecluding frequent nodes s trivil cses). 582

3 Reserch Trck Poster K S1 K S2 K 2 K 1 K 3 K Clss I C= {(k 2, k 3, ), (k 3, k 4, )} K 4 K S3 Before we proceed to detils out mining miml frequent sugrphs, we outline the enumertion scheme discussed so fr in Tle 1 nd Tle 2. Our strteg is quite strightforwrd: we first find ll frequent trees; trees re epnded to cclic grphs serching their serch spces; nd miml frequent sugrphs re constructed from frequent ones. We notice tht this lgorithm is correct, which mens we re gurnteed to find ll miml frequent sugrphs. However, it is not efficient in tht we still need to enumerte ll frequent sugrphs to construct miml ones. In the net section, we introduce optimiztion techniques to improve the serch for miml frequent sugrphs. Figure 3: Emple of enumerting grph s serch spce. We use dshed lines on the sugrph K S1 nd K S3 to denote the fct tht the will e pruned n optimiztion technique which is discussed in Section for the sme purpose. Therefore, in the following discussion, we focus on step 2, which is how to enumerte the equivlence clss of tree T nd how miml sugrph mining is relted to this enumertion. We wnt to point out tht the two-step division of the mining lgorithm is rtificil ut it mkes it es to eplin the ke ides of the lgorithm without introducing too mn detils. In our longer version of this pper [10], we discuss full optimized lgorithm which (1) uses modified FFSM lgorithm to enumerte trees from grph dtses nd (2) integrtes tree discover nd miml pttern mining for miml performnce. 2.2 Enumerting Grphs from Trees We first outline sic enumertion scheme to serch the equivlence clss of tree. We define joining opertion etween grph(tree) G nd hpotheticl edge connecting n two nodes i, j in G with lel e l such tht G (i, j, e l )=G where G is supergrph of G with one dditionl edge etween nodes i nd j with lel e l. If the grph G lred contins n edge etween nodes i nd j, the joining opertion fils nd produces nothing. If G is frequent, we denote the hpotheticl edge (i, j, e l ) s cndidte edge for G. The ove definition cn serve s the sis for recursive definition of the joining opertion etween grph G nd cndidte edge set E = {e 1,e 2,...,e n} such tht G E = (G e 1) {e 2,...,e n}. Let s ssume we lred clculted the set of cndidte edges C = {c 1,c 2,...,c n} from the set of ll possile frequent hpotheticl edges. We define the serch spce of G, denoted G : C, s the set of grphs which might e produced joining the grph G nd cndidte edge set in the powerset set of C (denoted 2 C ). Tht is: G : C = {G 2 C } (1) In the following discussion, the group of cndidte edges re sometimes referred to s the til of the grph G in its serch spce. We present recursive lgorithm in Tle 2 to enumerte the serch spce for grph G. The procedure we use to clculte the set of cndidte edges for tree pttern cn e found in [10]. EXAMPLE 2.3. in Figure 3, we single out the lrgest equivlence clss (Clss One) from Figure 2. We show tree K together with its til C = {(k 2,k 3,), (k 3,k 4,)}. K s serch spce is hence composed of four grphs {K, K S1,K S2,K S3} (K is lws included in its serch spce) nd is orgnized into serch tree in nlog to frequent item set mining. This tree structure follows the recursive procedure we present in Tle 2. Algorithm Miml Sugrph Mining(G,σ) egin 1. R {T T is frequent tree in G} 2. S {G G Epnsion(T ) nd T R} 3. return {G G S nd G is miml } Tle 1: An outline of the miml sugrph mining lgorithm Algorithm Epnsion(T ) egin 1. C {c c is cndidte edge for T } 2. S Serch Grphs (T,C) 3. return {G G S, G is frequent, nd G hs the sme cnonicl spnning tree s T hs} Algorithm Serch Grphs(G, C = {c 1,c 2,...,c n}) egin 1. Q 2. for ech c i C 3. Q Q Serch Grphs(G c i, {c i+1,c i+2...,c n}) 4. for 5. return Q Tle 2: An lgorithm for eploring the equivlence clss of tree T 2.3 Optimiztions: Glol nd Locl Miml Sugrphs In this section, we eplore severl techniques for fst miml frequent sugrph mining. These techniques ( pruning techniques) dnmicll remove set of frequent sugrphs tht cn not e miml from serch spce. To tht, we define frequent sugrph G to e locll miml if it is miml in its equivlence clss i.e. G hs no frequent supergrph(s) tht shre the sme cnonicl spnning tree s G; we refer to sugrph s gloll miml if it is miml frequent in grph dtse. Clerl, ever glol miml sugrph must e locll miml ut not ever locl miml sugrph is necessril gloll miml. Our pruning techniques im to void enumerting sugrphs which re not locll miml. Not surprisingl, the prolem of finding ll locll miml frequent sugrphs cn e trnsformed to the well-known miml frequent itemset mining prolem. Ech cndidte edge is n item; the joining opertion cn e viewed s the union opertion for itemsets; nd ech locl miml sugrph corresponds to miml frequent itemset in its serch spce. Hence, we dvocte the following pruning techniques, which re prtill dpted from the miml itemset mining nd prtill developed in the grph mining contet, for miml frequent sugrph mining. 583

