A High-level Strategy for C-net Discovery

Transcription

1 A High-lvl Strtgy for C-nt Disovry Mr Solé nd Josp Crmon Softwr Dprtmnt Univrsitt Politèni d Ctluny (UPC) Brlon, Spin msol@.up.du, jrmon@lsi.up.du Astrt Cusl nts hv n rntly proposd s suitl modl for pross mining, du to thir dlrtiv smntis nd ompt rprsnttion. Howvr, th disovry of usl nts from log is omplx prolm. Th urrnt lgorithmi support for th disovry of usl nts ompriss ithr fst ut inurt mthods (ompromising qulity), or urt lgorithms tht r omputtionl dmnding, thus limiting th siz of th inputs thy n pross. In this ppr high-lvl strtgy is prsntd, whih uss pproprit lustring thniqus to split th log into pis, nd nfits from th dditiv ntur of usl nts. This llows mlgmting struturlly th disovrd Cusl nt of h pi to driv vlul modl. Th lims in this ppr r ompnid with xprimntl rsults showing th signifin of th high-lvl strtgy prsntd. Kywords-Pross disovry, Cusl nts, High-lvl strtgy, Clustring; I. INTRODUCTION Th ontinuous growth of dt gnrtd y informtion systms hs origintd th dvnt of Pross Mining, disiplin tht sits in twn th dt mining nd softwr nginring filds. Th dt, oftn vill s st of trs (logs) tht informtion systms gnrt, n prossd in ordr to disovr forml pross modls, hk whthr th vill modls onform to th rlity osrvd in th log, nd xtnd th urrnt prosss for improving th qulity of th informtion providd [1]. Thr r svrl opn prolms in th r of Pross Mining. A ruil on is Control Flow Pross Disovry: givn log, to find forml modl (.g., Ptri Nt) for pross whih: i) rprsnts (most of) th usl rltions twn tivitis in th log, ii) onforms [2] with high dgr th log, nd iii) its grphil dsription is s muh struturd s possil [3]. Oviously, th sltion of givn trgt modl (Trnsition Systms, Ptri nts, Huristi nts, mong othrs) influns th typ of lgorithms on n us for disovry. Morovr, givn forml modl, thr xist mny possiilitis dpnding on th trgt sulss. For instn th omplxity of th α-lgorithm [4] to disovr workflow nts is vry low whn omprd to th thory of rgions [5], [6], [7], [8], whih llows to disovr gnrl Ptri nts. Rntly, formlism lld Cusl nts (C-nts) [9] hs n proposd s suitl modling lngug for () () Log () Figur 1. () A log. () Cusl nt dsriing th log. () Singl Ptri nt modling th sm log with th hlp of silnt trnsitions. pross mining. It is rthr ompt rprsnttion tht llows xprssing omplx hvior tht it is somtims diffiult to dsri using othr formlisms. For instn onsidr th log in Fig. 1(). In Fig. 1() Ptri nt tht ontins ll th squns in th log is dsrid. In ordr to rprsnt xtly th log (i.., to void inorporting in th nt xtr hvior), th us of silnt trnsitions is rquird, grphilly rprsntd s lk oxs. On th othr hnd, simpl C-nt rprsnting th log is shown in (), whih is quit ompt. Th smntis of tht C-nt n informlly dsrid s: Ativity must xutd initilly, sin no oligtions (input rs with dots) xist for. It n gnrt oligtions to ithr 1) tivity, or 2) tivity or 3) tivitis nd. Any of th ths thr possiilitis rquirs th xution of th orrsponding tivitis, onsuming th oligtion(s) from tivity nd gnrting oligtion(s) to tivity. Th finl xution of will mpty th st of oligtions nd thrfor will ld to vlid tr. Th prolm of C-nts disovry poss nw hllngs tht prvious lgorithms for disovry of othr modls lik Ptri nts did not hv to ddrss. First, th dlrtiv (or -postriori) smntis of C-nt is dfind on vlid firing squns (squns tht rsult in n mpty st of pnding

2 Log L lustring Figur 2. Log L 1. Log L i. Log L n disovry C-nt C 1. C-nt C i. C-nt C n Proposd disovry workflow. union n i=1 C i oligtions), nd thrfor, C-nts r not prfix-losd. Sond, ontrry to Ptri nts tht hv rstritiv ntur (i.., th ddition of pl n only rstrit th hvior), C-nts hv n dditiv ntur, tht is, th ddition of lmnts to th C-nt n only dd hvior. For instn, y dding n r with two dots twn nd in Fig. 1(), th squn is ddd to th lngug of th C-nt. This ppr dsris high-lvl strtgy to disovr C- nt from log. Although thr r urrntly fw disovry lgorithms for C-nts in th litrtur, w xpt thir numr to inrs in th nr futur. Th vill disovry lgorithms n roughly lssifid in two lsss: ithr thy r fst nd n hndl lrg inputs [10] ut my provid unstisftory rsults (for instn th C-nts gnrtd y [10] n ddlok) or thy provid high-qulity C-nts (in trms of i), ii) nd iii) ov) ut thy r omputtionlly xpnsiv [11]. Th dditiv ftur of C-nts mks thm spilly suitl for th divid-nd-onqur strtgy prsntd in this ppr (s Fig. 2). It is sd on struturlly omining svrl C-nts tht rsult on pplying disovry mthods to smll frtions of th log. Th prtitioning of th log is don y tilord lustring lgorithms whih n guidd y prtiulr ftors (frquny, similrity, ln, tim, mong othrs). Sin th frtions tnd to smll, thy r trtl for high-qulity disovry lgorithms lik [11]. Importntly, th union of th st of disovrd C-nts n omplishd struturlly, ruil ft tht mks C-nts vry suitl modl. If th sm pproh would hv n don using diffrnt modl lik Ptri nts, on hs to f th prolm of