A linear time algorithm for Shortest Cyclic Cover of Strings

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A linear time algorithm for Shortest Cyclic Cover of

Strings

Bastien Cazaux, Eric Rivals

To cite this version:

Bastien Cazaux, Eric Rivals. A linear time algorithm for Shortest Cyclic Cover of Strings. Journal of

Discrete Algorithms, Elsevier, 2016, 37, pp.56-67. �10.1016/j.jda.2016.05.001�. �lirmm-01525995�

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Contents lists available atScienceDirect

Journal

of

Discrete

Algorithms

www.elsevier.com/locate/jda

A

linear

time

algorithm

for

Shortest

Cyclic

Cover

of

Strings

Bastien Cazaux

a

,

b

,

Eric Rivals

a

,

b

,

∗

a_LIRMM,_CNRS_and_Université_de_Montpellier,₁₆₁_rue_Ada,₃₄₀₉₅_Montpellier_Cedex_5,_France

b_Institut_de_Biologie_{Computationnelle,}_CNRS_and_Université_de_Montpellier,₈₆₀_rue_Saint_Priest,₃₄₀₉₅_Montpellier_Cedex_5,_France

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory:

Availableonline19May2016

Keywords:

Greedy

Shortestcycliccoverofstrings Superstring

Overlap

Minimumassignment Graph

Mergingwordsaccordingtotheiroverlapyieldsasuperstring.Thisbasicoperationallows toinferlongstringsfromacollectionofshortpieces,asingenomeassembly.Tocapture amaximum of overlaps, the goalis to infer the shortest superstring ofa set ofinput words.TheShortestCyclicCoverofStrings(SCCS)problemasks,insteadofasinglelinear superstring,forasetofcyclicstringsthatcontainthewordsassubstringsandwhosesum oflengthsisminimal. SCCSis used asacrucial stepinpolynomial timeapproximation algorithms for the notably hardShortest Superstring problem, but it is solvedin cubic time.Thecyclicstringsarethencutandmergedtobuildalinearsuperstring.SCCS can alsobesolvedbyagreedyalgorithm.Here,weproposealineartimealgorithmforsolving SCCSbasedonaEuleriangraphthatcapturesallgreedysolutionsinlinearspace.Because thegraphisEulerian,thisalgorithmcanalsoﬁndagreedysolutionofSCCSwiththeleast numberofcyclicstrings.ThishasimplicationsforsolvingcertaininstancesoftheShortest linearorcyclicSuperstringproblems.

1. Introduction

The possibility ofmergingtwo words intoa longerone accordingto their overlap– forinstancemerging abcde with cdeba into abcdeba – allows to infer the sequence of a target molecule from the short reads produced by sequencing machines. Incomputerscience, mergingresultsina superstring oftheinput words.Theactionofmergingisheavily used in any genome assembler (although not necessarily on the complete reads, sometimes only on substrings ofthe reads) andgiventheredundancy ofthesequencing,whichyields ahighdensityofreads, theobjectiveistocompute ashortest superstringofasetofinputwords[10].Inbioinformatics,thetaskofgenomeassembly isworsenedby theinaccuracyof the sequencingandby thestructure ofthe genomewhichcontains repeatedsequences [10].However, progresseson the modelquestion, calledShortest(linear)SuperstringProblem (SLiS)–oroftenShortestCommon Superstring– stillshedlight on thediﬃcultyofassembly.Moreover,thetarget moleculecanbe alinearora circularstructure.Itisalsointeresting to addressthequestionoftheShortestCyclicSuperstring (SCyS).Findingsuperstringsalsohasapplicationindatacompression. Indeed,asasuperstringrepresentsacondensedrepresentationoftheinitiallanguage,itcanserveasacompressedversion oftheinputset.