4 Reserch Trck Poster Bottom-Up Pruning The serch spce of grph G is eponentil in the crdinlit of the cndidte edges set C. One heuristic to void such n eponentil serch spce is to check whether the lrgest possile cndidte G = G C is frequent or not. If G is frequent, ech grph in the serch spce is sugrph of G nd hence not miml. This heuristics is referred to s the Bottom-Up Pruning nd cn e pplied to ever step in the recursive serch procedure presented in Tle 2. B ppling ottom-up pruning to the equivlence clss I presented in Figure 2, grph K S1 nd K S3 re pruned. Dnmic Reordering: An importnt technique relted to the efficienc of the ottom-up pruning is the so-clled dnmic reordering technique, which works in two ws. First, it trims infrequent cndidte edges from the til of grph to reduce the size of the serch spce (n edge cndidte cn ecome infrequent fter severl itertions since other edges re incorported into the ptterns). Second, it rerrnges the order of the elements in the til ccording to their support vlue. For emple, given grph s til C, dnmic reordering, we sort the elements in C their support vlues, from lowest to highest. After this sorting, the infrequent heds re trimmed. At the of the remining til is fmil of elements individull hving high support nd hence the pttern otined grouping them together is likel to still hve high support vlue. This heuristics is widel used in mining miml itemsets to gin performnce. However, without the spnning tree frmework, ppling dnmic ordering is ver difficult in n of the current sugrph mining lgorithms, which intrinsicll hve fied order of dding edges to n eisting pttern for vrious performnce considertions Til Shrink Given grph G nd supergrph G of G, nemedding of G in G is sugrph isomorphism f from G to G. We prefer the term emedding to sugrph isomorphism, though the re interchngele, for the purpose of intuitive descriptions. In Figure 4, we show sugrph L nd its supergrph P. There re two emeddings of L in P : (l 1 p 1,l 2 p 2,l 3 p 3,l 4 p 4) nd (l 1 p 1,l 2 p 3,l 3 p 2,l 4 p 4). We define cndidte edge (i, j, e l ) to e ssocitive to grph G if it ppers in ever emedding of G in grph dtse. In other words, cndidte edge (i, j, e l ) of G is ssocitive if nd onl if for ever emedding f of G in grph G, G hs the edge (f(i),f(j)) with lel e l. One emple of ssocitive edge is edge (l 1,l 3,) to the tree L shown in Figure 4. If tree T contins set of ssocitive edges {e 1,e 2,...,e n}, n miml frequent grph G which is supergrph of T must contin ll such edges. Hence we cn remove these edges from the til of T nd ugment them to T without missing n miml ones. This technique is referred to s the til shrink technique. Til shrink hs two dvntges: (1) it reduces the serch spce nd (2) it cn e used to prune the entire equivlence clss in certin cses. To elorte the ltter point, we define set of ssocitive edges C of tree T to e lethl if the resulting grph G = T C hs cnonicl spnning tree other thn tht of T. For emple, in Figure 4, ssocitive edge e =(1, 3,) of L is lethl since G = L e hs different cnonicl spnning tree thn tht of L. In the sme emple, the lethl edge e cn e ugmented to ech memer of the clss II to produce supergrph with the sme support. Therefore the whole clss cn e pruned w once we detect lethl edge(s) to the tree L. Detecting group of lethl edges cn do further pruning other thn trimming off the whole equivlence clss. Those detils s well s the forml proof of the optimiztion re discussed in [10]. l 2 l 3 l 1 l 4 Clss II L L S1 L S2 p 1 p 2 p 3 (P) p 4 Figure 4: An emple showing how til shrink might e used to prune the whole equivlence clss. Edge e =(l 1,l 3,), denoted dshed line to e distinguished from other edges, is ssocitive to tree L nd lethl to L s well. The grph otined joining L nd e should elong to equivlence clss I shown in Figure Eternl-Edge Pruning In this section, we introduce technique to remove one equivlence clss without n knowledge out its cndidte edges. We refer to this