LoonyBin: Keeping Language Technologists Sane through Automated Management of Experimental (Hyper)Workflows

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1 LoonBin: Keeping Langage Technologi Sane hrogh Aomaed Managemen of Eperimenal (Hper)Workflo Jonahan H. Clark, Alon Laie Langage Technologie Inie Carnegie Mellon Unieri Abrac Man conemporar langage echnolog em are characeried b long pipeline of ool ih comple dependencie. Too ofen, hee orkflo are implemened b ad hoc crip; or, ore, ool are rn manall, making eperimen difficl o reprodce. Thee pracice are difficl o mainain in he face of rapidl eoling orkflo hile he alo fail o epoe and record imporan deail abo inermediae daa. Frher complicaing hee em are hperparameer, hich ofen canno be direcl opimied b conenional mehod, reqiring er o deermine hich combinaion of ale i be ia rial and error. We decribe LoonBin, an open-orce ool ha addree hee ie b proiding: 1) a ial inerface for he er o creae and modif orkflo; 2) a elldefined mechanim for racking meadaa and proenance; 3) a crip generaor ha compile ial orkflo ino hell crip; and 4) a ne orkflo repreenaion e call a HperWorkflo, hich iniiel and ccincl encode mall eperimenal ariaion ihin a larger orkflo. 1. Inrodcion Empirical reearch in naral langage proceing and compiling reorce o hi end hae become comple mli-age procee. Daa preparaion alone can reqire okeniaion, e normaliaion, re-encoding, cleaning of noi daa, ec. Eperimen on hi daa ofen inole raining and ning of mliple model, eing on a heldo e e, and ealaion of he rel all of hich are condced nder mliple eperimenal condiion (i.e. ih differen corpora and differen e of hperparameer). For eample, in nacic aiical machine ranlaion, a pical eperimen coni of oer 20 ool ih a comple neork of dependencie panning mliple machine or een cler of machine. Paring and phrae eracion migh be rn on a large cler of hndred of lo-memor machine, preproceing and ord alignmen migh be rn on a local erer here ofare inallaion i eaier, hile ning and decoding migh be done on a mall cler of large-memor machine. The managemen of ch orkflo preen a real challenge in erm of keeping rel organied, analing rel a eer age, and aomaing he orkflo. Some find hi ak o frraing ha he forgo aomaion alogeher, boh making eperimen difficl o reprodce and aing CPU ccle hen ak finih hile he er i aleep or oherie aa from a erminal. Thoe ho aomae heir ak ofen e ad hoc crip ha are brile o orkflo change, perform ani checking in an inconien a (if a all), and keep log file in diparae forma. Een pecial-prpoe orkflo managemen em (ee Secion 5.) are akard a be for rnning eperimen nder mliple condiion. LoonBin a deigned o handle medim-cale 1 arbirar cienific orkflo keeping he need of naral langage 1 B medim-cale, e mean orkflo haing hndred of erice. Mega-cale orkflo hae been idenified b Deelman a hoe haing hndred of hoand of erice. proceing in mind. Specificall, i accommodae orkflo ha: pan ario machine, cler, and chedler inole man eparae ool, hich can be inoked b arbirar UNIX command hae componen ha are rn mliple ime nder mliple condiion eole qickl ih ool freqenl being added, remoed, and apped LoonBin accomplihe hi b proiding he folloing adanage oer crren common pracice: aociaing ani check and logging direcl ih ool, eparaing hee from ad hoc rapper and aomaion crip mainaining a cleanl organied direcor rcre for each ep and each condiion nder hich a ep i rn proiding a reme-on-failre mechanim for eer age in he pipeline making i ea for hoe iho a deailed knoledge of each ool inernal o rn he em b proiding eal decripion of each parameer, inp file, and op file in a graphical orkflo deigner aomaicall coping reqired file beeen machine or cler ia SSH compiling orkflo ino hell crip, a medim alread in idepread e b NLP reearcher 2.1. Workflo Semanic 2. Workflo Creaion We no dic he repreenaion of orkflo in Loon- Bin. In heir mo baic form, LoonBin repreen orkflo a Direced Acclic Graph (DAG) a hon in Figre 1. In hi form, each ere repreen a TOOL, hich prodce op file gien inp file and parameer, and