driving Ptri nt whih is th union of st of Ptri nts, n importnt issu tht nnot tkld struturlly in gnrl ( stt-sd solution to this prolm whih is groundd on th thory of rgions n found in [12]). Th orgniztion of th ppr is s follows: in St. II w introdu som mthmtil nottion nd th forml dfinition of C-nts. Stion III xplins th proprtis of th union of C-nts. Ths proprtis r thn usd in St. IV to propos high-lvl strtgy to disovr urt C-nts from lrg inputs. Th strtgy is thn xprimntlly tstd on St. V. Stion St. VI inorports disussion on th onntion of this work with xisting ontriutions in th litrtur, whil St. VII onluds this ppr nd provids som rsrh dirtions for th futur. II. BACKGROUND A. Mthmtil prliminris A multist (or g) is st in whih lmnts of st X n ppr mor thn on, formlly dfind s funtion X N. W dnot s B(X) th sp of ll multists tht n rtd using th lmnts of X. Lt M 1, M 2 B(X), w onsidr th following oprtions on multists: sum (M 1 + M 2 )(x) = M 1 (x) + M 2 (x), sutrtion (M 1 M 2 )(x) = mx(0, M 1 (x) M 2 (x)) nd inlusion (M 1 M 2 ) x X, M 1 (x) M 2 (x). A log L is g of squns of tivitis. In this work w rstrit th typ of squns tht n form log. In prtiulr, w ssum tht ll th squns strt with th sm initil tivity nd nd with th sm finl tivity, nd tht ths two spil tivitis only ppr on in vry squn. This ssumption is without loss of gnrlity, sin ny log n sily onvrtd to stisfy ths rquirmnts y using two nw tivitis tht r proprly insrtd in h tr. Givn finit squn of lmnts σ = n, its lngth is dnotd σ = n. Th lpht of σ, dnotd A σ, is th st of lmnts in σ. W xtnd this nottion to logs, so tht A L is th lpht of th log L, i.., A L = σ L A σ. Finlly, th numr of ourrns of givn lmnt in σ, i.. { i i = }, is dnotd s #(σ, ). B. Cusl nts (C-nts) In this stion w introdu th min modl usd long this ppr. Dfinition 1 (Cusl nt [9]): A Cusl nt is tupl C = A, s,, I, O, whr A is finit st of tivitis, s A is th strt tivity, is th nd tivity, nd I (nd O) r th st of possil input (output rsp.) indings pr tivity. Formlly, oth I nd O r funtions A S A, whr S A = {X P(A) X = { } / X}, nd stisfy th following onditions: { s } = { I() = { }} nd { } = { O() = { }} ll th tivitis in th grph (A, rs(c)) r on pth from s to, whr rs(c) is th dpndny rltion indud y I nd O suh tht rs(c) = {( 1, 2 ) 1 X I( 2) X 2 Y O( 1) Y }. Dfinition 1 slightly diffrs from th originl on from [9], whr th st rs(c) is xpliitly dfind in th tupl. Th C-nt of Fig. 1() is formlly dfind s C = {,,, },,, I, O, with I() =, O() = {{}, {}, {, }}, I() = {{}}, O() = {{}}, I() = {{}}, O() = {{}}, I() = {{}, {}, {, }} nd O() =. Th dpndny rltion of C, whih orrsponds grphilly to th rs in th figur, in this s is rs(c) = {(, ), (, ), (, ), (, )}. Th tivity indings r dnotd in th figur s dots in th rs,.g.,

3 {} O() is rprsntd y th dot in th r (, ) tht is nxt to tivity, whil {} I() is th dot in r (, ) nxt to. Non-singlton tivity indings r rprsntd y rs onnting th dots: {, } O() is rprsntd y th two dots in rs (, ), (, ) tht r onntd through n r. Dfinition 2 (Binding, Binding Squn, Projtion): Givn C-nt A, s,, I, O, th st B of tivity indings is {(, S I, S O ) A S I I() S O O()}. A inding squn β B is squn of tivity indings. By rmoving th input nd output indings from inding squn β, w do otin n tivity squn dnotd s σ β. Two inding squns of th C-nt in Fig. 1() r: β 1 = (,, {})(, {}, {})(, {}, ) nd β 2 = (,, {, })(, {}, {})(, {}, ). Th projtion of β 1 is σ β 1 =. Th smntis of C-nt r sd on hrtrizing, mong ll th inding squns it hs, thos ons tht stisfy rtin proprtis nd thrfor thir orrsponding projtion (s Df. 2) will long to th lngug of th C-nt. Th nxt dfinition ddrsss this. Dfinition 3 (Stt, Vlid Binding Squn, Lngug): Givn C-nt C = A, s,, I, O, its stt sp S = B(A A) is omposd of stts tht rprsnt multists of pnding oligtions. Funtion ψ B S is dfind indutivly: ψ(ɛ) = nd ψ(β (, S I, S O )) = ψ(β) (S I {}) + ({} S O ). Th stt ψ(β) is th stt of th C-nt ftr th squn of indings β. Th inding squn β = ( 1, S1, I S1 O )... ( n, Sn, I Sn O ) is sid to vlid if: 1) 1 = s, n = nd k : 1 < k < n, k A \ { s, } 2) k : 1 k n, (S I k { k}) ψ(β k 1 ) 3) ψ(β) = Th st of ll vlid inding squns of C is dnotd s V CN (C). Th lngug of C, dnotd L(C), is th st of tivity squns tht orrspond to vlid inding squn of C, i.., L(C) = {σ β β V CN (C)}. For instn, in Fig. 1(), β 1 is vlid inding squn, whil β 2 is not, sin th finl stt is not mpty (ondition 3 is violtd). Th lngug of tht C-nt is {,,, }. Importntly, C-nts n nturlly rprsnt hvior tht nnot sily xprssd in th Ptri nt nottion unlss unosrvl (silnt) trnsitions r usd. Fig. 3 illustrts this point. In th C-nt, tivitis nd d n our onurrntly or xlusivly, vn in diffrnt itrtions of th loop rtd y tivity f,.g., dz or dfz. Howvr, thr is still nothr possiility tht riss from omining th AND-split nd th XOR-join: df df z. Not tht in this lst tr, tivity ould xut twi for singl Figur 3. nd d. Log L f d C-nt mixing onurrnt