Let P

:= {

s1

,

. . . ,

s|P|

}

beasetofinputwordsanddenotethesumoftheirlengthsby

P

,i.e. thenormof P .Without loss ofgenerality, we always assume that P isfactor free,i.e. for anytwostrings of P ,noneis asubstring of theother. From thecomputationalviewpoint,ﬁnding ashortestsuperstringisknowntobehardtosolve(NP-hardevenonabinary

*

Correspondingauthorat:LIRMM,CNRSandUniversitédeMontpellier,CC477,161rueAda,34095MontpellierCedex5,France.Tel.:+33467418664.

E-mailaddresses:[email protected](B. Cazaux),[email protected](E. Rivals).

http://dx.doi.org/10.1016/j.jda.2016.05.001

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alphabet) andto approximate(Max-SNP hard) [7,1].The research inthe last 25years hasmainlyfocused on polynomial timeapproximationalgorithmsforSLiS(see[8]forarecentlist).Despitetheseefforts,thealgorithm

greedy

(Algorithm 1), an algorithmthat iteratively updatestheinput setby mergingtwo maximally overlappingstrings untilone stringis left, wasconjectured toapproximatethelengthoftheshortestsuperstringwithratio2[17,1].Thisisbetterthanmostknown algorithms,andtheconjectureisstillvalid.Forlinearsuperstringsontheconstantsizealphabets,Ukkonenproposedalinear timeimplementationofthealgorithm

greedy

basedontheAho–Corasickautomaton[19];later, anotherimplementation usingthesuﬃxtreewasgivenin[12,13].

Thequestion ofapproximatingthe superstringcanbe reformulatedona completeweighteddigraph, theso-called dis-tancegraph:each input wordisavertexandan arcencodesthedirectional mergingofthetwo words.The weightofan arcfromu to v equalsthelengthofthepreﬁxofu thatisnot coveredbytheoverlapofv (e.g.,thecorrespondingpreﬁx ofabcde to cdeba isab, andtheweightwouldbe 2). Thena shortestlinearsuperstringisassociatedwithaHamiltonian path on this graph,and a shortestcyclic superstring witha Hamiltonian cycle[1]. Yet those problems are also hard to approximateandmostapproximationalgorithmsforSLiSresorttoarelaxedquestion,namelyShortestCyclicCoverofStrings (SCCS).ThesealgorithmssolveSCCSbycomputingaminimumassignmentonthedistancegraphinO

(

P

+ |

P

|

3

₎

_[16]_,_and

thentransformthecyclesintocyclicstrings.OurworkfocusesonSCCS.Weproposeanalgorithmthatusesonlylineartime in

P

,andforthissakeintroducethesuperstringgraph,whichcapturesallsolutionsof

greedy

forSCCS(seeSections4

and3). Thesegreedysolutions arecalledgreedysuperstrings. Exploringthe graphproperties,we show that itgives,fora subsetofinstancesoftheshortestlinearandcyclicsuperstringproblems,interestingapproximationsorevenexactsolutions (Section4).

1.1. Notationandproblemdeﬁnition

ForanyrootedtreeT ,VT isthesetofnodesofT .Foranode v ofVT,ChildrenT

(

v

)

isthesetofchildrenofv inT .We denoteby

[

i

,

j

]

thesetofintegersbetweeni and j.

Weconsidertwokindsofstrings:linear andcyclicstrings (seeFigs. 1a and 1b).Forastrings,thelengthofs is

|

s

|

.Fora linearstrings andi

≤

j in

[

1

,

|

s

|]

,s

[

i

,

j

]

isthelinearsubstringofs beginningatthepositioni andendingattheposition j, s

[

i

]

is thesubstring s

[

i

,

i

]

, s

[

1

,

j

]

isa preﬁx of s and s

[

i

,

|

s

|]

isa suffix of s. Aprefix (or suffix)s of s isproper ifs is differentofs. Foranotherlinearstringt,theoverlap froms tot,denotedby ov

(

s

,

t

)

, isthelongestsubstring whichisa propersuffixof s andaproper prefixoft.The prefixfroms tot,denotedby pr

(

s

,

t

)

,issuchthat s

=

pr

(

s

,

t

)

ov

(

s

,

t

)

.The merge ofs witht isthelinearwordpr

(

s

,

t

)

t ifs

=

t,andthecyclicwordpr

(

s

,

t

)

otherwise.Alinearsuperstring ofasetP of linearstringsisalinearstring w suchthateachstringofP isasubstringofw.Wesaythatalinearstringw isasubstring ofacyclicstringc ifthereexists wc alinearpermutation ofc such that w isa substringofw∞c (where w∞c

=

wcwc

. . .