2 Informaion Rerieal Corp Deermine Te Vocablar Inde Docmen for Te Se Ealae Accrac Te Qerie Preproce Te Qerie Figre 1: A imple orkflo repreened a a Direced Acclic Graph (DAG) a decribed in Secion 2.1. ih QA Corp A a b Bild Onolog {a,b} Epand ia WordNe {a,b} iho QA Corp B Inde Corp {a,b} Ealae on Te Qeion {a-ih,b-ih, a-iho, b-iho} Figre 2: A HperWorkflo ih mliple REALIZATIONS, repreened a a Semanic DAG a decribed in Secion 2.2. Hperedge are iall repreened a edge originaing from rianglar erice. direced edge indicae relaie emporal ordering of ool and informaion flo (file or parameer) b mapping he op of one ool o he inp of he ne. A TOOL DE- SCRIPTOR define he command necear o rn a ool gien inp, op, and parameer. Com ool decripor can be implemened ia imple er-defined Phon crip ha generae hell command. Thee ool decripor conain PRE-ANALYZERS o check he ani of he inp and o log informaion and POST-ANALYZERS o check he ani of he op file, log informaion abo he op, and erac log daa from an hird-par log file forma HperWorkflo Semanic LoonBin alo repreen he rnning of orkflo nder mliple eperimenal condiion (i.e. ih differen inp file or parameer). We call hi a HYPERWORKFLOW. A HperWorkflo conain REALIZATION VARIABLES, hich inrodce ariaion ino a hared orkflo. Each realiaion ariable can ake on a REALIZATION VALUE, hich i a e of file and parameer. For inance he realiaion ariable langage model file and order cold ake on he realiaion ale {englih., 4}. Finall, a REAL- IZATION INSTANCE i a reglar orkflo npacked from a HperWorkflo; i i a configraion of a HperWorkflo ch ha all realiaion ariable hae been aigned a pariclar realiaion ale. Noe ha ome realiaion ariable migh ake on a nll ale if he are irrelean for a gien inance (ee Figre 8 in he appendi). HperWorkflo are efl for performing eploraion of hperparameer, ablaion die, ariaion of inp corpora, and o forh. To mee hi reqiremen, e e a ne daa rcre baed on direced hpergraph 2. A hpergraph i a raighforard generaliaion of a graph in hich each endpoin of an edge can connec mliple erice (Gallo e al., 1993). For HperWorkflo, e e a HYPERDAG 3, he hpergraph formlaion of a DAG, hon in Figre 2. Since hi daa rcre i difficl o dra in a ial inerface, e preen i o he er b giing haped erice pecial emanic; e call hi repreenaion he SEMANTIC DAG forma. In LoonBin, a DIRECTED HYPEREDGE originae a an edge enering a PACKING VERTEX (diplaed a a riangle in Figre 2). A packing ere correpond eacl o a realiaion ariable. Each hperedge m hae a name and each hperedge inrodce a realiaion ale of he realiaion ariable. A hperedge ha mliple orce hen mliple edge ih he ame name ener he packing ere, and i ha mliple deinaion hen mliple edge ei he packing ere. A packing ere ac like a ich o elec one of i realiaion ale. Th, each niqe named edge enering a packing ere creae a ne realiaion inance in he orkflo. Thee realiaion ariable are hen propagaed hrogh he remainder of he orkflo. Where mliple realiaion ariable mee, LoonBin prodce he cro prodc beeen heir realiaion ale. A HperWorkflo i a packed repreenaion of mliple orkflo DAG, and a 2 Hpergraph are alread ed in NLP boh in paring (Klein and Manning, 2002) and nacic machine ranlaion (Zollmann and Vengopal, 2006). 3 In pracice, e e a MeaHperDAG, hich generalie a HperDAG b alloing mea-edge o ake hperedge a inp. See he appendi for deail.

