A Survey on Combinatorial Group Testing Algorithms with ...

1 DIMACSS eriesinDiscreteMathematicsandTheoretical ComputerScienceA SurveyonCombinatorialGroupTestingAlgorit hmswithApplicationstoDNA TRACT. Inthispaper, wegive focusesonseveralclassesofconstructionsno tdiscussedinprevioussurveys,providesa (aDNAsegment)fromthelibrarycontainswhich probefroma saidtobepositivefora probeif it containstheprobe, ,sincecheckingeachclone-probepairis expensive andusuallyonlyafew clonescontainany whenSequenced-TaggedSitemarkers(alsocall edSTSprobes)areused[OHCB89].If thetestresultfora pool(ofclones)isnegative, indicatingthatnocloneinthepoolcontainsth eprobe, justaninstanceofthegeneralgrouptestingpr oblem,inwhicha largepopulationofitemscontaininga smallsetofdefectivesaretobetestedtoident ifythedefectivesefficiently. We assumesometestingmechanismexistswhichif appliedtoanarbitrarysubsetofthepopulatio ngivesanegativeoutcomeif thesubsetcontainsnodefec-tive ,limitingnumberofpools,limitingpoolsizes totoleratinga isconceivablethattheseobjectivesareoften contradicting, categories:CombinatorialGroupTesting(CGT )andProbabilisticGroupTesting(PGT).

2 InCGT, it is oftenassumedthatthenumberofdefectivesamo ng itemsis equalto orat most forsomefixedpositiveinteger . InPGT, wefixsomeprobability ofhavinga thepoolsaresimul-taneouslytested times,withlatertestpoolscollectedbasedon previoustestresults,thentheCGTalgorithmi s saidtobean A grouptestingalgorithmis non-adaptive if ,05D05;Secondary05D40, , beingnon-adaptive isequivalenttobeing it candetectorcorrectsome constraintsto , -stagegrouptestingalgorithmswithsmall ( ) areoftenpreferable[BT96,BBKT96].Thecommo nrequirementis tohave anadaptive , DNAscreeningis errorpronesincethepoolshave ,toler-atingseveralerrorsis desirable[BT96].Lastly, asassemblingpoolsis costly, (suchasa grid) , Survey fromDyachkov andRykov (1983,[DR83]) doneinthecontext monographbyDuandHwang(1993,[DH93]),which gave a al.(1995,[BBKT96]),whichcomparativelysur veyedcertainclassesofnon-adaptive , wegive focusesonseveralclassesofcon-structionsn otdiscussedinprevioussurveys,providesa fixesupbasicdefinitionsandnotationsneede dfortherestofthepaper.

3 It alsogivesa providesa newgeneralperspective onconstructinga classofdeterministicpoolingdesigns,fromw hichseveralopenproblemspoppedupnaturally . Section5 presentsrandomalgorithms,andsection6 firstemphasizethatweareconcernedonlywith combinatoriallynon-adaptive grouptestingstrategies,forDNA libraryscreeningap-plicationspreferparal leltestsaswehave mentionedearlier. The Combinatorial partcomesfromtheassumptionthatthereareat most defectivesina populationof -matrix . Let and denoterow andcolumn ,wealsolet (resp. ) denotethesetofcolumn( )indicescorrespondingtothe -entriesofrow ( ). Theweightofa row ora columnis thenumberof s it has. is saidtobe -disjunctif theunionofany columnsdoesnotcontainanother. A -disjunct matrix canbeusedtodesigna non-adaptivegrouptestingalgorithmon thenitem is containedinpool (andthustest ).If therearenomorethan defectivesandthetestoutcomesareerror-fre e,thenit is simplyidentifytheitemscontainedinnegativ e poolsasnegatives(gooditems)andtherestasp ositives(defecteditems).