) –seeFig. 1c.Acycliccover of P isasetC ofcyclicstringssuchthateachlinearstringofP isasubstringofacyclicstring of C .

Fig. 1. (c) Shows how the linear word (a) is included in the cyclic word (b). Wedeﬁnethefollowingproblem:

Problem1(Shortestcycliccoverofstrings). Input: Asetoflinearstrings P .

Output: AcycliccoverC ofP thatminimises

C

.

Let

σ

be a permutation of

{

1

,

. . . ,

|

P

|}

, we can split up

σ

in cyclicpermutations:

σ

=

σ

1

◦ . . . ◦

σ

m. Foreach cyclic permutation

σ

ioflengthni,wedenotethecyclicstringby:

P

(

σ

i

)

:=

pr

(

sσi(1),sσi(2)) . . .pr

(

sσi(ni−1),sσi(ni))pr

(

sσi(ni),sσi(1)),

and

P

(

σ

)

:= {

P

(

σ

1

), . . . ,

P

(

σ

m

)}

(SeeFig. 2).

Itiseasytosee,thatforeachpermutation

σ

of

{

1

,

. . . ,

|

P

|}

,P

(

σ

)

isacycliccoverof P andforeachoptimalsolutionw ofProblem 1(SCCS),thereexists

σ

apermutationof

{

1

,

. . . ,

|

P

|}

suchthatw

=

P

(

σ

)

.

1.2. Relatedworks

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Fig. 2. Running examplewiththeinputsetP:= {ababb, aab, abba, abaa}.InstanceofacycliccoverofP obtainedwithapermutationσ.Weobtainthe cycliccoverP(σ) = {ababba, aba}(ababba andaba arecyclicwords).

SinceSLiSdoesnotadmitaPTAS,numerousworksonSLiShavereportedpolynomialtimeapproximationalgorithmswith aconstantratiorangingfrom4 to

>

2 (see[8, Table 1]).AmajorityofthesemodelSLiSasaminimumweightHamiltonian path problemonthedistancegraph G of P (whereanarc

(

si

,

sj

)

isweighted by

pr

(

si

,

sj

)

foranyvalidi

,

j).Notethat ifinstead itis weightedby

ov

(

si

,

sj

)

, weobtain theOverlapGraph,andSLiS becomesequivalent to findinga maximum weight Hamiltonianpath[1].Clearly, theweight ofacycliccoverof G (i.e.,the sumoftheweight ofall cycles)islower thanthat ofaHamiltoniancycle,whichitselfgivesalowerboundoftheweightofaHamiltonianpathonG.Hence,most approximationalgorithmsforSLiSrelyonsolvingSCCS,whichispolynomial.Blumetal.[1]werethefirsttoexhibitsuchan algorithmandtoprovethatitachievesaconstantratioof3 forSLiS.ThisalgorithmfirstsolvesSCCSandinasecondstep combinesthecyclicstringsbyrunningagreedyalgorithmonthem.TengandYaoimprovedthisratiobyanalgorithmthat recursivelysolvesSCCS[18].Afterthat,manyauthorshaveinvestigatedthecombinatorialpropertiesofwordsandofsome specificcycles,whichenabledthemtofurtherimprovetheapproximation(e.g.[3,14,15]),butallneedtosolveSCCS.Todo so,they computea MinimumAssignment onthedistancegraph.Anassignment(or cycliccover)ofa digraph G isasetof cyclessuchthatevery vertexofG belongstoexactlyonecycle.TosolveSCCSonaninstance P ,approximationalgorithms for SLiS first1/ build the distancegraphof P , with

|

P

|

nodes and

|

P

|(|

P

| −

1

)

arcs,which requires O

(

P

+ |

P

|

2

)

time andspaceusingtheGeneralisedSuﬃxTree(GST)of P [11],then2/computeaminimumcycliccoverusingtheHungarian algorithmin O