3 realiaion inance i a pariclar npacked inance of a orkflo. P anoher a, a realiaion inance i an eperimenal condiion (ih regard o inp file and parameer) nder hich a orkflo i rn. For eample, in Figre 2 edge a and b ener a packing ere and hen propagae realiaion a and b. Noice ha a he final Ealae on Te Qeion ere, he realiaion combine ih each oher o form a fll cro prodc of eperimenal condiion. B repreening orkflo in hi a, e can alo eploi he inheren hared brcre in hee orkflo in a dnamic programming fahion (Hang, 2008). We can boh cleanl repreen all of he ep reqired o reprodce he eperimen hile no rernning an ep haing he ame e of eperimenal condiion Deigning LoonBin proide a ial edior, hich li all ool (ee Figre 4) in broable ree. Tool can impl be dragged and dropped ino he orkflo a erice and edge can be dran b dragging arro beeen hee erice. The onl reqiremen on he deign machine i a recen erion of Jaa (Phon crip are eeced ia Jhon). Loon- Bin alo allo for aomaic generaion of imple docmenaion for ool b ing docmenaion ring ha are reqired for eer inp, op, and parameer a a par of eer ool decripor Deploing Once a orkflo ha been deigned, LoonBin can hen compile i ino an eecable hell crip. Th, he onl reqiremen on he machine ha eece he orkflo i bah. Before an ool are eer eeced, he generaed crip check ha all inp file and all direcorie conaining reqired ool ei. Becae LoonBin handle all filename oher han he iniial inp, hi eliminae he common ie of pipeline crahing de o po in file and direcor name. The generaed crip ill log ino remoe machine, coping file and eecing procee a necear. LoonBin alo offer he opion of eecing orkflo anchronol. In hi mode, a Jaa proce lanche indiidal bah crip hen dependencie hae been aified, and a broer-baed eb poral i proided o monior orkflo progre. LoonBin allo each ere o rn on a differen machine. A machine can be a arge on hich o impl rn command, or i can be he head node of a cler hrogh hich one can bmi job. LoonBin naiel ppor chedler ch a Torqe, Sn Grid Engine, and Condor. In hi repec, LoonBin can be hogh of a a mea-chedler: i primaril relie on oher chedler o enre ha reorce are effeciel allocaed. LoonBin alo proide inegraion ih he Hadoop Diribed File Sem. 3. Daa, Meadaa, and Proenance While being able o aomaicall eece and reprodce orkflo i good, impl compleing he job i no enogh. We alo an o kno ho he inp file a a gien ep came o be; hi hior of he ool anceor along ih heir parameer and inp i called PROVE- NANCE dbbed he bridge beeen daa and eperimen b Mile (2008). In addiion, e ma alo ih o ore METADATA, qaliie aociaed ih he daa ch a ho long i ook o rn a ool and on hich machine a ool ran. All he proenance and meadaa i ored in plaine log file in a andard forma: ab-delimied ke-ale pair and neline-delimied record, making i ea o proce hee log file ing andard command-line ool or crip. Finall, he log file for all aneceden ep for he ame realiaion are concaenaed ogeher a each ere o ha log for each realiaion can be proceed ia a ingle file. Since he er migh an o rn frher anali laer, i i imporan o be able o eail find he daa ielf. To accommodae hi, LoonBin mainain a highl organied direcor rcre for each orkflo. Under a maer direcor, LoonBin creae a direcor ih he name of each ere in he HperWorkflo ih child bdirecorie for each realiaion. 4. Real-World Uage The fir real-orld e of LoonBin ha been in he CMU SaXfer machine ranlaion em for GALE Phae 4 (ee Figre 3) and he 2010 Workhop on Machine Tranlaion. Aide from he eperimenal benefi proided b HperWorkflo, he ani-checking and logging capabiliie rned o o be ome of he mo efl feare of LoonBin for hi ak. For inance, he ord alignmen program GIZA++ i noorio for creaing blank op file, crahing, and hen rerning a ccefl error code. Or GIZA++ ool decripor ha a po-analer ha deec failre and log he Alignmen Error Rae. Dring corp preproceing, e log he nmber of pe, oken, ec. for he enire parallel corp afer each proceing ep. Thi ha proen efl boh for nderanding ho each ool affec he daa and for comparing differen em a laer dae. 5. Relaed Work In hi ecion, e decribe ario alernaie mehod of implemening orkflo (ee Table 1 for a mmar). Perhap he mo common ool for implemening orkflo are crip. Perhap he mo common ool for implemening orkflo are crip, hich can be hard o mainain a ne ep and ool ineiabl make heir a ino a orkflo. Oher hae ed bild iliie ch a GNU Make ince hi more eplicil model he dependencie beeen ep. Hoeer, a a bild ool, i i an akard fi for eperimen managemen, ince i doe no hae he noion of HperWorkflo and keeping eperimenal condiion eparae ia Make ariable can be error-prone. Workflo Managemen Sem (WMS) hae been a foc of reearch in he e-science commni for man ear, hogh he hae remained largel nnoiced b he NLP commni. The DAGMan (DAGMan Team, 2009) projec gre o of he Condor chedler and allo he er o decribe dependencie beeen Condor job in he form of a DAG. While cerainl a beer fi han a bild ili, DAG- Man doe no ppor handling of meadaa or proenance

