Vertex Unique Labelled Subgraph Mining

Transcription

1 Verte Unique Lbelled Subgrph Mining Wen Yu, Frns Coenen, Michele Zito, nd Subhieh El Slhi Abstrct With the successful development of efficient lgorithms for Frequent Subgrph Mining (FSM), this pper etends the scope of subgrph mining b proposing Verte Unique lbelled Subgrph Mining (VULSM). VULSM hs focus on the locl properties of grph nd does not require eternl prmeters such s the support threshold used in frequent pttern mining. There re mn pplictions where the mining of VULS is significnt, the ppliction considered in this pper is error prediction with respect to sheet metl forming. More specificll this pper presents formlism for VULSM nd n lgorithm, the Right-most Etension VULS Mining (RE- VULSM) lgorithm, which identifies ll VULS in given grph. The performnce of REVULSM is evluted using rel world sheet metl forming ppliction. The eperimentl results demonstrte tht ll VULS (Verte Unique Lbelled Subgrphs) cn be effectivel identified. 1 Introduction A novel reserch theme in the contet of grph mining [7, 15, 8, 16], Verte Unique lbelled Subgrph Mining (VULSM), is proposed in this pper. Given prticulr sub-grph g in single input grph G; this subgrph will hve specific structure, nd edge nd verte lbelling ssocited with it. If we consider onl the structure nd edge lbelling there m be number of different comptible verte lbellings with respect to G. A Verte Unique Lbelled Subgrph (VULS) is subgrph with specific structure nd edge lbelling tht hs unique verte lbelling ssocited with it. This pper proposes the Deprtment of Computer Science, The Universit of Liverpool Ashton Building, Ashton Street, Liverpool, L69 3BX, UK {uwen,coenen,michele,hsselsl}@liverpool.c.uk

2 Wen Yu, Frns Coenen, Michele Zito, nd Subhieh El Slhi Right-most Etension Verte Unique Lbelled Subgrph Mining Algorithm (REVULSM) to identif ll VULS. REVULSM genertes subgrphs (potentil VULS cndidtes) using Right Most Etension [3], in DFS mnner, s first proposed in the contet of gspn [14]; nd then identifies ll VULS using level-wise pproch (first proposed b Agrwl nd Sriknt in the contet of frequent item set mining [1, 2, 9]). VULSM is pplicble to vrious tpes of grph; however, in this pper we focus on undirected grphs. VULSM hs relevnce with respect to number of domins. The ppliction domin used to illustrte the work described in this pper is error prediction in sheet metl forming. More specificll error prediction in Asmmetric Incrementl Sheet Forming (AISF) [4, 6, 10, 12, 13]. In this scenrio the piece to be mnufctured is represented s grid, ech grid centre point is defined b Eucliden (X-Y-Z) coordinte scheme. The grid cn then be conceptulised s grph (lttice) such tht ech verte represents grid point. Ech verte (ecept t the edges nd corners) cn then be connected to its four neighbours b sequence of edges, which in turn cn be lbelled with slope vlues. An issue with sheet metl forming processes, such s AISF, is tht distortions re introduced s result of the ppliction of the process. These distortions re non-uniform cross the shpe, but tend to be relted to locl geometries. The proposed grph representtion cptures such geometries in terms of sub-grphs, prticulr sub-grphs re ssocited with prticulr locl geometries (nd b etension distortion/error ptterns). Given before nd fter shpes we cn crete trining set b deriving the error ssocited with ech verte in the grid. This trining dt, in turn, cn then be used to trin predictor or clssifier of some sort. There re vrious ws tht such clssifier cn be generted; but one mechnism is to ppl VULSM, s proposed in this pper, to identif sub-grphs tht hve unique error ptterns ssocited with them tht cn then be used for error prediction purposes (some form of mitigting error correction cn then be formulted). A simple emple grid nd corresponding grph re given in Figure 1. The grid (lefthnd side of Figure 1) comprises si grid squres. Ech grid centre is defined b X-Y -Z coordinte tuple. Ech grid centre point is ssocited with verte within the grph (right-hnd side of figure 1). The edges, s noted bove, re lbelled with slope vlues, the difference in the Z coordinte vlues ssocited with the two end vertices. Ech verte will be lbelled with n error vlues (e 1 to e 3 in the figure) describing the epected distortion t tht verte s obtined from trining set (derived from before nd fter grid dt). Identified VULS will describe locl geometries ech with prticulr ssocited error pttern. This knowledge cn then be used to predict errors in unseen grids so tht some form of mitigting error correction cn be pplied. The rest of this pper is orgnised s follows. In Section 2, we define the bsic concepts of VULS together with n illustrtive emple. The REVULSM lgorithm is then described in detil in Section 3. An eperimentl nlsis