nd xlusiv hvior for tivitis Figur 4. Binry Srh Ar Bound Struturl Equtions SMT Formul SMT Solvr SMT thniqu for C-nt disovry. z C nt ontining L nd with miniml numr of rs (lthough in th ovrll tr thy xut th sm numr of tims). C. C-nt disovry using SMT W informlly dsri th strtgy to driv C-nt from log sd on Stisfiility Modulo Thoris (SMT), prsntd in [11]. Th pproh is shown in Fig. 4. First, th log is usd to onstrut n SMT formul rprsnting th possil indings tht h tivity n hv in potntil C-nt tht inluds s vlid squns ny tr in th log. Thn th formul is ugmntd with n uppr ound on th numr of rs th drivd C-nt n hv, whih n lso odifid in th domin of SMT with th thory of quntifir-fr it-vtor rithmti [13]. This uppr ound n initilly omputd y ounting th rs of C-nt tht is uilt using simpl ordring rltion twn tivitis nd whos lngug is gurntd to ontin ll th squns in th log (s th forml dtils in [11]). On th othr hnd, simpl onntivity ritri n usd to lso driv simpl lowr ound, y using th lpht of th log A L : A L 1. Thn, if n uppr nd lowr ound on th numr of rs of th drivd C-nt r vill, inry srh strtgy n usd to sk for th miniml C-nt tht oth inluds th lngug of th log nd hs th miniml numr of rs. Th pproh itrtivly invoks n SMT solvr to dtrmin whthr th urrnt

4 r ound usd dos not hrm stisfiility of th formul. Hn, sd on th outom of th SMT solvr, th inry srh strtgy will updt th ounds ordingly. Th mthod in [11] gurnts tht i) th trs in th log r inludd in th st of vlid inding squns of th drivd C-nt (s Df. 2), i.. th modl drivd is fitting [2], nd ii) it hs th miniml numr of rs. To th st of our knowldg, thr is no othr thniqu in th litrtur tht ithr gurnts fitting modls or limits th numr of rs. Hn, this will th disovry thniqu usd in our high-lvl strtgy. III. C-NET UNION C-nts, ontrry to Ptri nts, hv n dditiv ntur. Tht is, whil dding pl to Ptri nt n only rstrit hvior, dding n r (or ny othr lmnt) to C-nt n only dd hvior. Th dditiv ntur of C-nts is formlly dfind with th hlp of Df. 4 nd Proprty 1. Dfinition 4: Givn two C-nts C 1 = A 1, 1 s, 1, I 1, O 1 nd C 2 = A 2, 2 s, 2, I 2, O 2, w sy tht C 1 is inludd in C 2, dnotd C 1 C 2, if: 1 s = 2 s 1 = 2, A 1 A 2, nd A 1, I 1 () I 2 () O 1 () O 2 () For instn th C-nt C 1 of Fig. 5() is inludd in th C-nt of Fig. 5(), ut it is not inludd in th C-nt of Fig. 1(). Proprty 1: Lt C 1 nd C 2 two C-nts. If C 1 C 2, thn V CN (C 1 ) V CN (C 2 ), L(C 1 ) L(C 2 ). Proof: Sin th input nd output inding sts of C 2 inlud th input nd output inding sts of oth C 1 (Df. 4), ny vlid inding squn in C 1 will lso vlid inding squn of C 2, thus V CN (C 1 ) V CN (C 2 ) whih ntils L(C 1 ) L(C 2 ) us th lngug of C-nt is otind y simply kping only th squns of tivitis xutd in th inding squns. As C 1 of Fig. 5() is inludd in Fig. 5(), its lngug, {}, is sust of th lngug of Fig. 5(). Not tht th union dos not nssrily giv th smllst C-nt (in trms of its orrsponding lngug) tht n ontin th union of lngugs of th unitd C-nts. For instn, th C-nt in Fig. 1() inluds oth L(C 1 ) nd L(C 2 ) ut its lngug is propr sust of th C-nt of Fig. 5(). In spit of th ft tht minimlity of hvior is not gurntd y th union oprtor (s th prvious xmpl dmonstrts), still Proprty 1 mks th union of C-nts vry simpl nd fftiv oprtion to gnrt C-nts tht inlud th hvior of th unitd C-nts. Dfinition 5: Givn two C-nts with idntil initil nd finl tivitis, C 1 = A, s,, I, O nd C 2 = A, s,, I, O, thir union, dnotd C 1 C 2, is th C-nt () () () Figur 5. () C-nt C 1 with lngug. () C-nt C 2 with lngug. () Union C-nt C 1 C 2. Its lngug (using rgulr xprssions) is () () + () () +. A A, s,, I, O, whr I() I () if A A I () = I() if A \ A I () othrwis. O() O () if A A O () = O() if A \ A O () othrwis. For instn C-nt C 1 nd C 2 of Fig. 5() nd (), hv th sm initil nd finl tivitis, thus thy n unitd. Thir union is th C-nt of Fig. 5(). Lmm 1: Givn two C-nts C 1 nd C 2, V CN (C 1 C 2 ) V CN (C 1 ) V CN (C 2 ), thus L(C 1 C 2 ) L(C 1 ) L(C 2 ). Proof: Sin y Df. 5 C 1 C 1 C 2 nd C 2 C 1 C 2, y Proprty 1 w know V CN (C 1 ) V CN (C 1 C 2 ) nd V CN (C 2 ) V CN (C 1 C 2 ), thus V CN (C 1 C 2 ) V CN (C 1 ) V CN (C 2 ). Th union will ruil oprtor in th pproh prsntd in this ppr, nling th splitting of log into pis nd mlgmting th individul rsults y using th union of C-nts. Th nxt stion is dvotd to prsnt this high-lvl strtgy. IV. A DIVIDE-AND-CONQUER STRATEGY A. A lustring lgorithm for C-nts Clustring is n tiv rsrh r in pross disovry nd thr r numrous squn lustring lgorithms vill (s St. VI for disussion on rltd work). Th pprohs r vry divrs, rnging from lgorithms tht ssign to th sm lustr squns tht r nr whn thy r mppd to n n-dimnsionl sp, to pprohs tht rly on squn lignmnt thniqus. Any of ths thniqus ould usd in th lustring phs of our strtgy. Howvr, non of thm is tilord to produ lustrs prtiulrly suitl for C-nt disovry. In this stion w propos simpl lustring mthod tht tris to tk dvntg of th C-nt spifiitis.

