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Algorithms for computing approximate repetitions in musical sequences
Emilios Cambouropoulos, Maxime Crochemore, Costas S. Iliopoulos, Laurent Mouchard, Yoan J. Pinzón Ardila
To cite this version:
Emilios Cambouropoulos, Maxime Crochemore, Costas S. Iliopoulos, Laurent Mouchard, Yoan J.
Pinzón Ardila. Algorithms for computing approximate repetitions in musical sequences. Australasian Workshop On Combinatorial Algorithms, 1999, Australia. pp.129-144. �hal-00452240�
Repetitions in Musial Sequenes
EmiliosCambouropoulos 1?
,Maxime Crohemore 2??
, CostasS.Iliopoulos 3???
,
LaurentMouhard 4 y
,andYoanJ.Pinzon 3
1
AustrianResearhInstitutefor ArtiialIntelligene,Shottengasse3,1010Wien,
Austria
emiliosai.univie.a.at
www.ai.univie.a.at/emilios
2
InstitutGaspardMonge,UniversitedeMarne-la-Vallee,77454Marne-la-Vallee
CEDEX2,Frane.
mauniv-mlv.fr
www-igm.univ-mlv.fr/ma
3
Dept.ComputerSiene,King'sCollegeLondon,LondonWC2R2LS,England,
andShoolofComputing,CurtinUniversityofTehnology,GPOBox1987U,WA.
fsi,pinzongds.kl.a.uk,
www.ds.kl.a.uk/staff/si, www.ds.kl.a.uk/pg/pinzon
4
LIFAR-ABISS,UniversitedeRouen,76821MontSaintAignan,Frane.and
ShoolofComputing,CurtinUniversityofTehnology,GPOBox1987 U,WA.
lmdir.univ-rouen.fr
www.dir.univ-rouen.fr/lm
Abstrat. Hereweintroduetwonewnotionsofapproximatemathing
withappliationinomputerassisted musianalysis. Wepresent algo-
rithmsforeahnotionofapproximation:forapproximatestringmathing
andforomputingapproximatesquares.
Keywords: String algorithms,approximate stringmathing, dynamiprogram-
ming,omputer-assistedmusianalysis.
1 Introdution
This paperfouses on a set of stringpattern-mathing problems that arise in
musialanalysis,andespeiallyinmusialinformationretrieval.Amusialsore
anbeviewedasastring:ataveryrudimentarylevel,thealphabetouldsimply
betheset ofnotesin thehromatiordiatoninotation,ortheset ofintervals
thatappearbetweennotes(e.g.pithmayberepresentedasMIDInumbersand
?
SupportedbytheSTARTprogrammeY99-INF,AustrianFederalMinistryofSiene
andTransport
??
PartiallysupportedbytheC.N.R.S.Program\Genomes"
???
PartiallysupportedbytheEPSRCgrantGR/J17844.
y
PartiallysupportedbytheC.N.R.S.Program\Genomes"
musial works play a ruial role in disovering similarities between dierent
musial entities and may be used for establishing \harateristi signatures"
(see [6℄). Suh algorithms an be partiularly useful for melody identiation
andmusialretrieval.
Bothexatandapproximatemathingtehniqueshavebeenusedforavariety
of musial appliations (see overviews in MGettrik [23℄ ; Crawford et al [6℄;
Rolland et al [28℄;Cambouropoulos et al [4℄). Thespei problem studied in
this paperis pattern-mathing for numeri stringswhere aertaintolerane is
allowedduringthemathingproedure.Thistypeofpattern-mathinghasbeen
onsideredneessaryforvariousmusialappliationsandhasbeenusedbysome
researhers(see,forinstane,Cope[5℄).AnumberofeÆientalgorithmswillbe
presentedin thispaperthattaklevariousaspetsofthis problem.