3 Verte Unique Lbelled Subgrph Mining 2 1 e 1 e 2 e e 3 e 3 e 2 Fig. 1: Grid representtion (left) with corresponding grph/lttice (right) feturing slope lbels on edges of the pproch is presented in Section 4, nd Section 5 summrises the work nd the min findings, nd presents some conclusions. 2 The problem formultion This section presents forml definition of the concept of VULS. Assume connected lbelled grph G comprised of set of n vertices V, such tht V = {v 1, v 2,..., v n }; nd set of m edges E, such tht E = {e 1, e 2,..., e m }. The vertices re lbelled ccording to set of p verte lbels L V = {l v1, l v2,..., l vp }. The edges re lbelled ccording to set of q edge lbels L E = {l e1, l e2,..., l eq }. A grph G cn thus be conceptulised s comprising k one-edge subgrphs: G = {P 1, P 2,..., P k }, where P i is pir of vertices linked b n edge, thus P i = v, v b (where v, v b V ). The size of grph G ( G ) cn thus be defined in terms of its one edge sub-grphs, we refer to 1-edge subgrphs, 2-edge subgrphs nd so on up to k-edge subgrphs. For undirected grphs, the edge v, v b is equivlent to v b, v (in this pper we ssume undirected subgrphs). We use the nottion P i.v nd P i.v b to indicte the vertices v nd v b ssocited with prticulr verte pir P i, nd the nottion P i.v.lbel nd P i.v b.lbel to indicte the lbels ssocited with P i.v nd P i.v b respectivel. We indicte the sets of lbels which might be ssocited with P i.v nd P i.v b using the nottion L Pi.v nd L Pi.v b (L Pi.v, L Pi.v b L V ). We indicte the edge lbel ssocited with P i using the notion P i.lbel (P i.lbel L E ). We cn use this nottion with respect to n subgrph G sub of G (G sub G). For trining purposes the grphs of interest re required to be lbelled. However, we cn lso conceive of edge onl lbelled grphs nd subgrphs. Given some edge onl lbelled subgrph (G subedgelb ) of some full lbelled grph G (G subedgelb G) comprised of k edges, there m be mn different verte lbelings tht cn be ssocited with such subgrph ccording to the nture of G. We thus define function, getv ertelbels, tht returns the potentil list of lbels S tht cn be ssigned to the vertices in G subedgelb ccording to G:

4 Wen Yu, Frns Coenen, Michele Zito, nd Subhieh El Slhi getv ertelbels(g subedgelb ) S where G subedgelb = {P 1, P 2,..., P k } nd S = [[L P1.v, L P1.v b ], [L P2.v, L P2.v b ],..., [L Pk.v, L Pk.v b ]] (recll tht L Pi.v nd L Pi.v b re the sets of potentil verte lbels for verte v nd v b ssocited with one-edge subgrph P i ). Thus ech element in S comprises two sub-sets of lbels ssocited respectivel with the strt nd end verte for ech edge in G subedgelb ; there is one to one correspondence between ech element (pir of lbel sets) in S with ech element in G subedgelb, hence the re both of the sme size k (recll tht k is the number of edges). We lso ssume tht some cnonicl lbelling is dopted. b b b Fig. 2: Undirected emple lttice b b b Fig. 3: One edge VULS generted from lttice in Figure 2 b Fig. 4: Two edge VULS generted from lttice in Figure 2 b b b Fig. 5: Three edge VULS generted from lttice in Figure 2 According to the bove, the forml definition of the concept of VULS is s follows. Given: (i) k-edge edge lbelled subgrph G subedgelb = {P 1, P 2,..., P k } (G subedgelb G), (ii) list of lbels tht m be ssocited with the vertices in G subedgelb, S = [[L P1.v, L P1.v b ], [L P2.v, L P2.v b ],..., [L Pk.v, L Pk.v b ]], nd (iii) the proviso tht G subedgelb is connected. If [L i, L j ] S, L i = 1, L j = 1 then G subedgelb is k-edge VULS with respect to G. So s to provide for full nd complete comprehension of the concept of VULS n emple lttice is presented in Figure 2. The VULS tht eist in this lttice re itemized in Figures 3 to 5. If we consider one-edge subgrphs first, there re two possibilities: (i) grphs feturing edge, nd (ii) grphs feturing edge. The list of possible vertices S ssocited with the first, obtined using the getv ertelbels function, is [[{}, {}]], while the list ssocited with the second is [[{, b}, {b}]] (this cn be verified b inspection of Figure 2). Considering edge first, [L i, L j ] S, L i = 1 nd L j = 1,