(

|

P

|

3

)

time[16].Oncethecycliccoverisknown,thealgorithmin[18]alsorequirestocomputeaparticular assignment calledacanonicalassignment, whichtakesanadditional O

(

P

)

usingthesuﬃx treebasedalgorithmof[9]. However, thecomplexitybottleneckofalgorithmsusingacycliccoverremainsthe O

(

P

+ |

P

|

3

)

time tobuildthegraph andcomputethecover.Itwasalsoshownthat

greedy

(Algorithm 1)solvesSCCSexactly[4].Tothebestofourknowledge, thereiscurrentlynolineartimealgorithmforsolvingSCCS.

Algorithm 1: Thealgorithm

greedy

forShortestCyclicCoverofStrings.

1 Input: P asetoflinearwords;

2 Output: S asetofcyclicstringsthatcoverP ;

3 S:= ∅

4 while P is notempty do

5 u andv twoelementsofP havingthelongestoverlap(u canbeequaltov)

6 w isthemergeofu andv

7 P:=P\ {u, v}

8 if u =v (i.e.w isacyclicstring) then S:=S∪ {w}else P:=P∪ {w}

9 return S

Theorem1([6]).ForShortestCyclicCoverofStrings,Algorithm

greedy

yieldsanoptimalsolution. 1.3. Explanationsaboutalgorithm

greedy

An overlapissaidtobegreedy-feasible ifagreedysolutiontakesit.Letu1, u2, v1 and v2 befourstrings.Wesaythat

the overlapsov

(

u1

,

u2

)

andov

(

v1

,

v2

)

aredependent ifandonlyifu1

=

v1 oru2

=

v2.WhenAlgorithm

greedy

merges

u withv,itchoosestheoverlapbetweenu andv andforbidsanyoverlapbeginningbyu orendingbyv,i.e. itforbidsall overlaps that aredependent ofov

(

u

,

v

)

.Wesaythat a setofoverlapsis independent ifeach overlapisnot dependentof another.

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Thus,Algorithm

greedy

makesaseriesofchoicestoselectasequenceofgreedy-feasibleandindependentoverlaps:

•

IntheﬁrststepoftheAlgorithm

greedy

choosesanoverlapamongthelongestoverlaps.

•

Inthesecondstep,itchoosesanoverlapamongthelongestoverlapswhichareindependentoftheﬁrstoverlap.

•

Ateachfurtherstep,ittakesamaximumoverlapthatisindependentofthepreviouslyselectedoverlaps.

•

Intheend,itoutputsasetofoverlapsthataremaximal(fortheinclusionasabaseinamatroid).Thissetofoverlaps isindependent.

Asthesetofoverlapsattheendof

greedy

ismaximal,thereexistsapermutation

σ

of

{

1

,

. . . ,

|

P

|}

,suchthatforalli in

{

1

,

. . . ,

|

P

|}

,ov

(

si

,

sσ(i)

)

isanoverlapofoursetofoverlaps.Thus,weobtainthefollowingpropertywhichissimilartoa well-knownpropertyonthesolutionofthegreedyalgorithmforSLiS[20, Def. of Induced

(

π

,

S

)

and Def. 2]:

Proposition2.Letw agreedysolutionofSCCS.Thereexistsapermutation

σ

of

{

1

,

. . . ,

|

P

|}

suchthatw

=

P

(

σ

)

. 2. GeneralisedSuﬃxTree,overlap,merge,andRed–Bluepaths

Thesuffixtreeofawordw isadatastructurethatindexesallsubstringsofw.Eacharcislabelledwithasubstringofw suchthattheconcatenationofthelabelsalongabranchfromtheroottoaleafspellsoutasuffixofw.Aconstraintrequires that thelabelsof arcsfromanode to itsoffspring start withdifferentsymbols.Thus, each noderepresents aprefix ofa suffix,i.e. asubstringof w.Infact,eachinternalnode isthelongestcommonprefixofallleavesinitssubtree.Traversing an arcextends therepresented stringon itsright. The worddepth of a node isthelength of thestringit represents.An exampleof(generalised)suffixtreeisshowninFig. 3:thesolesuffixtreestructureisdisplayedinFig. 3a.Forinstance,the node 3 representsthestringaa.Goingfromnode 3 tonode 1 extends thestringrepresentedbya b toyieldthestring aab.