4 Noticethat -disjunctpropertyimpliesthateachsetof defectivescorrespondsuniquelytoa testoutcomevector, thusdecodingtestoutcomesinvolvesonlya -disjunctmatrixis thusalsonaturallycallednon-adaptivepooli ngdesign. We shallusethisterminterchangeablywiththelo ngphrase non-adaptivecombinatorialgrouptestingalg orithm . denotesthesetofallsubsetsof items(orcolumns)withsizeat most ,calledthesetofsamples. For , let denotetheunionofallcolumnscorre-sponding to . Apoolingdesignis -error-detecting(correcting)if it candetect(correct)upto ,if a designis -error-detectingthenthetestoutcomevector sforma -dimensionalbinarycodewithminimumHamming distanceatleast . Similarly, if a designis -error-correctingthenthetestoutcomevecto rsforma -dimensionalbinarycodewithminimumHamming distanceat least . Thefollowingremarksaresimpletosee, hasthepropertythatforany , and viewedasvectors haveHammingdistance .Inotherwords, where denotesthesymmetricdifference. Then, is -error-detectingand !#"$&% being -disjunctisequivalenttothefactthatforany setof distinctcolumns (' *)*) ) ,+withonecolumn(say ,') designated, ,'hasa insomerowwhere all ,- s.

5 Contain given itemswithatmost defectives,atleasthow many testsareneededtoidentifythedefectives? Thebestasymptoticanswertothisquestionis datedbacktoDyachkov andRykov (1982,[DR82]) andDyachkov, Rykov andRashad(1989,[DRR89]), denotetheminimumnumberof poolsneededforthe problem,thenas 0/21and /21 $ 436587$ :9 ; 36587$ < $36587$ :9 = 36587$ now give a tentative tax-onomyofnon-adaptive poolingdesigns,fromwhichlatersectionsareorganized.(1)DeterministicDesigns. Thisreferstothefactthateverypoolis :(i)Set-packingdesigns.(ii)Transversaldesigns.(iii)Designswhose -disjunctmatricesaredirectlyconstructed.(2)RandomDesigns. Inthesedesigns,someoralloftheentriesarerandomlyde-terminedwithparameterizedprobabilities,whichcouldbeoptimizedbasedoncertainobjective function(s).Thecategoriesare:(i)Randommatrices.(ii)Randomweight->designs.(iii)Randomsize- designs.(iv)Randomdesignswhichcomefromde terministicdesigns.(3)ErrorToleranceDesi gns. Althoughthesedesignsareeitherdeterminist icorran-dom,they [KS64] backin1964, A@ packingdesignis a collectionBof -subsets(calledblocks) ofC 8D E GF )*) )H <Isuchthatany?

6 -subsetofC JDis containedinat Oneusefulsituationforusis when@ , inwhichcasethepackingiscalleda ? @ meansnotwo membersofBhave? ,byRemark2 if ? a -disjunctmatrix canbeconstructedfroma ? -packingbysimplyindexing s columnsbytheblocksand s rowsbymembersofC JD. Moreover, byRemark1 weseethatif ?4 ( ) then is -errordetectingand $J% , thebasicproblemofpackingdesignis to findthepackingnumber ? ,thesizeofa maximum?- @ write ? insteadof " ? when@ . Maximumsized ? -packingsinduceverygoodpoolingdesigns[BB KT96].Unfortunately, verylittleis and?. MillsandMullin[MM92] gave a give thereadera senseofhow difficultthisproblemis,wequotea resulton ? ,it is conceivablethatfindingoptimalsetpackingi s justashardasthemaincodingtheoryproblem[R om92]. > denotethesizeofa maximumconstant>-weightbinary -code, then ? J ?# Let ? ) )*) ? ?# @ thenSch onheim[Sch66] observedthat ? ? . Equalityholdswhenthedesignis any?- @ , sincewewant@ , SteinerTripleSystems( - designs)andSteinerQuadrupleSystems( - designs)couldbeusedtoconstructdisjunctma triceswithsmall planesandaffineplanesarealso?

7 -designswith@ butthey don t give goodpoolingdesigns(toomany tests).Theonlyothernoticeableresultwhich concernsusis fromBrouwer[Bro79], whodeterminesallvaluesof . Fora comprehensive treatmentondesigntheory, thereaderisreferredtoa nicebookbyBeth,Jungnickel andLenz[BJL86]. calledthegriddesign. To facilitatetheuseofrobotsforpoolassemblin g,theclonescanbearrangedintorowsandcolum nsofa setof grids, , wecanassume . Clearly, ambiguitycanoccurif therearemorethanonepositive whentherearetwo positives,say and , lyingondifferentrowsandcolumnsofa grid . Inthiscase, Testing aloneis notenoughtoidentify and becausethetwo clones and collinearwithboth and resolve ambiguity, wewishtorearrange intoanothergrid(givingadditionalpools)so that and arenotcollinearwithboth and thereare ormorepositive ,if werequirea strongerconditionthatnotwo clonesarecollineartwice,calledtheuniquec ollinearitycondition, thenHwang[Hwa95] showedthattheexistenceofthegridsis [BLC91] generalizedthisideato -dimensionalgrids,whereeachin-tersection pointcouldbeviewedasa vertex ofthe new grid canbeobtainedfromtheoldgrid bya lineartransformationrepresentedbya matrix.