5 Figur 6. d nd. C-nt with optionl hvior (tivity ) nd hoi twn Idlly w would lik to void situtions lik th following: onsidr th log {, }. If w prtition this log into two singlton lustrs, nmly {} nd {}, nd th simplst C-nt for h lustr is gnrtd, w otin th C-nts C 1 nd C 2 of Fig. 5() nd Fig. 5(), rsptivly. Th union of ths two C-nts is shown in Fig. 5(). Clrly th C-nt of Fig. 5() ontins mny dditionl hvior in th lngug of th nt with rspt to th originl log. In gnrl, this phnomnon will our whn prtiulr ordrings of onurrnt hvior r ssignd to diffrnt lustrs. On th othr hnd, lt us onsidr th log L = {df, f, df, f}, whih n rprsntd y th C-nt of Fig. 6. This log xhiits typil onstruts lik hoi (twn tivitis d nd ) nd optionl hvior (tivity ). In this s, rting four singlton lustrs, omputing thir simplst C-nts nd mrging thm will yild prisly th C-nt of Fig. 6, thus inluding no dditionl hvior. This is du to th ft tht C-nts rprsnt hoi nd optionl hvior with dditionl rs. Not, nvrthlss, tht in gnrl lustring trs in this wy my lso introdu dditionl hvior: for instn onsidr th sm log L ut now without th squn f, lt us ll this log L. Th rsult of splitting L into thr singlton lustrs nd mrging th orrsponding C-nts will yild xtly th sm C-nt s for, thus squn f will still long to th lngug of th nt, whil it is possil to onstrut C-nt whos lngug is only L (xluding f). Howvr, not lso tht th simplst C-nt for L is gin th C-nt of Fig. 6, sin lrgr input/output inding sts (involving dditionl rs) r rquird to xlud squn f from th lngug. Considring this prtiulrity of C-nts, w propos th rursiv lustring mthod shown in Algorithm 1. Th lgorithm is invokd with two prmtrs, th log L nd thrshold t, nd will rturn th st of lustrd logs tht ithr hv lss tht t squns or ould not furthr split. First of ll (lin 2), it omputs th st of tivitis A s tht ould usd to prtition L. Ths r th tivitis tht oth ppr in t lst on squn of L, nd do not ppr in t lst on squn of L. Thn it hks if th stop onditions r mt (lins 3-5). In suh s, it simply rturns th singlton st ontining L. Othrwis, it d f Algorithm 1 C-nt orintd lustring 1: funtion RECURSIVESPLIT(L,t) 2: A s { A L σ L : / A σ } 3: if ( L < t) (A s = ) thn 4: rturn {L} 5: nd if 6: sltsplitativity(a s, L) 7: S 1 rursivsplit({σ L A σ }, t) 8: S 2 rursivsplit({σ L / A σ }, t) 9: rturn S 1 S 2 10: nd funtion prforms th rursiv prt of th lgorithm, y slting first on tivity mong th st of ndidt tivitis A s (lin 6). Th r svrl possil huristis to did whih ndidt is ttr. In our urrnt implmnttion th sltsplitativity funtion simply hooss th tivity tht will yild mor lnd prtitions. Finlly, two lustrs r formd y onsidring th squns in whih th sltd tivity is prsnt or not, nd th funtion is lld rursivly on thm. With rspt to our initil ojtiv of lustring togthr onurrnt hvior, this issu is firly rltd with th onpt of synhroni distn [14], whih provids dgr of mutul dpndn twn two tivitis. In our stting, th synhroni distn of tivitis 1 nd 2 in log L n formlly dfind s th mximl vlu of #(σ, 1 ) #(σ, 2 ) whr σ = σ γ is tr of L. Whn two tivitis r totlly indpndnt, lik it hppns whn tivitis r in onflit or on of thm is optionl, thir synhroni distn is lrg (it n s lrg s th lngth of tr). On th othr hnd, onurrnt or uslity-rltd tivitis oftn hv n smll synhroni distn. Noti tht slting tivitis tht ppr in on tr nd do not ppr in nothr tr (s it is don in Algorithm 1), is light pproximtion to slt ths tivitis with lrg synhroni distn, nd hn ndidts to hv low dpndny with t lst som othr tivity in th log. Thus thy r good ndidts for splitting. On th othr hnd, two onurrnt tivitis r likly to hv low synhroni distn, nd thrfor thy will oth ppr in th squn or non of thm will ppr. If on of thm is sltd s splitting tivity, thn with high proility th lustrs tht will ris from th splitting will kp th synhroni distn twn ths tivitis. Howvr, in gnrl s th rursiv splitting progrsss th synhroni distn twn onurrnt tivitis my hng in th smllr logs gnrtd, nd thrfor it is not dvisl to prform too muh rursiv splitting. This is th rson to rquir in Algorithm 1 th miniml siz lustr must hv.