Mostomputer-aidedmusialappliationsadopt anabsolute numeripith
representation(mostommonlyMIDIpithandpithintervalsinsemitones;du-
rationisalsoenodedinanumeriform).Theabsolutepithenoding,however,
maybeinsuÆientfor appliationsin tonal musi asit disregardstonal quali-
tiesofpithesand pith-intervals(e.g. atonal transposition from amajortoa
minor keyresultsin adierentenoding ofthemusialpassageandthus exat
mathing annot detet the similaritybetween the two passages). Oneway to
aountforsimilaritybetweenloselyrelatedbutnon-identialmusialstringsis
to usewhatwill bereferredto asÆ-approximatemathing (and-approximate
mathing). In Æ-approximate mathing, equal-length patterns onsisting of in-
tegers math if eah orresponding integer diers by not more than Æ- e.g. a
C-majorf60;64;65;67gandaC-minorf60;63;65;67gsequeneanbemathed
ifatoleraneÆ=1isallowedinthemathingproess(-approximatemathing
is desribedin the next setion). Twosimple musial examplesthat illustrate
theusefulnessoftheproposedpattern-mathingtehniquesarepresentedinAp-
pendiesI andII.
Exatrepetitionshavebeenstudiedextensively.Therepetitionsanbeeither
onatenatedwiththeoriginal substringortheymayoverlaportheymaynot.
Algorithmsforndingnon-overlappingrepetitionsinagivenstringanbefound
in [1,8,15,21,18,26℄ andalgorithms foromputingoverlappingrepetitionsan
befound in [3,13,14,25℄. A natural extension of therepetitions problem is to
allow the presene of errors; that is, the identiation of substrings that are
dupliated to within a ertain tolerane k (usually edit distane orHamming
distane).Moreover,therepeatedsubstringmaybesubjettootheronstraints:
itmayberequiredtobeofat leastaertainlength,andertainpositionsin it
mayberequiredtobeinvariant.
Furthermore,eÆientalgorithmsforomputingtheapproximaterepetitions
arealsodiretlyappliabletomoleularbiology(see[11,17,24℄)andinpartiular
in DNA sequening by hybridization ([27℄), reonstrution of DNA sequenes
from known DNA fragments (see [29,30℄), in human organand bonemarrow
transplantationaswellasthedeterminationofevolutionarytreesamongdistint
speies([29℄).
thatofndingevolutionaryhains:givenastringt(the\text")andapatternp
(the\motif"),ndwhetherthereexistsasequeneu
1
=p;u
2
;:::;u
`
ourring
in the text t suh that u
i+1
ours to therightof u
i
in t andu
i and u
i+1 are
\similar" for 1 i < ` (i.e. they dier by a ertain number of symbols). In
[9℄ and[7℄ algorithms foroverlappingand non-overlappingevolutionary hains
werepresentedandseveralvariantsoftheproblemwerestudied: omputingthe
longesthain,omputingthehain withtheleastnumberoferrors.
Thepaperisorganisedasfollows.Inthenextsetionwepresentsomebasi
denitionsforstringsandbakgroundnotionsforapproximatemathing.InSe-
tion3wepresentanalgorithmforÆ-approximate(therstnotionofapproxima-
tion)patternmathing.Insetion4wepresentanalgorithmforÆ;-approximate
(theseond notionofapproximation)patternmathing.Insetion5wepresent
algorithmsforomputingallÆandfÆ;g-approximatesquaresin agiventext.
FinallyinSetion 6wepresentouronlusionsandopenproblems.
2 Bakground and basi string denitions
A string isasequeneof zeroormoresymbolsfrom analphabet;thestring
withzerosymbolsisdenoted by.Thesetofallstringsoverthealphabet is
denotedby
.Astringxoflengthnisrepresentedbyx
1 :::x
n
,wherex
i 2
for 1 i n. A string w is a substring of x if x = uwv for u;v 2
; we
equivalentlysaythatthestringwoursatpositionjuj+1ofthestringx.The
position juj+1 is said to be the starting position of w in x and the position
jwj+juj the end position of uin x.A stringw is a prex of x if x =wu for
u2
.Similarly,wisasuÆxofx ifx=uwforu2
.
Thestringxyisaonatenationoftwostringsxandy.Theonatenations
ofkopiesofxisdenotedbyx k
.Fortwostringsx=x
1 :::x
n
andy=y
1 :::y
m
suhthatx
n i+1 :::x
n
=y
1 :::y
i
forsomei1,thestringx
1 :::x
n y
i+1 :::y
m
isasuperpositionofx andy.Wesaythat xandy overlap.