8 Thusa " )*)*) of is mappedto vertex of . A thirdgridcouldeitherbeobtainedbyusing twice(withtransformationmatrix $) orbyusinga differenttransformationmatrix (withtransformationmatrix ). They alsoextendedthe ,however [BBKT96].Basicallya poolingdesignis transversalif thepoolscanbepartitionedintoparts,eachof whichis a a alsospecifiedin[BBKT96]. [Mac96, Mac99] gave thefollowingconstruc-tionofa < bea -matrixwhoserowsareindexedbythe -subsetsofC Dandwhosecolumnsareindexedbythe -subsetsofC Dwhere $ & . < iff the -subsetis containedinthe is easytoseethat < is -disjunctwith rowsand < is , whichis , Maculashowedthatwithhighprobability couldsolve the H problem,effectivelyconvertinga deterministicconstructiontoa probabilistic(random) , and couldbechosencarefullyincertaincasestosu itone s , -disjunctMatricesInsetpackingdesigns,the matrix wasrowindexedbyallelementsofC 8D, andcolumnindexedbyselected -subsetsofC 8D. Lookingatthisfroma differentangle,therowswereindexedbyallpo intsat rank1 andcolumnsbysampledpointsat rank oftheBooleanAlgebralattice (seeFigure1).

9 Ontheotherhand,Macula s constructioninvolvestakingallpointsat rank asrowsandrank ascolumnsofour s designratewasn t sogoodbecausenumberofpointsat level is , if wepickpointsat lowerlevelsthan tobetherows,thenthematrixis not ,onemighthopetosomehow take sampledpointsatdifferentranksof , (1999,[ND99]) tookthisapproachandgave . A matchingofsize in is calledan bea -matrixwhoserowsareindexedbythesetofall -matchingson $ , andwhosecolumnsareindexedbythesetofall -matchingson $ . Allmatchingsareto beorderedlexicographically. hasa inrow andcolumn if andonlyif the -matchingis containedinthe is -disjunctis notdifficultto -matchingisa subsetof $ $ , becausethesetofedgesof $ isexactly $ $ . Fromtheabove observation,thisconstructioncouldbeseena stakingfrom C Dsampledpointsat rank asrowsandsampledpointsat rank is -error-detectingand $% ,theBooleanAlgebrais clearlynottheonlylatticewehave Figure1:TheBooleanAlgebraLattice(1)Besid estheBooleanAlgebra , whatareotherlatticeswecanuse?

10 Forexam-ple,onecandidateis , thelatticeofall tuplesof , whichis a generaliza-tionof , since $.(2)Whichconditionsmustholdtopicksometw o levelsofthelatticetoconstruct -disjunctmatrices?To avoidbeingtoovagueandfortheeaseofanalysi s, at rank andthenumberofpointscoveredby dependonlyon .(3)Intermsoferrortoleranceproperties,ca nwefromthelatticeinfersomeinfor-mationab outtheerrorcorrectinganddetectingcapabil ityofthematrixbeingconstructed?Withrespe cttoquestion1,NgoandDu[ND99] foundthatpickingpointsat levels and ofthelatticeofallsubspacesof ,infact,isthe -analogofMacula s thedesignswhosematricesarerandomlydeterm inedin somemanner. Thefactthata designis nondeterministicmeansthatit is bea random matrix,ouralgorithmofidentifyingthedefec tivesis thesameasbefore,namelypointingthoseitems containedinnegative Clearly, aniteminapositive itemsaresaidto beresolvedpositives. Let ( ) denotethenumberofunresolvednegatives(pos itives).Baldinget J , J , , and , where is theprobabilityof and is theexpectedvalueofa randomvariable.