6 B. A flxil divid-nd-onqur strtgy Th niv pproh to th disovry phs of th mthodology would to simply run th disovry lgorithm on h lustr. Howvr y doing so w my miss importnt optimiztion opportunitis. Sin ths tuning strtgis r vry dpndnt on th tul disovry lgorithm usd, lt us ntr th disussion on th disovry lgorithm proposd in [11]. This disovry lgorithm silly trnslts th disovry prolm to Stisfility Modulo Thoris (SMT) formul, nd thn uss n SMT solvr to otin solution whih n onvrtd into C-nt (s St. II-C). Using suh strtgy, it is possil to gurnt tht ll th squns in h lustr will long to th lngug of th orrsponding C-nt gnrtd, nd tht thr is no othr C-nt with lss rs tht inluds th lustr. Lt us ll disovrminarcnt th funtion tht, givn lustr L i, rturns th C-nt C i with th prvious proprtis. Algorithm 2 shows th divid-nd-onqur strtgy uilt round this funtion. Algorithm 2 Divid-nd-onqur indpndnt strtgy 1: funtion DIVIDEANDCONQUER(L,t) 2: {L 1,..., L n } lustrlog(l, t) 3: for L i {L 1,..., L n } do 4: C i disovrminarcnt(l i ) 5: nd for 6: {C 1,..., C m} sltcnts({c 1,..., C n }) 7: rturn m i=1 C i 8: nd funtion First of ll, th lgorithm rivs two prmtrs: th log L nd th thrshold t tht dtrmins th siz of th squn lustrs gnrtd y th lustring lgorithm. Th lgorithm prisly strts with th lustring phs, with th ll to th lustrlog funtion. W hv dlirtly voidd th dirt us of th rursivsplit funtion (Algorithm 1) to rinfor th id tht ny lustring lgorithm (or omintion) n usd. In ordr to tst th nfits of th lustring thniqu prsntd in th prvious stion, in our xprimnts w hv usd two lustring ltrntivs: Th rursivsplit funtion from Algorithm 1. A rndom lnd lustring, otind y dividing th initil log into givn numr of frgmnts of th sm siz (with xption of th lst on tht ould hv lss trs). For h lustr L i in th st {L 1,..., L n } of omputd lustrs, th funtion disovrminarcnt is lld, whih produs C-nt C i whos lngug inluds L i nd hs th minimum numr of rs. Not tht this stp n trivilly prlllizd nd timouts n lso st so tht C-nt is gnrtd in givn mximum mount of tim (y uniting th C-nts of th lustrs whos prossing finishd for th timout, thus yilding lso fulttolrnt pproh to pross disovry), trding fitnss (i.., pility for rplying th squns in th log) for spd. Finlly, from th st of gnrtd C-nts, sust is sltd (lin 6). Hr dditionl qulity huristis n introdud. For instn, it is possil to slt only th st C-nts in trms of th rtio: ovrd squns pr C-nt r, or th sust of C-nts whos union yilds th st rtio, t. In gnrl, to gurnt tht ll th squns long to th lngug of th finl C-nt, ll th gnrtd C-nts must sltd. Howvr, it is somtims not dqut to us th whol st of C-nts for th union, sin othr ftors my mor importnt thn fitnss. Among th possil ftors to onsidr, thr r two whih r oftn ontmpltd. Nois: whn high prntg of nois is dttd in lustr, it my dvisl to not us th orrsponding C-nt. Nois n somtims dttd with trditionl dt mining thniqus [15]. Rdility: this is sujtiv ftor tht my stimtd on th grph strutur of th C-nt. C. An r minimizing divid-nd-onqur strtgy Algorithm 2 offrs flxil frmwork to dpt th C- nt disovry lgorithms to diffrnt rquirmnts. Howvr, in its most strightforwrd implmnttion (i.., using rursivsplit or th rndom lnd lustring s th lustrlog funtion nd slting ll th C-nts in th sltcnts funtion), th finl C-nt omputd dos not nssrily hv th minimum numr of rs, lthough h on of its omponnt C-nts hs. Th rson is tht ll th minimiztions r lol to h lustr nd do not tk into ount th rs of th C-nts found y pplying th disovry thniqu to th othr lustrs. A possil shm to llvit this inonvnint is shown in Algorithm 3. Th si id in this s is to sort th Algorithm 3 Divid-nd-onqur inrmntl strtgy 1: funtion INCDIVIDEANDCONQUER(L,t,α) 2: (L 1,..., L n ) sortlogs(lustrlog(l, t)) 3: C 1 disovrminarcnt(l 1 ) 4: C C 1 5: mx( A L 1, rs(c) ) 6: for L i (L 2,..., L n ) do 7: l A Li \ i 1 j=1 A L j + 1 8: u α rs(c) 9: C i disovrminnwarcnt(l i, C, l, u) 10: C C C i 11: nd for 12: rturn C 13: nd funtion lustrs ording to som ritrion (.g., y th numr of squns thy rprsnt, y th siz of thir lpht, t.)