5 Verte Unique Lbelled Subgrph Mining so this is VULS; however, considering edge, [L i, L j ] S, L i 1 nd L j = 1 hence this is not VULS. We now consider the two edge subgrphs b etending the one edge subgrphs. We cn not enumerte ll two edge subgrphs here due to spce limittions but the two edge VULS re shown in Figure 4. Tking the first VULS in Figure 4, {P 1, P 2 }, s n emple, here P 1.v =, P 1.v b =, P 2.v = nd P 2.v b =, furthermore the edge lbels ssocited with P 1 nd P 2 re P 1.lbel = nd P 2.lbel = respectivel. In this cse S = [[, ], [, ]] thus [L i, L j ] S, L i = 1, L j = 1 therefore this is two-edge VULS with respect to G. Algorithm 1 REVULSM 1: Input: 2: G input = Input grph 3: M=M subgrph size 4: Output: 5: R = Set of VULS 6: Globl vribles: 7: G = set of subgrphs (VULS cndidtes) in G input 8: procedure REV ULSM(G input, M) 9: R = 10: G = 11: G = Subgrph Mining(G input ) (Algorithm 2) 12: k=1 13: while (k < M) do 14: for ll G sub G k (where G k is the set of k-edge subgrphs in G) do 15: if IdentifV ULS(G sub, G k ) == true (Algorithm 3) then 16: R = R G sub 17: end if 18: end for 19: k++ 20: end while 21: Return R 22: end procedure 3 The REVULSM lgorithm The proposed REVULSM lgorithm is defined in this section. The lgorithm is founded on the VULS properties presented bove nd mkes use of grph representtion technique borrowed from gspn. The pseudo code for REVULSM is presented in Algorithms 1, 2 nd 3. Algorithm 1 presents the high level control structure, while Algorithm 2 presents the detil for generting ll subgrphs (VULS cndidtes), nd Algorithm 3 the detil for determining whether specific sub-grph is VULS or not.

6 Algorithm 2 Subgrph Mining 1: Input: 2: G input = Input grph 3: Output: 4: G = set of subgrphs in G input 5: Globl vribles: 6: G temp=set of subgrphs generted so fr 7: procedure Subgrph Mining(G input ) 8: G = 9: G temp = 10: G 1 =the set of one-edge subgrphs in G input 11: sort G 1 in DFS leicogrphic order 12: for ech edge e G 1 do 13: G temp = Subgrph(e,1,M) 14: G = G G temp 15: G input = (G input e) (remove e from G input ) 16: end for 17: Return G 18: end procedure Wen Yu, Frns Coenen, Michele Zito, nd Subhieh El Slhi 19: procedure Subgrph(e, size, M) 20: if size > M then 21: return 22: end if 23: generte ll e s potentil etension subgrphs c in G input with one edge growth b right most etension 24: for ech c do 25: if c is miniml DFSCode then 26: G temp = G temp c 27: Subgrph(G temp,size+1,m) 28: end if 29: end for 30: Return G temp 31: end procedure Considering Algorithm 1 first, the lgorithm commences with n input grph G input nd prmeter M tht defines the mimum size of the VULS. If we do not limit the size of the serched-for VULSs the entire input grph m ultimtel be identified s VULS which, in the contet of the sheet metl forming trget ppliction, will not be ver useful. The output is set of VULS R (the set R m include overlps). Note tht ll grphs re encoded using Depth First Serch (DFS) leicogrphicl ordering (s used in gspn [14]). The globl vrible G (line 7 in Algorithm 1) is the set of ll subgrphs in G input. At the strt of the REVULSM procedure, the sets G nd R will be empt. We proceed in depth first mnner to generte ll subgrphs (VULS cndidtes) G b clling lgorithm 2 (line 11). Then we identif VULS from ll subgrphs G strting from one-edge subgrphs (k = 1), then two edge sub-grphs (k = 2), nd so on. We continue in this mnner until k = M