Letxbeastringoflengthn.A prexx
1 :::x
p
, 1p<n,ofx is aperiod
of x ifx
i
=x
i+p
forall1in p.Theperiod of astringxis theshortest
period ofx.A stringy isaborder ofx ifyisaprexandasuÆxofx.
LetbeanalphabetofintegersandÆaninteger.Twosymbolsa;bofare
saidtobeÆ-approximate,denoted a=
Æ
bifandonlyif
ja bjÆ
Wesaythattwostringsx;y areÆ-approximate,denoted x Æ
=y ifandonlyif
jxj=jyj; andx
i
=
Æ y
i
; 8i2f1::jxjg (2:1)
Let beaninteger.Twostringsx;y aresaidto be-approximate,denoted
x
=y ifandonlyif
jxj=jyj; and jxj
X
jx
i y
i
j< (2:2)
Furthermore,wesaythattwostringsx;yaref;Æg-approximate,denotedx = y,
ifandonlyifx andy satisfyonditions(2.1)and(2.2).
3 Æ-Approximate Pattern Mathing
The problem of Æ-approximate pattern mathing isformally dened as follows:
givenastringt=t
1 :::t
n
andapatternp=p
1 :::p
m
omputeallpositionsj of
t suhthat
p Æ
=t[j::j+m 1℄
Thealgorithm isbasedontheO(1)-time omputationofthe\Deltastates"
DState
j
;j2f1::ngby usingbitoperationsunder theassumptionthat mw,
wherewisthenumberofbitsinamahineword.Thebasistepsofthealgorithm
areas follows:
1. Firstweomputethe\Deltatable"DT:wesetDT()=r,wheredenotes
asymbolourringin t andr=r
1 :::r
m
isabinarywordwithr
i
equalto
1ifj p
i
jÆ,otherwiser
i
isequalto 0fori2f1::mg.
2. LetLeftShiftbeabit-wiseoperationthatshiftsthebitsofabinaryword
byoneposition totheleft.Wedene
DState
j
=(LeftShift(DState
j 1
) OR 1) AND DT[t
j
℄ (3:1)
forj=1:::nandDState
0
=0;henethisproedureisalled\Shift-And".
OnewehaveomputedtheDT table,weanuseittoomputetheDState
j
forj=1 :::n,usingthereursiveformula(3.1).
3. WesaythatthereisaÆ-approximatemath(orsimplyÆ-math)atposition
j m+1ifandonlyifthem-th bitof DState
j
is1orequivalentlyifand
onlyifDState
j
, isgreaterorequalto 2 m 1
whenitis viewedasadeimal
integer.
Example. For=f1, :::, 9g let us onsider p=3,4,6,2, t=3,4,6,2,8,2,4,5,7,1
andÆ=1.Inthepreproessingtable,DT()denotesthepositionswherej p
i j
Æ. Forexample,DT[3℄=1011beausej3 p
i
j1fori=1;2;4.
i
pi DT[1℄ DT[2℄ DT[3℄ DT[4℄ DT[5℄ DT[6℄ DT[7℄ DT[8℄ DT[9℄
4 2 1 1 1 0 0 0 0 0 0
3 6 0 0 0 0 1 1 1 0 0
2 4 0 0 1 1 1 0 0 0 0
1 3 0 1 1 1 0 0 0 0 0
Table 1.ThetableDT forpatternp=2;6;4;3andalphabet=f1;:::;9g.
Thetable belowevaluatesDState usingtherelation(3.1).Forexample,
4 3 4
=(LeftShift(0100)OR1)ANDDT[2℄
=(1000OR1)AND1001
=1001AND1001
=1001
whihimpliesthatthereisamathstartingatposition1oft,sinethe4-th
bitofDState
4 is1.
j
1 2 3 4 5 6 7 8 9 10
tj 3 4 6 2 8 2 4 5 7 1
LeftShift(DState
j 1
)OR1 0001 0011 0111 1001 0011 0001 0011 0111 1101 1001
DT[t
j
℄ 1011 0011 0100 1001 0000 1001 0011 0110 0100 1000
DStatej 0001 0011 0100 1001 0000 0001 0011 0110 0100 1000
[DState
j
℄
10
1 3 4 9 0 1 3 6 4 8
Table 2.ComputingtheDstatesandndingtheÆ-approximatemathes.