STare equippedwithadditionalarcs calledSuffixLinks (SL):a SL linksanode representing au (witha aletter andu a string)to thenode representingu. LetuslabeltheSL by thelettera.Thenotion ofSLis usuallyrestrictedto internal nodes,butitextendsnaturallytoleaves.Weconsiderthateachnode hasitsSL.TraversingaSL removesthefirstletterof theword:itshortenstherepresentedwordatthefront.ToeachSL,weassociatealabel,whichistheletteritremoves.For instance,inFig. 3,traversingtheSLofnode 1 shortensthewordaab toab,whichisthestringrepresentedbynode 2.The labelofthatSLisa (labelsofSLarenotrepresentedinthisfigure).Itisknownthat,ifSLsarereversed(consideredinthe otherdirection),theyformatreerootedintherootoftheST,andallleavesareonthesamebranchofthistree[5].Note thatthechainofSLstartingfromanynodegoesuptotherootofT (Forsimplicity,assumetheroothasaSLtoitself).The treecomposedbythesuffixlinksisdisplayed onFig. 3b.Moreover,inthefigures,theSLaredrawnasanarcfromachild toitsparentinL (seeFig. 3).

Forsimplicity,wesaythata suﬃxtreearcisblue,whileSLarered.So,wehavethebluetree,denoted T ,andthered tree,denotedL,onthesamenodes.

The GST generalises thisprinciple toseveral words. From now on,the generalised suﬃx tree of P isthe pair

(

T

,

L

)

. AnextensiveintroductiontoGSTandtheirapplicationscanbefoundin[10].Thedistinctionbetweenexplicitandimplicit nodesisskippedhere,butexplainedin[10,2].

2.1. Overlaps

Basically,allsuffixesandallsubstringsofwordsof P arerepresentedintheGST. Ifawordu overlapsaword v,then a suffixof u equals toa prefixof v. Hence, oneexpectsthe overlapto berepresented somewhereinthe GST. Indeed,if aninternalnoderepresentingthesuffixofu appearsonthebranchspelling v,thenu overlaps v.Thisinternalnodeexists sincethissubstringoccursatleasttwice(inu andinv), andrepresentstheoverlap.Letuscallsuchnodean overlapnode fromu to v.Inourrepresentation,theinternalnodeislabelledwiththestarting positionofthatsuffixinu onthebranch leading totheleafof v.Forexample,inFig. 3 thenode 2 islocatedonthebranchof 1.Itisalsoonthebranchesof 1 and 1 .Itrepresentsthestringab.Soweknowthataaboverlaps

abba

,abaaandababb,withtheoverlapab.

Clearly,alloverlapnodesfromu tov areancestorsofanother.Moreover,theworddepthofan overlapnode givesthe lengthofthecorrespondingoverlap.Hence,themaximumoverlapbetweentwowordsisthedeepestoverlapnodeofthose words.Forinstance,both nodes 4 and 3 areancestorsof 1 meaningthatabaaoverlapsaab withtwopossibleoverlaps: a andaa,oflength1 and2 respectively.

Allthesenotionsareknown.Indeed,itisknownthatonecancomputeeﬃcientlyalloverlapsforeachpairofwordsin P usingtheGSTof P .Infact,one usuallycomputespairwise themaximumoverlapsfor P [11],tobuild thepreﬁxorthe overlapgraphofP (asmentionedinintroduction).

2.2. MergeandRed–Bluepaths

Abovetheoverlapnode was deﬁnedonlyusing T ;theSL(hence L)were notinvolved.Letusnowgive analternative deﬁnitionusingT and L.