7 Verte Unique Lbelled Subgrph Mining Algorithm 3 IdentifVULS 1: Input: 2: g = single k-edge subgrph (potentil VULS) 3: G k = set of k-edge subgrphs to be compred with g 4: Output: 5: true if g is VULS, flse otherwise 6: procedure IdentifV ULS(g, G k ) 7: isv ULS = true 8: S = the list of potentil verte lbels tht m be ssigned to g 9: for ll [L i, L j ] S do 10: if either L i = 1 or L j = 1 then 11: isv ULS = flse 12: brek 13: end if 14: end for 15: return isv ULS 16: end procedure (line 13-20). On ech itertion lgorithm 3 is clled (line 15) to determine whether G sub is VULS or not with respect to the k-edge subgrphs G k. If it is VULS, it will be dded to the set R. Algorithm 2 comprises two procedures. The first, Subgrph Mining(G input ), is similr to tht found in gspn. We re itertivel finding ll subgrphs, up to size of M. We commence (line 10-11) b sorting ll the one-edge subgrphs, contined in input grph G input, into DFS leicogrphic order nd storing them in G 1. Then (lines 12-16), for ech one edge subgrph e in G 1 we cll the Subgrph procedure (line 13), which finds ll super grphs for ech one edge grph e up to size M in DFS mnner, nd stores the result in G temp ; which is then dded to G (line 14). Finll, we remove e from G input (line 15) to void generting gin n duplicte subgrphs contining e. The Subgrph(e, size, M ) procedure genertes ll the super grphs of the given one edge subgrph e b growing e b dding edges using the right most etension principle. For ech potentil subgrph c, if c is described b miniml DFSCode (line 25) the process is repet (in DFS stle) so s to generte ll the super grphs of e (line 27). The process continues in this recursive mnner until the number of edges in the super grphs to be generted (size) is greter thn M, or no more grphs cn be generted. Algorithm 3 presents the pseudo code for identifing whether given subgrph g is VULS or not with respect to the current set of k-edge sub-grphs G k from which g hs been removed. The lgorithm returns true if g is VULS nd f lse otherwise. The process commences (line 8) b generting the potentil list of verte lbels S tht cn be mtched to g ccording to the content of G k (see previous section for detil). The list S is then processed nd tested. If there eists verte pir whose possible lbelling is not unique

8 Wen Yu, Frns Coenen, Michele Zito, nd Subhieh El Slhi (hs more thn one possible lbelling tht cn be ssocited with it) g is not VULS nd the procedure returns flse, otherwise g is VULS nd the procedure returns true. 4 Eperiments nd Performnce Stud This section describes the performnce stud tht ws conducted to nlse the genertion nd ppliction of the concept of VULS. The reported eperiments were ll pplied to rel ppliction of sheet metl forming, more specificll the ppliction of AISF [11, 5] to the fbriction of flt-topped prmid shpes mnufctured out of sheet steel. This shpe ws chosen s it is frequentl used s benchmrk shpe for conducting eperiments in the contet of AISF (lthough not necessril with respect to error prediction). Nine grphs were generted from this dt using three different grid sizes nd different numbers of edge nd verte lbels; in ddition rnge of vlues were used for the M prmeter. The rest of this sub-section is orgnised s follows. The performnce mesures used with respect to the evlution re itemised in Section 4.1, more detil concerning the dt sets used for the evlution is given in Section 4.2, nd the obtined results re presented nd discussed in Section Eperimentl performnce mesurement Four performnce mesures were used to nlse the effectiveness of the proposed REVULSM : (i) run time (seconds), (ii) number of VULS identified, (iii) discover rte nd (iv) coverge rte. The lst two merit some further eplntion. The discover rte is the rtio of VULS discovered with respect to the totl number of subgrphs of size less thn M (Eqution 1). The coverge rte is the rtio of the number of vertices covered b the detected VULS compred to the totl number of vertices in the input grph (Eqution 2); with respect to the sheet steel forming emple ppliction high coverge rtes re desirble. discover rte (%) = coverge rte (%) = number of V ULS number of subgrphs number of vertices covered b V ULS number of vertices in input grph (1) (2)