7 nd thn disovr th smllst C-nt for th first lustr in th ordr, i.., L 1. Thn, for th rmining lustrs L 2,..., L n, instd of srhing for th C-nt with th miniml mount of rs, th ojtiv will to minimiz th numr of dditionl rs (with rspt to th urrnt C-nt C). This tsk is prformd y funtion disovrminnwarcnt, whih is quit sy to implmnt in th disovry lgorithm of [11], sin w must simply rmov from th prt of th SMT formul tht ounds th numr of rs (s Fig. 4 in St. II-C) th rs lrdy in C. This will minimiz th numr of nw rs introdud. Rmmr tht th disovry lgorithm of [11] is l to minimiz th numr of rs y prforming inry srh, thus w hv to provid som nw ounds whn minimizing th numr of dditionl rs. Lins 5, 7 nd 8 tkl this. Lt us onsidr first th lowr ound in lin 7. This is omputd s th diffrn twn th lpht of th urrnt lustr L i nd ll prvious lustrs. By Df. 1 ny C-nt tivity (sids s nd, tht, in ny s, nnot long to this st diffrn us thy lwys ppr in ll lustrs) hv t lst on outgoing nd on inoming r. Thus th smllst strutur in whih k nw tivitis n insrtd in C-nt, rquirs t lst k + 1 rs (for instn onnting ll k tivitis in row, nd thn onnting th ndings of th row to som tivitis lrdy in th nt). For th uppr ound, similr thortil limit n found. Howvr, it is mor prtil to dfin usr uppr ound on th rltiv numr of rs tht th usr wnts. This prmtr, nmd α in th lgorithm, indits th frtion of dditionl rs llowd, ing α th totl numr of rs tht th finl C-nt n hv. Th vlu for th vril is omputd s th mximum twn th minimum numr of rs tht C-nt with lpht A L n hv ( A L 1) or th numr of rs found in th first lustr. In s thr is no C-nt in th givn ounds, th disovrminnwarcnt funtion rturns th mpty C-nt, whih is th nutrl lmnt with rspt to C-nt union. V. EXPERIMENTS Th min purpos of this stion is to illustrt th nfits of using th high-lvl thniqu prsntd in this ppr with rspt to th monolithi pplition of th thniqu in [11]. For tht, w hv sltd smll st of nhmrks for whih th formntiond thniqu hs prolms to tkl, nd w provid hr th rsults for ths nhmrks using th thniqus dsrid in this ppr. Also, w ompr th lustring thniqu proposd in Algorithm 1 with th niv strtgy of rndom lnd lustring, dsrid informlly in Stion IV-B. Tl I shows som si informtion on th logs usd in our xprimnts. Th first fiv nhmrks r wllknown logs from [6]. Thy wr sltd us thy r diffiult to tkl with th monolithi pproh of [11]. For th sk of rdility, th nms of th logs hv n shortnd y rmoving f0n00 for th undrsor, i.. 22_5 rfrs to 22f0n00_5. Th lst on log is th msw nhmrk, whih is sltion of squns from th Mirosoft Anonymous W Dt dts in th UCI KDD rpository 1. Th tl ontins th following informtion for h nhmrk: L is th numr of (nondistint) squns in th log, L u is th numr of distint squns, σ is th lngth of th lrgst squn nd A L is th siz of th lpht of tivitis. Log L L u σ A L t msw Tl I BENCHMARK INFORMATION. Tls II nd III giv informtion on th xution of h on of th lustring lgorithms xplind in St. IV-A: t is th thrshold vlu usd to prtitioning th log, l is th numr of lustrs produd. For h nhmrk, w rport prtiulr informtion on h on of th lustrs providd in olumns L 1 to L 8 (th mximum numr of lustrs hivd in th nhmrks is 8). For h lustr, th numr of squns, numr of distint squns (if diffrnt), nd th siz of th lpht is shown. Fousing on this tl, th first imprssion is th good ln in siz for th drivd lustrs, oth in numr of trs nd siz of th log. This is dsirl ftur of ny divid-ndonqur thniqu, nling signifint rdution oth in pk mmory onsumption nd CPU tim. Tls IV nd V show th rsults of th divid-ndonqur strtgy of Algorithm 3 on th nhmrk logs omprd with th monolithi pproh of [11], using for lustring Algorithm 1 nd th rndom lnd lustring, rsptivly. For th lustr initil sorting of Algorithm 3 w hos no prtiulr ordring, thus lustrs wr squntilly prossd (first L 1, thn L 2, t.). Th tls ontin th following informtion: rs is th numr of rs of th finl C-nt, CPU is th lpsd tim (in sonds) rquird to omplt th disovry pross, α is th α prmtr of Algorithm 3, l indits th frtion of lustrs sussfully prossd. For ths xprimnts w hv limitd th mximum mount of mmory to usd to 1G, nd th mximum mount of lpsd tim to on hour. Th xprssions mm nd tim in th CPU olumns indit whih of oth limits ws rhd. Th α prmtr ws st in h s to vlu tht llowd us to unit ll th rsulting C-nts. 1 Avill t:

8 Log t l L 1 L 2 L 3 L 4 L 5 L 6 L 7 L L L u A L t L L u A L L A L L A L L A L msw L L u A L Tl II RESULTS OF THE RECURSIVESPLIT CLUSTERING FUNCTION (ALGORITHM 1). Log t l L 1 L 2 L 3 L 4 L 5 L 6 L 7 L L L u A L t L L u A L L A L L A L L A L msw L L u A L Tl III RESULTS OF A BALANCED RANDOM CLUSTERING. To rport on th qulity of th drivd C-nts, Tls IV nd V dditionlly hv th lst thr olumns. For h nhmrk, ths thr vlus r mnt to stimt two min fturs of th modl drivd: i) fitnss [2] nd ii) similrity with th modl drivd using th monolithi pproh. Informlly, in th fitnss dimnsion th pility of th log in rplying log trs is msurd. Column f is th ost-sd fitnss pr s mtri of [16], whr 1.0 indits tht ll squns in th log long to th lngug of th C-nt, nd th smllr th vlu is, th lss squns r