Theconceptoflongestcommonprexescanbegeneralizedforsets:Denition1.7:F - PDF document

Theconceptoflongestcommonprexescanbegeneralizedforsets:Denition1.7:ForastringSandastringsetR,denelcp(S;R)=maxflcp(S;T)jT2Rglcp(R)=XT2Rlcp(T;RnfTg)InthealgorithmforinsertingSintoR(Algorithm1.3),therearetwowhileloops.Therstloopfollowsexistingedgesaslongaspossibleandthesecondloopthencreatesthenecessarynewnodesandedges.Thenumberofroundsintherstloopisexactlylcp(S;R).ThusthenumberofnewnodesandedgesaddedisjSj�lcp(S;R).Basedonthisobservation,wewillnextderiveanexpressionforthesizeoftrie(R).Thisisnotdirectlybasedthevaluelcp(R);weneedamorerenedmeasure.18 Theorem1.10:Thenumberofnodesintrie(R)isexactlyjjRjj�L(R)+1,wherejjRjjisthetotallengthofthestringsinR.Proof.Considertheconstructionoftrie(R)byinsertingthestringsonebyoneinthelexicographicalorder.Initially,thetriehasjustonenode,theroot.Asobservedearlier,thenumberofnodesaddedwheninsertingSiisjSij�LCPR[i].Summingup,wegettheresult.Thevaluelcp(R)isperhapsaconceptuallysimplermeasurethanL(R).Thefollowingresultshowsthatitisasymptoticallyequivalent.Lemma1.11:L(R)lcp(R)2L(R).Theproofisleftasanexercise.WewilllaterseethatthearrayLCPRisusefulasanactualdatastructure.20 Example1.12:PathcompactedtriesforR=fpot$;potato$;pottery$;tattoo$;tempo$g. tery$tempo$attoo$ato$$opt tery$tempo$attoo$ato$$potTheegdelabelsarefactorsoftheinputstrings.Iftheinputstringsarestoredseparately,theedgelabelscanberepresentedinconstantspaceusingpointerstothestrings.Thetimecomplexityofthebasicoperationsonthecompacttrieisthesameasforthetrie(anddependsontheimplementationofthechildoperationinthesameway),butprexandrangequeriesarefasteronthecompacttrie(exercise).22 TernaryTrieThebinarytreeimplementationofatriesupportsorderedalphabetsbutawkwardly.Ternarytrieisasimplerdatastructurebasedonsymbolcomparisons.Ternarytrieislikeabinarysearchtreeexcept:Eachinternalnodehasthreechildren:smaller,equalandlarger.Thebranchingisbasedonasinglesymbolatagivenposition.Thepositioniszero(rstsymbol)attherootandincreasesalongthemiddlebranches.Ternarytriehasvariantssimilarto-arytrie:Abasicternarytrieisafullrepresentationofthestrings.Compactternarytriesreducespacebycompactingbranchlesspathsegments.23 Aternarytrieisbalancedifeachleftandrightsubtreecontainsatmosthalfofthestringsinitsparenttree.Thebalancecanbemaintainedbyrotationssimilarlytobinarysearchtrees. dDEbABbABCdDECrotationWecanalsogetreasonablyclosetobalancebyinsertingthestringsinthetreeinarandomorder.25 Inabalancedternarytrieeachstepdowneithermovesthepositionforward(middlebranch),orhalvesthenumberofstringsremaininginthesubtree(sidebranch).Thus,inabalancedternarytriestoringnstrings,anydownwardtraversalfollowingastringSpassesatmostjSjmiddleedgesandatmostlognsideedges.Thusthetimecomplexityofinsertion,deletion,lookupandlcpqueryisO(jSj+logn).Incomparisonbasedtries,wherethechildfunctionisimplementedusingbinarysearchtrees,thetimecomplexitiescouldbeO(jSjlog),amultiplicativefactorO(log)insteadofanadditivefactorO(logn).Prexandrangequeriesbehavesimilarly(exercise).26 ThefollowingtheoremshowsthatwecannotachieveO(nlogn)symbolcomparisonsforanysetofstrings(when=no(1)).Theorem1.14:LetAbeanalgorithmthatsortsasetofobjectsusingonlycomparisonsbetweentheobjects.LetR=fS1;S2;:::;Sngbeasetofnstringsoveranorderedalphabetofsize.SortingRusingArequires (nlognlogn)symbolcomparisonsonaverage,wheretheaverageistakenovertheinitialordersofR.Ifisconsideredtobeaconstant,thelowerboundis (n(logn)2).NotethatthetheoremholdsforanycomparisonbasedsortingalgorithmAandanystringsetR.Onlytheinitialorderisrandomratherthan\any".Anyordercouldbethecorrectorder,inwhichcaseanalgorithmthatrstchecksiftheorderiscorrectwouldneedtodoonlyO(n+L(R))symbolcomparisons.Anintuitiveexplanationforthisresultisthatthecomparisonsmadebyasortingalgorithmarenotrandom.Inthelaterstages,thealgorithmtendstocomparestringsthatareclosetoeachotherinlexicographicalorderandthusarelikelytohavelongcommonprexes.28 