9 Verte Unique Lbelled Subgrph Mining 4.2 Dt sets The dt sets used for the evlution consisted of before nd fter coordinte clouds ; the first generted b CAD sstem, the second using n opticl mesuring technique. These were trnsformed into grid representtions, referenced using X-Y-Z coordinte sstem, such tht the before grid could be correlted with the fter grid nd error mesurements obtined. A frgment of the before grid dt, with ssocited error vlues (mm), is presented in Tble 1. The before grid dt ws then trnslted into grph such tht ech grid squre ws represented b verte linked to ech of its neighbouring squres b n edge. Ech verte ws lbelled with n error vlue while the edges were lbelled ccording to the difference in Z of the two end vertices (the slope connecting them). Furthermore, the verte nd edge lbels were discretised so tht the were represented b nominl vlues (otherwise ever edge pir ws likel to be unique). This ws then the input into the REVULSM lgorithm. Tble 1: Formt of rw input dt z error As noted bove, from the rw dt, different sized grid representtions, nd consequentl grph representtions, m be generted. For eperimentl purposes three grid formts were used 6 6, nd We cn lso ssign different numbers of edge lbels to the vertices nd edges, for the evlution reported here vlues of two nd three were used in three different combintions. In totl nine different grph dt sets were generted, numbered AISF1 to AISF9. AISF1 to AISF3 were generted using 6 6 grid, while AISF4 to AISF6 were generted using grid, nd AISF7 to AISF9 were generted using grid. Some sttistics concerning these grph sets re presented in Tble 2.

10 Wen Yu, Frns Coenen, Michele Zito, nd Subhieh El Slhi Tble 2: Summr of AISF grph sets grph # # edge # verte grph # # edge # verte set vertices lbels lbels set vertices lbels lbels AISF AISF AISF AISF AISF AISF AISF AISF AISF Eperimentl results nd nlsis For eperimentl purposes REVULSM ws implemented in the JAVA progrmming lnguge. All eperiments were conducted using 2.7 GHz Intel Core i5 with 4 GB 1333 MHz DDR3 memor, running OS X (12B19). The results obtined re presented in Figures 6 to 17. Figures 6 to 8 give the run time comprisons with respect to the nine grph sets. Figures 9 to 11 give the number of discovered VULS in ech cse. Figures 12 to 14 present comprison of the recorded discover rtes with respect to the nine grph sets considered. Finll Figures 15 to 17 give comprison of the coverge rtes. From Figures 6 to 8 it cn be seen, s might be epected, tht s the vlue of the M prmeter increses the run time lso increses becuse more subgrphs nd hence more VULS re generted. The sme observtion is true with respect to the size of the grph; the more vertices the greter the required runtime. From Figures 9 (6 6 grid), 10 (10 10 grid) nd 11 (21 21 grid) it cn be observed tht s M increses the number of VULS will lso increse, gin this is s might be epected. Compring AISF1, 4 nd 7 with AISF2, 5 nd 8 respectivel, it cn be seen tht s the number of edge lbels increses while the number of verte lbels is kept constnt the number of VULS lso increses (AISF1 nd AISF2 hve the sme number of verte lbels; s do AISF4 nd AISF5, nd AISF7 nd AISF8). This is becuse the likelihood of VULS eisting increses s the input grph becomes more diverse. However, compring AISF2, 5 nd 8 with AISF3, 6 nd 9 respectivel it cn be seen tht s the number of verte lbels increses, while the number of edge lbels is kept constnt, the number of VULS generted will decrese (AISF2 nd AISF3 hve the sme number of edge lbels; s do AISF5 nd AISF6, nd AISF8 nd AISF9); becuse, given high number of verte lbels, the likelihood of VULS eisting decreses. Figure 12 (6 6 grid), 13 (10 10 grid) nd 14 (21 21 grid) show the recorded discover rte vlues. Compring AISF1, 4 nd 7 with AISF2, 5 nd 8 respectivel; when the number of edge lbels increses, while the number of verte lbels is kept constnt, the discover rte increses. This is

11 Verte Unique Lbelled Subgrph Mining Fig. 6: Run time comprison using 3 edge nd 2 verte lbels (AISF1, AISF4 nd AISF7) Fig. 10: Comprison of number of VULS generted (AISF4, AISF5 nd AISF6) Fig. 7: Run time comprison using 2 edge nd 2 verte lbels (AISF2, AISF5 nd AISF8) Fig. 11: Comprison of number of VULS generted (AISF7, AISF8 nd AISF9) Fig. 8: Run time comprison using 2 edge nd 3 verte lbels (AISF3, AISF6 nd AISF9) Fig. 12: Comprison of discover rte (AISF1, AISF2 nd AISF3) Fig. 9: Comprison of number of VULS generted (AISF1, AISF2 nd AISF3) Fig. 13: Comprison of discover rte (AISF4, AISF5 nd AISF6)