4 Reme on Failre Sani Checking Manal Eecion Manal Scriping Make DAGMan Kepler Pega Drad LoonBin Sppor Hper- Workflo Timeconming Difficl No ao- No ao- No ao- No ao- No ao- for large maic maic maic maic maic orkflo χ Manal Manal eaminaion of daa Uall no Depend on er Uall primiie Record Meadaa χ Uall no Record Proenance χ Uall no Progre Monioring Conole Conole op op Mli-Cler Sppor Mli-Schedler Sppor χ χ χ χ ia Condor Flocking χ χ χ χ ia Condor- G Workflo Definiion Sna Reprodcible Rel Fleible Pipeline Change For imple orkflo Wih com configraion Uall primiie Uall no Uall no Conole op χ Scrip langage M be embedded in job No aomaic No aomaic Condor Log/ Conole M be M be Unknon Embedded embedded in embedded in cenrall in ool job job Unknon Unknon In GUI (Eperimenal) Propriear Propriear Vial Vial / XML Table 1: Tool commonl ed o implemen orkflo. Ye Conole op and eb inerface Unknon χ χ C++ Vial moe Moe Phrae Table Training Parallel Corp Targe Langage Corp Filer Corp Bild Langage Model Word Alignmen Sanford Parer Charniak Parer ch na Bild Snacic Tranlaion Model {,ch} Minimm Error Rae Training {na-, na-ch, moe} Decode Senence {na-, na-ch, moe} Figre 3: A implified erion of he CMU SaXfer em HperWorkflo for he GALE Phae 4 Machine Tranlaion Ealaion hoing he mliple eperimen ha ere rn on i on. The Pega Projec (Deelman e al., 2003) aim o ole hi b bilding on op of DAGMan and proiding a mechanim for decribing orkflo in a more abrac a hile proiding comprehenie ppor for meadaa and proenance managemen. Similarl, Kepler (Alina e al., 2004) proide a and-alone em for deigning and rnning orkflo. Drad (Iard e al., 2007) alo allo conrcing orkflo on Windo HPC cler ing a C++ librar. In erm of ra performance, Drad i a clear inner among all of hee em, inclding LoonBin. Hoeer, none of hee em allo he er o eplicil encode mliple eperimen a H-

5 Toolbo Workflo Deigner Parameer Enr Figre 4: A creenho of he LoonBin er inerface perworkflo iho reoring o looping conrc. In conra, LoonBin eplicil aoid Tring-compleene o enre deerminim hile ill alloing he er o eail encode eperimenal ariaion. For a general diion of erminolog ed in WMS, e recommend McPhilip (2008). For a more complee dicion of he crren ae of cienific orkflo managemen em, e recommend Deelman (2008) or Ldächer (2009). 6. Fre Work Thogh LoonBin aemp o encode reprodcible orkflo, hi i onl achieable p o a poin. Change in ofare erion, enironmen ariable, and hardare reorce cold cae orkflo o fail in ne enironmen. While i i no pracical o conider hardare limiaion, i i ea o generae orkflo ch ha all enironmen ariable m be capred ihin he orkflo definiion raher han eernall. Hoeer, ince hi old place a large brden on he er, e hae no e pred hi. On hi ie of ofare erion, e plan o link each ool in a orkflo o a pariclar reiion of a orce code managemen em o ha he original ofare erion can be rebil if deired. Wih hi addiion, e foreee he abili o pblih and hare LoonBin orkflo online. Beide HperWorkflo in hich cerain porion of a orkflo are rn mliple ime gided b packing erice, cerain porion ma alo be hared a differen poin ihin he ame realiaion. For eample, a grop of erice migh creae a ranlaion model, filer i o a pecific e, and hen prne i. Crrenl, one m dplicae hee porion of he orkflo. Programming langage, in conra, deal ih hi cleanl ih he noion of fncion. Eenall, e old like o inegrae hi ino LoonBin ih he abili o e orkflo (b no HperWorkflo) a erice. LoonBin gie he er he abili o eail eece pah hrogh a HperWorkflo eiher ehaiel or b elecing a be of realiaion. Ye ehaie eploraion of hee orkflo configraion i no ideal ih regard o ime and reorce limiaion hile he er ofen e 1) imple crieria for elecing hich pah o elec ne or 2) doe no hae a good idea of ha o elec ne. Thi gge ha e cold do aomaic opimiaion of hperparameer. Since a HperWorkflo alread define a earch pace oer a orkflo, e old need onl o define an objecie fncion o chooe he ne realiaion o eplore ne. Anoher apec of cienific orkflo i analing and recording rel. While LoonBin record hee in organied log file, e old like o aomaicall generae char, graph, and able from hee. Frher, i old be deirable o hae a mehod of archiing eperimen from a collecion of orkflo rn oer a long period of ime ihin a reearch grop. Some er are er eperienced and comforable ih e edior ch a im and Emac. Therefore, e alo plan o proide a hman-ediable forma for oring orkflo o he ial edior can be bpaed. Crrenl, LoonBin generae bah crip ha implemen orkflo. Hoeer, i hold alo be poible o generae DAGMan file o rn on a Condor cler (alhogh Loon- Bin alread ppor Condor a a chedler). Anoher opion old be generaing Pega DAX file o be mapped on o he Grid. Finall, e cold conider generaing Drad proce rapper o be mapped on o a Windo HPC cler b adding a ne WorkfloViior in LoonBin.