rproduil y th C-nt. A mtri omplmntry to fitnss is prision: it is usd to dtrmin th mount of xtr hvior tht th modl ontins ut ws not osrvd in th log [2], [17]. Sin th prision msurs for C-nts r not yt dvlopd, thr is no wy to quntify th mount of dditionl hvior of th gnrtd modl with rspt to th C-nt of th monolithi pproh. For this rson w hv dfind two similrity msurs to ompr two C-nts gnrtd for monolithi [11] d&-rsplit (St. IV) Log rs CPU f l α rs CPU f s ios / t32 1 mm 2/ / / mm 8/ msw tim 7/ Tl IV RESULTS OF THE DIVIDE-AND-CONQUER STRATEGY COMPARED TO THE MONOLITHIC APPROACH. d&-rnd (St. IV) Log l α rs CPU f s ios / t32 1 0/ / / / msw 7/ Tl V RESULTS OF THE DIVIDE-AND-CONQUER STRATEGY USING A RANDOM BALANCED CLUSTERING. th sm log. Ths two similrity msurs, nmly r similrity nd input/output inding st similrity, ppr in olumns s nd ios in Tl IV, rsptivly. Givn two C- nts C 1 = A, s,, I 1, O 1 nd C 2 = A, s,, I 2, O 2, ths similrity msurs r dfind s follows: ios(c 1, C 2 ) = s(c 1, C 2 ) = rs(c 1) rs(c 2 ) rs(c 1 ) rs(c 2 ) A\{ } ( A\{ s} I 1 () I 2 () I 1 () I 2 () + O 1 () O 2 () O 1 () O 2 () ) 1 2 ( A 1). Ths r normlizd similrity msurs, whr vlus rng twn 1.0 (idntil) to 0.0 (ompltly diffrnt). In prtiulr th input/output inding st similrity is mor urt inditor thn th r similrity in th sns tht ios(c 1, C 2 ) = 1.0 ntils tht C 1 is qul to C 2, whil s(c 1, C 2 ) = 1.0 dos not. For instn th C-nts C 1 of Fig. 1() nd C 2 of Fig. 7() hv s(c 1, C 2 ) = 1.0 ut ios(c 1, C 2 ) = Not lso tht low vlus do not nssrily imply tht oth C-nts hv diffrnt lngug, spilly if th nts ontin rdundnt indings. For instn th C-nts of Fig. 5() nd Fig. 7() hv th sm lngug, i..,, ut thir similrity vlus r vry low (0.5 for th s mtri, nd 1 3 for ios). Th gnrl onlusion tht n drwn from th xprimntl rsults of Tl IV is th pility of th pproh in hndling nhmrks for whih th monolithi

9 () () Figur 7. () C-nt with r similrity qul to 1.0 with C-nt of Fig. 1(). () C-nt with low similritis with C-nt of Fig. 5() ut th sm lngug. thniqu of [11] fils. Morovr, for thos nhmrks wr th monolithi pproh suds, th rdution in CPU tim is onsidrl (roughly x6 rdution). This rdution in tim oms with no signifint pnlty in th qulity of th drivd C-nts: th s nd ios mtris omputd for ths C-nts r los to 1.0, whih mns tht sin th C-nt drivd for th monolithi pproh is n optiml modl in trms of numr of rs, thn th pproh of this ppr is pl in finding modls los to th optiml. It is lso worthwhil to ompr th rsults of Tls IV nd V, to hv som insight on th nfits of th nw lustring lgorithm proposd in this ppr. Th niv pproh of rndom lnd lustring provd to vlid mthod for most of th nhmrks ut t32_1, whih ould not hndld du to mmory prolms in two of th thr lustrs, nd CPU tim xpird in th rmining lustr. Compring th running tims twn th two ltrntivs, thy r on th sm ordr of mgnitud, ut with diffrns somtims rhing n x2 ftor. On of th lmnts tht hlps in xplining ths diffrns is th α ftor usd in h s. In gnrl, th rndom lnd lustring strtgy rquirs smllr vlus, whih hs dirt impt on th numr of SMT prolms tht hv to solvd. On th othr hnd, sin this lustring is not spifilly tilord for SMT, th tim tkn y h on of ths itrtions is usully lrgr thn with th nw lustring. This fstr prossing of th SMT prolms whn Algorithm 1 is usd for lustring n xplind y looking t th informtion of Tls IV nd V. First of ll, givn SMT formul, th numr of vrils nd qution it ontins r good prditors of th running tim tkn to did its stisfiility. Sin it ws shown in [11] tht ths quntitis r proportionl to th numr of distint squns ( L u ), th siz of th lpht ( A L ) nd th lngth of th squns, it is sy to s from ths tls tht lustrs produd y Algorithm 1 r mor sily prossd thn th ons rtd y th rndom strtgy. On th othr hnd, th ft tht ll tivitis r prsnt in vry lustr produd y th rndom strtgy (notly with th xption of msw) tnds to gnrt n initil numr of rs tht is losr to th finl on, thus llowing to rdu th α prmtr (xpt, gin, in th msw log). All in ll th lustring proposd fftivly rdus th urdn on th SMT solvr, whih hs mjor impt for instn whil prossing th t32_1 log, ut wys in whih th glol numr of itrtions ould drsd should invstigtd. VI. RELATED WORK Th thniqus prsntd in this ppr r groundd on th trditionl id of divid nd onqur omplx prolm, y splitting th input into svrl pis nd omining th individul solutions into th finl output. W will fous on th prtiulr pplition of this strtgy in th r of pross disovry. To th st of our knowldg, [18] ws th first ttmpt to tkl pross disovry y using lustring. Th pproh ws xtndd furthr in [19]. For th lustring thniqu prsntd in [19], n strtion is omputd s prpross: h tr is projtd into th most rlvnt fturs (omputd prviously) nd ssoitd with vtor of vlus. Thn th k-mns lgorithm is usd to prtition th vtoril sp dfind y th trs. A similr strtgy to this on ws prsntd in [20]. A totlly diffrnt pproh is dsrid in [21], [22], whr th prolm of multipl tr lignmnt is onsidrd. Ths pprohs r sd on omining tr lignmnt thniqus togthr with hirrhil lustring lgorithm. Although thir omplxity my high, th us of lignmnt mthods my ruil lmnt to improv