12 Wen Yu, Frns Coenen, Michele Zito, nd Subhieh El Slhi Fig. 14: Comprison of discover rte (AISF7, AISF8 nd AISF9) Fig. 16: Comprison of coverge rte (AISF4, AISF5 nd AISF6) Fig. 15: Comprison of coverge rte (AISF1, AISF2 nd AISF3) Fig. 17: Comprison of coverge rte (AISF7, AISF8 nd AISF9) becuse regrdless of the number of edge lbels grph hs (ll other elements being kept constnt) the number of subgrphs contined in the grph will not chnge, while (s indicted b the eperiments reported in Figures 9, 10 nd 11) the number of identified VULS increses s the number of edge lbels increses. Conversel, compring AISF2, 5 nd 8 with AISF3, 6 nd 9 respectivel, when the number of verte lbels increses while the number of edge lbels is kept constnt, the discover rte will decrese becuse (s lred noted) the number of VULS generted decreses s the number of verte lbels increses. It cn lso be noted tht s the M vlue increses, the discover rte does not lws increse, s shown in the cse of AISF4, 5 nd 6. This is becuse s the M vlue increses, the number of VULS goes up s does the number of subgrphs, but the m not both increse t the sme rte. Figures 15, 16 nd 17 show the coverge rte. From the figures it cn be observed tht s M increses the coverge rte lso increses. This is to be epected, however it is interesting to note tht the coverge rte in some cses reches 100% (when M = 6 with respect to AISF1, ASF2 nd AISF4). One hundred percent coverge is desirble in the contet of the sheet metl forming ppliction so tht unique ptterns ssocited with prticulr error distributions (verte lbels) cn be identified for ll geometries. Compring AISF1, 4 nd 7 with AISF2, 5 nd 8 respectivel, the more edge lbels grph hs the more VULS will be generted (see bove); s result more vertices will be covered b VULS nd hence the coverge rte will go up. On the other hnd, compring AISF2, 5 nd 8 with AISF3, 6 nd 9 respectivel,

13 Verte Unique Lbelled Subgrph Mining the more verte lbels grph hs the less VULS will be generted (see bove); s result fewer vertices will be covered b VULS vertices nd hence the coverge rte will go down. 5 Conclusions nd further Stud In this pper we hve proposed the mining of VULS nd presented the RE- VULSM lgorithm. The reported eperimentl results demonstrted tht the VULS ide is sound nd tht REVULSM cn effectivel identif VULS in rel dt. Hving estblished proof on concept there re mn interesting reserch problems relted to VULSM tht cn now be pursued. For instnce, currentl, when the M prmeter is high REVULSM will run out of memor, lthough for the purpose of error prediction in sheet metl forming it cn be rgued tht there is no requirement for lrger VULS, it m be of interest to investigte methods whereb the efficienc of REVULSM cn be improved. Finll, t present, REVULSM finds ll VULS up to predefined size, it is conjectured tht efficienc gins cn be mde if onl miniml VULS re found. 6 Acknowledgements The reserch leding to the results presented in this pper hs received funding from the Europen Union Seventh Frmework Progrmme (FP7/ ) under grnt greement number References 1. Agrwl, R., Sriknt, R.: fst lgorithms for mining ssocition rules. In: Proceedings of the 20th Interntionl Conference on Ver Lrge Dt Bses(VLDB 94), pp (1994) 2. Agrwl, R., Sriknt, R.: Mining sequentil ptterns. In: Proceedings of the Eleventh Interntionl Conference on Dt Engineering(ICDE 95), pp (1995) 3. Asi, T., Abe, K., Kwsoe, S., Skmoto, H., Arikw, S.: Efficient substructure discover from lrge semi-structured dt. In: In Proc.2002 SIAM Int.Conf. Dt Mining, pp (2002) 4. Cfut, G., Mole, N., tok, B.: An enhnced displcement djustment method: Springbck nd thinning compenstion. Mterils nd Design 40, (2012) 5. El-Slhi, S., Coenen, F., Dion, C., Khn, M.S.: Identifiction of correltions between 3d surfces using dt mining techniques: Predicting springbck in sheet metl forming. In: Reserch nd Development in Intelligent Sstems XXIX, pp (2012)

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