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Fig. 3. (c) TheGeneralisedSuffixTree(GST)ofP:= {ababb, aab, abba, abaa}.TheGSTofP isinfactcomposedoftwotrees:(a)thetreeT ,whichisthe basicstructureofthesuffixtree–inbluearcs,and(b)L,thetreemadebythesuffixlinksofT –inredarcs.Squarenodesrepresentwordsthatoccuras asuffixofsomeinputword,circlenodesaretheothernodesofT .EachsquarenodestoresitspositionsofoccurrencesinS;forsimplicity,wedisplaythe startingpositionasanumberandthewordofS inwhichitoccursasitscolour,insteadofshowingthelistofpairs

(

i, j).Thesquarenodelabelledbya red2,whichrepresentsthestringab,meansthatthisstringoccursasthesuﬃxstartingatposition2 intheredwordaab.A squarenodelabelledwitha 1 representsaninputword.(Forinterpretationofthereferencestocolourinthisﬁgurelegend,thereaderisreferredtothewebversionofthisarticle.)

Ifu overlapsv,thenasuffixofu equalsaprefixof v.Wehaveseenthattheoverlapsisanodeonthebranchspelling v. We canalsofindthisprecisesuffix ofu startingatthenode representingu byfollowingenough suffixlinks. Thismeans going upintheredtreefromthenodeu.Hence, oneseesthatanoverlapnodefromu to v isan ancestorofu in L (the red tree),andanancestorofv in T (thebluetree).Weobtainanalternativedefinition:anoverlapnodefromu to v isa red ancestorofu andablue ancestorofv.Forsimplicity,letussayitisared–blue ancestoroftheorderedpair

(

u

,

v

)

.

Now,weseethatthemaximumoverlapof

(

u

,

v

)

isthelowestred–blueancestorof

(

u

,

v

)

.Also, iftheredpathreaches theroot,thentheoverlapistheemptyword(andtheirmergeissimplytheirconcatenation).

Inotherwords,anyoverlapforapair

(

u

,

v

)

correspondstoaRed–Bluepath fromu tov in

(

T

,

L

)

.

The merge of u and v usingtheir maximumoverlap ov

(

u

,

v

)

is the word w

:=

pr

(

u

,

v

)

ov

(

u

,

v

)

su

(

u

,

v

)

, where u

:=

pr

(

u

,

v

)

ov

(

u

,

v

)

and v

:=

ov

(

u

,

v

)

su

(

u

,

v

)

.We state this deﬁnitionwith the maximum overlap,but it is validwith any overlap; thelargertheoverlap,theshorterthemerge.Wedenoteby RB-path

(

u

,

v

)

theRed–Bluepathwhichcorresponds tothemaximumoverlapbetweenu andv.

Assumewewanttospelloutthe(linear)mergeofu and v usingtheoverlapnode w in

(

T

,

L

)

.Itcanbedoneinthree steps:

1. traverseT fromtheroottou:thisspellsoutu withthelabelsofbluearcs;

2. goup L fromu to w followingtheredarcs:itispossiblesince w isaredancestorofu (here weskiponebyone all lettersofpr

(

u

,

v

)

);

3. godownT from w to v usingbluearcs,whichispossiblesincev isadescendantof w;thisspellsoutsu

(

u

,

v

)

. A mergeof

(

u

,

v

)

isa linearsuperstringof

{

u

,

v

}

.The process abovecan begeneralised tospellout anycyclic super-string of a cyclicpermutation of words

(

w1

,

w2

,

. . . ,

wn

)

using some givenpairwise overlaps: one starts from thenode representing w1,then performs steps2and3usingthe appropriateRed–Blue pathfrom w1 to w2 tomergethem, then

iteratesonlysteps2and3foreachfollowingworduntilonereaches w1 again.ThiscircuitontheGSTiscalledaRed–Blue

circuit.