AÆ-approximatemathoursatpositionj m+1oftif[DState
j
℄
10 2
m 1
,
where[DState
j
℄
10
denotestheDState
j
asadeimalinteger.Therefore,thereis
onemath ending atposition 4oft (f3,4,6,2g) andanother oneat position 10
oft(f4,5,7,1g)sinefDState
4
;DState
10 g2
3
.
3.1 Pseudo-ode
Fig.1givesaomplete speiationofthealgorithm.Intheline3wehavethe
preproessingphasewhihomputetheDT table.Inline6weusethereursive
formulatoomputetheDStates.Finally,inline7weapplythemathingriteria
tosee whetherthereisaÆ-approximatemathornot.
1. proedure Shift-And(p,t,Æ) fn=jtj; m=jpjg
2. begin
3. DT
i [℄
1 if j pijÆ
0 otherwise
8i2f1::mg; 82
4. DState
0 0
5. forj 1to n do
6. DStatej (LeftShift(DStatej
1
)OR 1 ) AND DT[tj℄
7. if DStatej2 m 1
then write j-m+1
8. od
9. end
Fig.1.TheShift-AndProedure.
Assuming that the pattern length is nolonger than the memory word size of
themahine(thusO(1)size), thetimeomplexityofthepreproessingphaseis
O(n)(sine weneedto evaluateDT only forthesymbolsthat ourin t) and
thetimeomplexityofthesearhingphaseinO(n).Figure2showsthetiming 1
fordierenttextsizes.
0 0.2 0.4 0.6 0.8 1
0 200 400 600 800 1000
Text Size (k)
Time (in secs.)
Pattern Size = 15 δ = 2
0 0.2 0.4 0.6 0.8 1
0 200 400 600 800 1000
Text Size (k)
Time (in secs.)
Pattern Size = 20 δ = 2
Fig.2.Timingurvesfor theShift-AndProedure.
4 fÆ;g-Approximate Pattern Mathing
Theproblem of fÆ;g-approximate pattern mathingis formally dened asfol-
lows:givenastringt=t
1 :::t
n
andapatternp=p
1 :::p
m
omputeallpositions
j oft suhthat
p Æ;
= t[j::j+m 1℄
InordertosolvethisproblemwerstmakeuseoftheShift-Andalgorithm
to nd the Æ-approximate mathes of the pattern p in t. One we nd a Æ-
approximatemathwewantto knowwhetheritisalsoa-approximatemath.
Todo so,weseektoomputesuessive\DeltaStates"DState
j
and\Gamma
States"GStates
j
in O(1) timeusing bitoperationsunder theassumption that
mwwherewisthenumberofbitsinamahineword.Themainstepsofthe
algorithmareas follows:
1. Weneedtoomputethe\DeltaTable"DTaswedidbeforeandthe\Gamma
Table" GT table; we set GT() = r, where denotes a symbol in the
alphabetandr=r
1 :::r
m
isawordwithr
i
equaltoj p
i
jifj p
i jÆ,
otherwiser
i
isequalto 0fori2f1::mg. Eahr
i
,i 2f1::mg isstored asa
binarynumberofdbitswhered=dlog(Æm)e.
1
UsingaSUNUltraEnterprise300MHzrunningSolarisUnix.
oneposition to theleft and R ightShift shiftsthe bits of abinary wordd
positions to theright.One wehaveomputed theDT and GT tables, we
anusethemto omputetheDState
j
andGState
j
forj=1:::n,usingthe
reursiveformulas
DState
j
=(LeftShift(DState
j 1
) OR 1) AND DT[t
j
℄ (4:1)
GState
j
=R ightShift(GState
j 1
;d)+GT[t
j
℄ (4:2)
Wealso needto dene the seeds DState
0
=0 and GState
0
=0. We allthis
proedure \Shift-Plus" beauseweuse the \shift" and \plus" operators
toomputeeahnewstate.