Theprecedinglowerbounddoesnotholdforalgorithmsspecializedforsortingstrings.Theorem1.15:LetR=fS1;S2;:::;Sngbeasetofnstrings.SortingRintothelexicographicalorderbyanyalgorithmbasedonsymbolcomparisonsrequires (L(R)+nlogn)symbolcomparisons.Proof.Ifwearegiventhestringsinthecorrectorderandthejobistoverifythatthisisindeedso,weneedatleastL(R)symbolcomparisons.Nosortingalgorithmcouldpossiblydoitsjobwithlesssymbolcomparisons.Thisgivesalowerbound (L(R)).Ontheotherhand,thegeneralsortinglowerbound (nlogn)mustholdheretoo.Theresultfollowsfromcombiningthetwolowerbounds.NotethattheexpectedvalueofL(R)forarandomsetofnstringsisO(nlogn).Thelowerboundthenbecomes (nlogn).Wewillnextseethattherearealgorithmsthatmatchthislowerbound.SuchalgorithmscansortarandomsetofstringsinO(nlogn)time.30 Inthenormal,binaryquicksort,wewouldhavetwosubsetsRandR,bothofwhichmaycontainelementsthatareequaltothepivot.Binaryquicksortisslightlyfasterinpracticeforsortingsets.Ternaryquicksortcanbefasterforsortingmultisetswithmanyduplicatekeys(exercise).Thetimecomplexityofboththebinaryandtheternaryquicksortdependsontheselectionofthepivot(exercise).Inthefollowing,weassumeanoptimalpivotselectiongivingO(nlogn)worstcasetimecomplexity.32 Example1.18:Apossiblepartitioning,when`=2. al p habetal i gnmental l ocateal g orithmal t ernativeal i asal t ernateal l =) al i gnmental g orithmal i as al l ocateal l al p habetal t ernativeal t ernate Theorem1.19:StringquicksortsortsasetRofnstringsinO(L(R)+nlogn)time.Thusstringquicksortisanoptimalsymbolcomparisonbasedalgorithm.Stringquicksortisalsofastinpractice.34 RadixSortThe (nlogn)sortinglowerbounddoesnotapplytoalgorithmsthatusestrongeroperationsthancomparisons.Abasicexampleiscountingsortforsortingintegers.Algorithm1.20:CountingSort(R)Input:(Multi)setR=fk1;k2;:::kngofintegersfromtherange[0::).Output:RinnondecreasingorderinarrayJ[0::n).(1)fori 0to�1doC[i] 0(2)fori 1tondoC[ki] C[ki]+1(3)sum 0(4)fori 0to�1do//cumulativesums(5)tmp C[i];C[i] sum;sum sum+tmp(6)fori 1tondo//distribute(7)J[C[ki]] ki;C[ki] C[ki]+1(8)returnJThetimecomplexityisO(n+).Countingsortisastablesortingalgorithm,i.e.,therelativeorderofequalelementsstaysthesame.36 TheLSDradixsortalgorithmisverysimple.Algorithm1.21:LSDRadixSort(R)Input:(Multi)setR=fS1;S2;:::;Sngofstringsoflengthmoverthealphabet[0::).Output:Rinascendinglexicographicalorder.(1)for` m�1to0doCountingSort(R,`)(2)returnRCountingSort(R,`)sortsthestringsinRbythesymbolsatposition`usingcountingsort(withkireplacedbySi[`]).ThetimecomplexityisO(jRj+).Thestabilityofcountingsortisessential.Example1.22:R=fcat;him;ham;batg.cathimhambat=) hi m ha m ca t ba t =) h a mc a tb a th i m =) b at c at h am h im 38 MSDradixsortresemblesstringquicksortbutpartitionsthestringsintopartsinsteadofthreeparts.Example1.24:MSDradixsortpartitioning. al p habetal i gnmental l ocateal g orithmal t ernativeal i asal t ernateal l =) al g orithm al i gnmental i as al l ocateal l al p habet al t ernativeal t ernate 40 Theorem1.26:MSDradixsortsortsasetRofnstringsoverthealphabet[0::)inO(L(R)+nlog)time.Proof.Consideracallprocessingasubsetofsizek:ThetimeexcludingtherecursivecallbutincludingthecalltocountingsortisO(k+)=O(k).Theksymbolsaccessedherewillnotbeaccessedagain.Atmostlcp(S;RnfSg)+1symbolsinSwillbeaccessedbythealgorithm.ThusthetotaltimespentinthiskindofcallsisO(L(R)+n).Thisstillleavesthetimespentinthecallsforasubsetsofsizek,whicharehandledbystringquicksort.Nostringisincludedintwosuchcalls.Therefore,thetotaltimeoverallcallsisO(L(R)+nlog).ThereexistsamorecomplicatedvariantofMSDradixsortwithtimecomplexityO(L(R)+n+). (L(R)+n)isalowerboundforanyalgorithmthatmustaccesssymbolsoneatatime.Inpractice,MSDradixsortisveryfast,butitissensitivetoimplementationdetails.42