th qulity of urrnt pross mining thniqus [1]. Finlly, in [23] n pproh tht is sd on rursivly projting th trs of th log is prsntd. Th prtitioning of th log is not horizontl (.g., slting trs of th log) lik th on of this ppr, ut vrtil (.g., slting vnts of th log nd projting vry tr onto ths vnts). This llows nding up with trs tht r mnl for pross disovry sin th finl rltions twn vnts tnd to simplr. VII. CONCLUSIONS AND FUTURE WORK This ppr prsnts high-lvl frmwork for th prolm of C-nt disovry. By using n spilly tilord lustring lgorithm, nd dpting monolithi C-nt minr tht rlis on SMT, this ppr dmonstrts tht th pproh is l to hndl inputs for whih th monolithi pplition of th SMT-sd disovry will fil. Rmrkly, th pproh is not srifiing fitnss, nd n guidd to driv nts whos siz is low som thrshold. Morovr, whn ompring th qulity of th drivd C-nts with rspt to

10 th monolithi pproh, thr r no signifint diffrns. Th thniqus of this ppr r implmntd in tool, nd xprimntl rsults on pross disovry nhmrks dmonstrt th fsiility of th ontriution proposd to tkl ths logs in limitd tim. As futur work, w pln to tst th thniqus of this ppr on lrgr nhmrks. Morovr, th thniqus of this ppr n xtndd in som dimnsions. First, furthr xtnsions of th lustring lgorithm will xplord, to nl guiding th disovry prolm with prtiulr ojtivs (.g. rdility, prision, hndl nois, mong othrs). Sond, th prolm of C-nt simultion will xplord. This my llow for instn to llvit th omplxity of th lustring pproh prsntd in this ppr: h tim C-nt is drivd from givn lustr, rmov from th rmining lustrs th trs tht r inludd y this C-nt. Finlly, othr strtgis to omin th C-nts otind in th lustrs whih onsidr dditionl informtion will onsidrd. ACKNOWLEDGMENT This work hs n supportd y projts FORMALISM (TIN ) nd TIN REFERENCES [1] W. M. P. vn dr Alst, Pross Mining - Disovry, Conformn nd Enhnmnt of Businss Prosss. Springr, [2] A. Rozint nd W. M. P. vn dr Alst, Conformn hking of prosss sd on monitoring rl hvior, Informtion nd Systms, vol. 33, no. 1, pp , [3] W. M. P. vn dr Alst nd C. W. Günthr, Finding strutur in unstruturd prosss: Th s for pross mining, in ACSD, T. Bstn, G. Juhás, nd S. K. Shukl, Eds. IEEE Computr Soity, 2007, pp [4] W. M. P. vn dr Alst, T. Wijtrs, nd L. Mrustr, Workflow mining: Disovring pross modls from vnt logs, IEEE TKDE, vol. 16, no. 9, pp , [5] R. Brgnthum, J. Dsl, R. Lornz, nd S.Musr, Pross mining sd on rgions of lngugs, in Businss Pross Mngmnt, Sp. 2007, pp [6] J. M. E. M. vn dr Wrf, B. F. vn Dongn, C. A. J. Hurkns, nd A. Srrnik, Pross disovry using intgr linr progrmming, in Ptri Nts, sr. LNCS, vol. 5062, 2008, pp [7] J. Crmon, J. Cortdll, nd M. Kishinvsky, Nw rgionsd lgorithms for driving oundd ptri nts, IEEE Trns. Computrs, vol. 59, no. 3, pp , [8] M. Solé nd J. Crmon, Pross mining from sis of stt rgions, in Ptri Nts, sr. LNCS, vol. 6128, 2010, pp [9] W. Vn Dr Alst, A. Adrinsyh, nd B. Vn Dongn, Cusl nts: modling lngug tilord towrds pross disovry, in CONCUR, 2011, pp [10] A. J. M. M. Wijtrs nd J. T. S. Riiro, Flxil huristis minr (FHM), in CIDM. IEEE, 2011, pp [11] M. Solé nd J. Crmon, An SMT-sd disovry lgorithm for C-nts, (sumittd to ATPN 12) lso s th. rp. t UPC, Th. Rp. LSI-12-2-R, [Onlin]. Avill: [12], Inrmntl pross mining, T. Ptri Nts nd Othr Modls of Conurrny, 2012 (To Appr). [13] S. Jh, R. Limy, nd S. Sshi, Bvr: Enginring n ffiint SMT solvr for it-vtor rithmti, in Computr Aidd Vrifition, 2009, pp [14] T. Murt, Ptri Nts: Proprtis, nlysis nd pplitions, Prodings of th IEEE, pp , Apr [15] L. Mrustr, A. J. M. M. Wijtrs, W. M. P. vn dr Alst, nd A. vn dn Bosh, A rul-sd pproh for pross disovry: Dling with nois nd imln in pross logs, Dt Min. Knowl. Disov., vol. 13, no. 1, pp , [16] A. Adrinsyh, B. vn Dongn, nd W. vn dr Alst, Conformn hking using ost-sd fitnss nlysis, in Entrpris Distriutd Ojt Computing Confrn (EDOC), 2011, pp [17] J. Munoz-Gm nd J. Crmon, A Frsh Look t Prision in Pross Conformn, in Businss Pross Mngmnt (BPM), [18] G. Gro, A. Guzzo, L. Pontiri, nd D. Sà, Disovring xprssiv pross modls y lustring log trs, IEEE Trns. Knowl. Dt Eng., vol. 18, no. 8, pp , [19] A. K. A. d Mdiros, A. Guzzo, G. Gro, W. M. P. vn dr Alst, A. J. M. M. Wijtrs, B. F. vn Dongn, nd D. Sà, Pross mining sd on lustring: A qust for prision, in Businss Pross Mngmnt Workshops, sr. Ltur Nots in Computr Sin, A. H. M. tr Hofstd, B. Bntllh, nd H.-Y. Pik, Eds., vol Springr, 2007, pp [20] M. Song, C. W. Günthr, nd W. M. P. vn dr Alst, Tr lustring in pross mining, in Businss Pross Mngmnt Workshops, sr. Ltur Nots in Businss Informtion Prossing, D. Ardgn, M. Mll, nd J. Yng, Eds., vol. 17. Springr, 2008, pp [21] R. P. J. C. Bos nd W. M. P. vn dr Alst, Contxt wr tr lustring: Towrds improving pross mining rsults, in Prodings of th SIAM Intrntionl Confrn on Dt Mining, SDM 2009, April 30 - My 2, 2009, Sprks, Nvd, USA. SIAM, 2009, pp [22], Pross dignostis using tr lignmnt: Opportunitis, issus, nd hllngs, Inf. Syst., vol. 37, no. 2, pp , [23] J. Crmon, Projtion pprohs to pross mining using rgion-sd thniqus, Dt Min. Knowl. Disov., vol. 24, no. 1, pp , 2012.