Example1. For instancein Fig. 3,the Red–Bluecircuit corresponding to thecyclic mergeof thewords ababb,abba,and thenaabinthisorder,containsthreeRed–Bluepaths:forthepairs

(

ababb

,

abba

)

,

(

abba

,

aab

)

andthenfor

(

aab

,

ababb

)

.This

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circuitstarts atnode 1 goesup withSLtonode 2 andthen 3 whichisared–blue ancestorof

(

ababb

,

abba

)

,then goes down withbluearcsto 1.Thisisthe Red–Bluepath for

(

ababb

,

abba

)

, whichspells outab with the SLlabels.Then the circuitgoesupwithSLto 2, 3,and4,whichisanancestorof 1.Thisspellsoutabb withSLlabels.Then,itgoesdownto 1.ThismakesuptheRed–Bluepathfor

(

abba

,

aab

)

.Then,thecircuitgoesupto 2,whichisanancestorof 1 ;thisspells outa withone redlabel.Then, itgoesdownandbackto 1 ,whichwas thestartingnode.Altogetherthesequenceofred labelsspellsoutababba,whilethesequenceofbluelabelsyields

aababb

:bothwritethesamecyclicword.

Now,letusstateamoresubtleproperty.ForaRed–Bluecircuit,writingallbluelabelsinorderorwritingallredlabels inorderspellout thesamesuperstring(aredlabelisalabelofaSL).Hence, thenumberofbluearcsissmallerthanthe numberofredarcs,whichisthesuperstringlength.Aproof ofaslightlydifferentversionofthispropertycanbefoundin

[5, Section 4.3 on p. 18]. 3. Thesuperstringgraph

Here, weare going to show that thechoicesmade byAlgorithm

greedy

forSCCS are not reallychoices. Infact, we willshowthateachgreedysolutioncanberepresentedbyauniquegraphembeddedintheGeneralisedSuﬃxTree.Asthis graphturnsouttobecomputableinlineartime,onecanderiveagreedysolutionforSCCSinlineartime.

3.1. Equivalenceofgreedysolutions

Letw beagreedysolutionofSCCS.ByProposition 2,thereexistsapermutation

σ

suchthat w

=

P

(

σ

)

.Foreachcyclic permutation

σ

iof

σ

,wedenotebyc

(

σ

i

)

theRed–Bluecircuitobtainedfrom

σ

i intheGST.

We denoteby GP

(

σ

)

= (

V

,

R

,

B

)

thedirected multigraphbuiltwith thearcsof theRed–Blue circuitsof allthe c

(

σ

i

)

, whereR denotesthesetofRed arcs,andB denotesthesetofBlue arcsofallthec

(

σ

i

)

.

AgraphissaidtobeEulerian,ifthereexistsaEulerianpathonthisgraph,i.e. apath thatvisitseacharcofthegraph exactlyonce.As GP

(

σ

)

isacollectionofRed–Bluecircuits,whicharemadeofRed–Bluepaths,eachconnectedcomponent ofGP

(

σ

)

isEulerian.(Witheachadditionalcircuit,thegraphremainsEulerian.)

Foreachpermutationof

{

1

,

. . . ,

|

P

|}

,we canbuildthecorresponding multigraphGP

()

.Wehavethefollowing proposi-tion.

Proposition3.Letw1andw2betwogreedysolutionsofSCCSandlet

σ

1and

σ

2betheirrespectivepermutationsof

{

1

,

. . . ,

|

P

|}

.

ThenGP

(

σ

1

)

=

GP

(

σ

2

)

.

Proposition 3statesakeypropertyofgreedysolutionsforSCCS.Below,wegivetwolinesofproof:theﬁrstoneexplains whythepropertyisvalid(andbetterconveystheintuitiontothereader),whilethesecondgivesamoreformal,butless explanatory,demonstration.

Fig. 4. Example withthreeoverlaps:twodependentones,betweenu1andv1andbetweenu1andv2,andanotheroverlapbetweenu2andv2:shownon

thestrings(a),intheoverlapgraph(b),andintheSuperstringGraph(c).