technique s the eternl-edge pruning. We define n edge to e n eternl edge for grph G if it connects node in G nd node which is not in G. We represent n eternl edge s three-element tuple (i, e l,v l ) to stnd for the fct tht we introduce n edge with lel e l incident on the node i in grph G nd node which does not elong to G with node lel v l. An eternl edge (i, e l,v l ) is ssocitive to grph G if nd onl if:. for ever emedding f of G in grph G, G hs node v with the lel v l, v connects to the node f(i) with n edge lel e l in G, nd node j V [G] such tht v = f(j). EXAMPLE 2.4. We show two emples of ssocitive eternl edges in Figure 5. One is (m 1,,) for the tree M nd nother one is (n f 1,,) for the tree N.IftreeT hs t lest one ssocitive eternl edge, the entire equivlence clss of T cn e pruned since the sme edge cn e ugmented to ever memer of the clss. In this emple, oth equivlence clsses IV nd V cn e eliminted due to the eternl-edge pruning. Once we find tree T hs n ssocitive eternl edge, the sme edge cn e ugmented to ech memers in T s equivlence clss nd therefore none of them re miml. Figure 5: Emples showing eternl edges nd ssocitive eternl edges. In rief summr, we present three pruning techniques to speed up miml sugrph mining. For the grph P shown in Figure 1, there re totl of twent five sugrphs of P, including itself 584

5 nd ecluding the null grph. These sugrphs re prtitioned into five non-singleton clsses, shown in Figure 2, nd twelve singleton clsses (not shown). There is onl one miml sugrph, nmel, grph P itself. We hve successfull pruned ever one of the five non-singleton equivlence clsses (P of the equivlence clss I is left untouched since it is miml). Wht we do not show further is tht we cn ppl the sme techniques to the remining twelve singleton equivlence clsses to eliminte ll of them. Interested reders might verif tht themselves. Tle 3 nd Tle 4 integrte these optimiztions into the sic enumerte technique we presented in Tle 1 nd Tle 2. Algorithm MSugrph-Epnsion(T ) egin 1. C {c c is cndidte edge for G} 2. A {c c C nd c is ssocitive } 3. ifa is lethl return #til shrinking 4. S Serch Grphs (T A, C A) 5. return {G G S, G is frequent, nd G hs the sme cnonicl spnning tree s T hs} Algorithm Serch Grphs(G, C = {c 1,c 2,...,c n}) egin 1. if G C is frequent, return G C #ottom-up pruning 2. Q 3. for ech c i C 4. Q Q Serch Grphs(G c i, {c i+1,c i+2...,c n}) 5. for 6. return Q Tle 3: An lgorithm for eploring the equivlence clss of tree T for miml sugrph mining Algorithm Miml Sugrph Mining(G,σ) egin 1. R {T T is frequent tree in G} 2. S {G G Epnsion(T ), T hs no eternl ssocitive edge, nd T R}#eternl-edge prunning 3. return {G G S nd G is miml } Tle 4: An outline of the miml sugrph mining lgorithm Due to the spce limittion, severl importnt detils re omitted which include: (1) how to enumerte frequent trees from grph dtse using modified FFSM lgorithm, (2) how to interleve the tree mining lgorithm nd the miml sugrph mining lgorithm nd deliver the finl optimized lgorithm, (3) how we gurntee tht ech reported pttern is () frequent, () miml, nd (c) unique, (4) how to clculte the edge cndidtes for tree, nd (5) how to determine ssocitive eternl edges. Those cn e found in [10]. 3. EXPERIMENTAL STUDY We performed our empiricl stud using single processor of 2.8GHz Pentium Xeon with 512KB L2 cche nd 2GB min memor, running RedHt Linu 7.3. The lgorithm is implemented using the C++ progrmming lnguge nd compiled using g++ with O3 optimiztion. We compred with two lterntive sugrph mining lgorithms: FFSM ([9]) nd gspn [19]. Ever miml sugrph reported in sntheticl nd rel dt sets re cross vlidted using results from FFSM nd gspn to mke sure it is () frequent, () miml, nd (c) unique. 3.1 Snthetic Dtset To evlute the performnce of the lgorithm, we first generte set of snthetic grph dtses using snthetic dt genertor [13]. In Figure 6, we represent the performnce comprison of, FFSM, nd gspn lgorithms for snthetic dt set with different support vlues. When the support is set to prett high vlue e.g. 5%, the performnce of ll three lgorithms re prett close. scles much etter thn the other two lgorithms s we decrese the support vlues. At support vlue 1%, provides si nd ten fold speed-up over FFSM nd gspn, respectivel. We do not show dt with support vlue gret thn 5% since there is little difference mong the three methods. More testing results on sntheticl dt sets cn e found in [10]. Run time (s) 10 1 FFSM gspn Totl identified sugrphs Reserch Trck Poster FFSM/gSpn Figure 6: Left: performnce comprison under different support vlues for dt set D10kT30L200I11V 4E4 using, FFSM nd gspn. Here we follow the common convention of encoding the prmeters of snthetic grph dtse s string. Right: Totl frequent ptterns identified the lgorithms. 