6 7. Conclion We hae preened LoonBin, hich e are releaing 4 a an open-orce ool for managing orkflo in langage echnolog em. In or on eperience ih he SaXfer MT em, e hae fond he ool o be eremel efl. To encorage adopion, e releae he orce nder he nonrericie LGPL licene and proide qickar ideo creenca a orial. We hope ha b proiding hi ool o he commni, eperimen ill become more reprodcible and reearcher become more prodcie. 8. Acknoledgemen Thank o o Kenneh Heafield, Alok Parlikar, and Nahan Schneider for heir man inigh in riing hi paper. Thank o Michael Denkoki and Greg Hanneman for paienl helping o ork he bg o of earl erion of LoonBin. We are graefl o Ondřej Bojar, Philipp Koehn, and Lane Schar for haring heir eperience in riing eperimen managemen em for heir machine ranlaion orkflo. Thank o Yahoo! for proiding acce o he M45 reearch cler, hich e ed o e he Hadoop/HDFS inegraion of LoonBin. Finall, hank o o he anonmo reieer for heir efl commen. 9. Appendi: MeaHperDAG In hi ecion, e kech he eqialence of MeaHper- DAG and he emanic DAG preened in Secion 2.2. We alo preen a comple eample ha highligh a fe imporan bondar cae and ho he poer of hi rcre. Thi informaion i no inended for caal er and ill likel be of inere onl o er ho ih o kno he bondarie of LoonBin epreie poer or hoe ihing o reimplemen he core algorihm. While emanic DAG proide a conenien a of incremenall creaing hperedge, he are no o conenien hen npacking HperWorkflo ino heir npacked conerpar. Therefore, i i eaier o hink abo iiing a HperWorkflo in erm of a MeaHperDAG in hich each hperedge (or mea-edge) repreen an aomic deciion. We m generalie a HperDAG o a METAHY- PERDAG b alloing mea-edge, hich ma ake mliple hperedge a heir inp 5, o ha a raeral of he HperWorkflo ii each ere he minimm nmber of ime. Conider Figre 5. If e ere o model he necearil join deciion of hich realiaion ale o elec for a hperedge alone, e arrie a hperedge ha conflae he deciion of chooing he ale of realiaion ariable A, B, and C. Thi old prodce o hperedge, one for a1-b1-c1 and one for a1-b1-c2. Boh of hee hperedge old hae a a deinaion, caing o ge iied ice. Ye hold be iied onl once. B inrodcing he large bold mea-edge in Figre 6, e can aomicall chooe a combinaion of realiaion ale for hile no caing prio ambigi for. We iner a mea-edge heneer 4 LoonBin i aailable for donload a hp://.c.cm.ed/ jhclark/loonbin 5 Alernaiel, e cold hae inered no-op erice a he orce of he mea-edge, making he mea-edge normal edge. he paren of a ere are inflenced b differing e of realiaion ariable. In hi a, e can conrc an eqialen MeaHperDAG for a emanic DAG. The remaining figre ho he npacking proce, aring a a emanic DAG and ending ih eparae npacked DAG. 10. Reference Ilka Alina, Chad Berkle, Efra Jaeger, Mahe Jone, Berram Ldächer, and See Mock Kepler: an eenible em for deign and eecion of cienific orkflo. In Scienific and Saiical Daabae Managemen. DAGMan Team Dagman docmenaion, hp://.c.ic.ed/condor/dagman/. Ea Deelman, Jame Blhe, Yolanda Gil, Carl Keelman, Garang Meha, Karan Vahi, Ken Blackbrn, Alber Laarini, Adam Arbree, Richard Caanagh, and Sco Koranda Mapping abrac comple orkflo ono grid enironmen. In Jornal of Grid Comping. Ea