3. Wesaythatthereisamath(fÆ;g-approximatemath)atpositionj m+1
ifandonlyifthem-thbitofDState
j
is1andthem-thblokofdbitstaken
asanintegeris.
Example.Forourexamplelet=f1;:::;9g, thepattern p=3;4;6;2;the
text t =3;4;6;2;8;2;4;5;7;1, Æ =1 and =3. We will use bloks of size 3
(d =3) to store thej p
i
j valueswhere j p
i
j Æ. Forexample,GT[3℄=
000 100000100 beausej3 p
i
j 1for i=1,2,4 and thedierenes are 0,1,1
respetively.(seeleft handtableoftable 3).
i p
i
1 2 3 4 5 6 7 8 9
0 1 0 1 0 0 0 0 0
1 3 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0
0 0 1 0 1 0 0 0 0
2 4 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0
0 0 0 0 1 0 1 0 0
3 6 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0
1 0 1 0 0 0 0 0 0
4 2 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0
j 1 2 3 4 5 6 7 8 9 10
tj 3 4 6 2 8 2 4 5 7 1
0 1 0 1 0 1 1 0 0 0
3 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0
1 0 1 0 1 0 1 0 0 0
4 0 0 0 0 0 0 0 1 0 0
0 0 0 0 0 0 0 0 0 0
0 1 0 1 0 1 0 0 1 0
6 0 0 0 0 0 0 0 1 1 0
0 0 0 0 0 0 0 0 0 0
1 0 1 0 1 0 1 0 0 0
2 0 0 0 0 0 0 0 0 1 0
0 0 0 0 0 0 0 0 0 1
Table 3.Thelefthandsidetableisthe\GammaTable"GT andtherighthandside
tableisthetableforndingf;Æg-approximatemathes.
Therighthand tableaboveshowstheomputation oftheDStatesandthe
GStatesusing(4.2).Forexample,
GState
9
=R ightShift(000010010000,3)+000000100000
=000000010010+000000100000=000000110010
Wealreadyknowthat therearetwoÆ-approximatemathesendingat posi-
tions4and10oft.NowweanusethelastthreebitsofGState
4
andGState
10
tondoutthevaluesof,whih are0and4respetively(seerighthandtable
ofFig.3).
Fig.3belowgivesaompletedesriptionofthealgorithm.Inthelines3and4are
thepreproessingphasewhihomputetheDT tableandGT tablerespetively.
Inlines8and9weomputethenextDStateandGStaterespetively.Finally,
inline10weapplythemathingriteriatoseewhetherthereisamathornot.
1. proedure Shift-Plus(p,t,Æ,) fn=jtj; m=jpjg
2. begin
3. DT
i [℄
1 if j pijÆ
0 otherwise
8i2f1::mg;82
4. GT
di d:::di 1 [℄
j pij if DTi[℄=1
0 otherwise
8i2f1::mg;82
5. DState
0 0
6. GState0 0
7. for j 1to n do
8. DState
j
(LeftShift(DState
j 1
)OR 1) AND DT[t
j
℄
9. GStatej R ightShift(GStatej 1,d)+ GT[tj℄
10. if DState
j 2
m 1
AND GState
dm d:::dm 1
then write j-m+1
11. od
12. end
Fig.3.TheShift-PlusAlgorithm.
4.2 Runningtime
Assumingthat Æm2 d
1thetimeomplexity ofthepreproessingphase
isO(Æm+jj)andthetimeomplexityofthesearhingphasein O(n),thus
independentfromthealphabetsize andthepatternlength.Figure4showsthe
timingfordierenttext sizes.
!htb
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
0 20 40 60 80 100
Text Size (k)
Time (in secs.)
Pattern Size = 20 δ = 2
γ = 20
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
0 20 40 60 80 100
Text Size (k)
Time (in secs.)
Pattern Size = 15 δ = 2
γ = 20
Fig.4.TimingurvesfortheShift-PlusAlgorithm.