3.2 Chemicl Dt Set We lso pplied to two widel used chemicl dt sets to test its performnce. The dt sets re otined from the DTP AIDS Antivirl Screen test, conducted U.S. Ntionl Cncer Institute. In the DTP dt set, chemicls re clssified into three sets: confirmed ctive (CA), confirmed modertel ctive (CM) nd confirmed inctive (CI) ccording to eperimentll determined ctivities ginst the HIV virus. There re totl of 423, 1083, nd chemicls in the three sets, respectivel. For our own purposes, we used ll compounds from CA nd from CM to form two dt sets, which re susequentl referred to s DTP CA nd DTP CM, respectivel. The DTP dt cn e downloded from dt.html. In Figure 7, we show the performnce comprison of, FFSM, nd gspn using the DTP CA dt set. We report tht is le to epedite the progrm up to five(eight) fold, compring with FFSM(gSpn) t support vlue 3.3%. Mining onl miml sugrphs cn reduce the totl numer of mined ptterns fctor up to three orders of mgnitude in this dt set. We lso pplied the sme lgorithms to the dt set DTP CM. In this cse, hs performnce ver close to FFSM nd oth re round eight fold speed-up over gspn. However, if we impose n dditionl constrint to let FFSM output the miml ptterns it finds mong the set of frequent ptterns, offers three fold speed-up from FFSM. 585

6 Reserch Trck Poster Run time (s) 10 1 FFSM gspn Totl identified sugrphs FFSM/gSpn Run time (s) 10 1 FFSM gspn Totl identified sugrphs FFSM/gSpn Figure 7: Left: performnce comprison under different support vlues for DTP CA dt set using, FFSM nd gspn. Right: Totl frequent ptterns identified the lgorithms. 4. RELATED WORK Knowledge discover from semi-structured dt sets is n ctive topic in the dt mining/mchine lerning communit. Mn different pttern definitions were proposed from different perspectives such s finding ptterns from single lrge network [14], finding pproimtel mtched ptterns [17], mining ptterns using domin knowledge from ioinformtics [8], nd finding frequent sugrphs. The lter one is the focus of our pper. Recent sugrph mining lgorithms cn e roughl clssified into two ctegories. Algorithms in the first ctegor use levelwise serch scheme sed on the Apriori propert to enumerte the recurrent sugrphs [12, 13]. Rther thn growing grph one single node/edge t time, Vnetik et l. recentl proposed n Apriori-sed lgorithm using pths s uilding locks with novel support definition [18]. Algorithms in the second ctegor use depth-first serch to enumerte cndidte frequent sugrphs [19, 20, 2, 9]. As demonstrted in these ppers, depth first lgorithms provide dvntges over level-wise serch for (1) etter memor utiliztion nd (2) efficient sugrph testing, e.g. it usull permits the sugrph test to e performed incrementll t successive levels during the serch [9]. Our current work enefits etensivel from eisting lgorithms for miml itemset mining such s [3, 6] nd frequent sutree mining lgorithms [1, 22]. 5. CONCLUSION AND FUTURE WORK In this pper we present, n lgorithm to mine miml frequent sugrphs from grph dtse. A new frmework, which prtitions frequent sugrphs into equivlence clsses is proposed together with group of optimiztion techniques. Compred to current stte-of-the-rt sugrph mining lgorithms such s FFSM nd gspn, offers ver good sclilit to lrge grph dtses nd t lest n order of mgnitude performnce improvement in snthetic grph dt sets. The efficienc of the lgorithm is lso confirmed enchmrk chemicl dt set. The lgorithm of compressing lrge numer of frequent sugrphs to much smller set of miml sugrphs will help us to investigte demnding pplictions such s finding structure ptterns from proteins in the future. Acknowledgement We thnk Dr. Jck Snoeink t the Universit of North Crolin for helpful discussions out the pper. 6. REFERENCES [1] T. Asi, K. Ae, S. Kwsoe, H. Arimur, nd H. Skmoto. Efficientl sustructure discover from lrge semi-structured dt. SDM, [2] C. Borgelt nd M. R. Berhold. Mining moleculr frgments: Finding relevnt sustructures of molecules. In Proc. Interntionl Conference on Dt Mining 02. [3] D. Burdick, M. Climlim, nd J. Gehrke. Mfi: A miml frequent itemset lgorithm for trnsctionl dtses. ICDE, [4] Y. Chi, Y. 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