Deelman, Denni Gannon, Mahe Shield, and Ian Talor Workflo and e-cience: An oerie of orkflo em feare and capabiliie. In Fre Generaion Comper Sem. Giorgio Gallo, Giino Longo, Sang Ngen, and Sefano Palloino Direced hpergraph and applicaion. In Dicree Applied Mahemaic. Liang Hang Adanced dnamic programming in emiring and hpergraph frameork. In Coling 2008 Torial Noe. Michael Iard, Mihai Bdi, Yan Y, Andre Birrell, and Denni Feerl Drad: diribed daa-parallel program from eqenial bilding block. In Proceeding of he 2nd ACM SIGOPS/EroS Eropean Conference on Comper Sem. ACM. Dan Klein and Chri Manning Paring and hpergraph. In Bn, Carroll, and Saa, edior, Ne Deelopmen in Paring Technolog. Kler Academic Pbliher. Berram Ldächer, Ilka Alina, Shan Boer, Jlian Cmming, Terence Crichlo, Ea Deelman, Daid De Rore, Jliana Freire, Carole Goble, Mahe Jone, Sco Klak, Timoh McPhillip, Norber Podhorki, Cladio Sila, Ian Talor, and Mladen Vok Scienific proce aomaion and orkflo managemen. In Scienic Daa Managemen: Challenge, Eiing Technolog, and Deplomen, Compaional Science Serie. Chapman & Hall. Timoh McPhillip, Shan Boer, Daniel Zinn, and Berram Ldaecher Scienific orkflo deign for mere moral. In Fre Generaion Comper Sem. Simon Mile, Pal Groh, Ea Deelman, Karan Vahi, Garang Meha, and Lc Morea Proenance: The bridge beeen eperimen and daa. In Comping in Science and Engineering. Andrea Zollmann and Ahih Vengopal Sna agmened machine ranlaion ia char paring. In Workhop on Machine Tranlaion (WMT) a he Aociaion for Compaional Lingiic (ACL).

7 a1 A d2 b1 B c1 C d1 d2 D Figre 5: A comple emanic DAG. The packing erice A and B are for illraing hperedge; all, here i lile poin in haing a packing ere ih a ingle inp edge. There i a 1-1 mapping beeen a LoonBin Semanic DAG and a correponding MeaHperDAG. b1 c1 a1 d1 d2 Figre 6: The correponding MeaHperDAG. Noice ha packing erice hae been ranformed ino hperedge and mea-edge. See ha he o d2 edge enering C creaed a hperedge ih mliple orce hile he mliple edge leaing A creae a hperedge ih mliple deinaion. When here are mliple packing erice a he direc paren of a ere, a cro prodc i implied a hon a. Finall, heneer he paren of a ere are inflenced b differing e of realiaion ariable, e creae mea-edge hon in bold a een a and a1-b1-c1-d1 a1-b1-d2 Figre 7: A broken HperDAG-onl concepion of he aboe emanic DAG. Noice ha o aif he conrain ha each deciion be aomic, compoie hperedge ha joinl decide a lea A, B, and C a he ame ime ere inrodced. Hoeer, hi cae o hae an o hperedge (one for d1 and one for d2), hich inclde realiaion ariable ha do no een affec. Thi iolae he conrain ha each ere be iied a minimal nmber of ime.

8 Pah of Realiaion a1-b1-c1-d1 a1 Pah of Realiaion a1-b1-d2 a1 b1 c1 d1 b1 d2 Figre 8: The correponding pah hrogh hi MeaHperDAG. Noe ha he realiaion ariable C become irrelean for realiaion ale d2 o ha e no longer need o pick a ale for i hen eploring ha pah. In hi form, i i naral o hae mliple edge ih he ame orce and deinaion erice ince he migh be linking differen file or parameer. Unpacked Realiaion a1-b1-c1-d1 Unpacked Realiaion a1-b1-d2 Figre 9: The correponding npacked DAG, conaining no hperedge or mea-edge.