Proof. Let w

=

P

(

σ

)

beagreedysolutionofSCCS.Letusshow thatthechoicesmadebyAlgorithm

greedy

forSCCSdo notinﬂuenceGP

(

σ

)

.

greedy

facesachoicewhenseveraloverlapsofthesamelengtharedependent(Clearly,independent overlapscanbeswappedinapermutationandthusdonotinﬂuencethegraph).

Let ov

(

u1

,

v1

)

and ov

(

u1

,

v2

)

be two dependent andgreedy-feasible overlaps. Assume that ov

(

u1

,

v1

)

is the overlap

choosenby

greedy

toobtain w,andletu2 inP bethesuccessorofv2in

σ

.Itimpliesthatov

(

u2

,

v2

)

istheoverlapthat

mergesu2 andv2in w.Bythedeﬁnitionof

greedy

,wehavethat

|

ov

(

u1

,

v1

)

| ≥ |

ov

(

u2

,

v2

)

|

.Letusshowthatthereexits

(8)

As

|

ov

(

u1

,

v1

)

| ≥ |

ov

(

u2

,

v2

)

|

,wehavethat ov

(

u2

,

v2

)

isapreﬁx ofov

(

u1

,

v2

)

,andthus apreﬁxofov

(

u1

,

v1

)

.Hence

intheGST,thenodeov

(

u2

,

v2

)

isanancestorofov

(

u1

,

v1

)

,andov

(

u1

,

v1

)

isanodeintheRed–Bluepathsfromu1 to v1

andfromu2 tov2.Thus,wehavebothRed–Bluepathsfromu1 tov2 andfromu2to v1areinGP

(

σ

)

(seeFig. 4).

2

Analternativeproof. Let

σ

1and

σ

2 betwopermutationsofP correspondingtotwodistinctgreedysolutionsforSCCS.We

setGp

(

σ

1

)

:= (

V1

,

R1

,

B1

)

andGp

(

σ

2

)

:= (

V2

,

R2

,

B2

)

.

WeproceedbycontrapositionandassumethatGp

(

σ

1

)

=

Gp

(

σ

2

)

;then

((

R1

∪

B1

)

\(

R2

∪

B2

))

∪((

R2

∪

B2

)

\(

R1

∪

B1

))

= ∅

.

As the role of bothpermutations issymmetrical, we assume that

(

R1

∪

B1

)

\ (

R2

∪

B2

)

= ∅

.Lete

:= (

u

,

v

)

be an arcof

(

R1

∪

B1

)

\ (

R2

∪

B2

)

suchthatmax

(

|

u

| ,

|

v

|)

ismaximalamongallpossiblearcsof

(

R1

∪

B1

)

\ (

R2

∪

B2

)

.Accordingtothe

deﬁnitionofGp

(

σ

1

)

,thereexisti and j suchthat

σ

1

(

i

)

=

j ande isanarcontheRed–Bluepathlinking si tosj (formally e

∈

RB-path

(

si

,

sj

)

).Twocasesarise.

Case e

∈

B1:

Let i2 denotethepredecessor of j in

σ

2, inotherwords, theelementof

σ

2 satisfying j

=

σ

2

(

i2

)

. But,one gets that

ov

(

si2

,

sj

)

isapreﬁxofov

(

si

,

sj

)

,sinceotherwisetherewouldexist j1

:=

σ

1

(

i2

)

,suchthat

ov

(

si

,

sj

)

<

ov

(

si2

,

sj1

)

,

whichwouldimplythate isnotmaximal.Itfollowsthate belongstoRB-path

(

si2

,

sj

)

,andthusto B2,which

contra-dictsourhypothesis. Case e

∈

R1:

Let j2 denote the successorof i in

σ

2, that is j2

:=

σ

2

(

i

)

. Clearly, ov

(

si

,

sj2

)

must be a suﬃx of ov

(

si

,

sj

)

, since

otherwise therewouldexisti1 suchthat j2

=

σ

1

(

i1

)

and

ov

(

si

,

sj

)

<

ov

(

si1

,

sj2

)

,whichwouldmeanthate is not

maximal.Oneobtainsthate

∈

RB-path

(

si

,

sj2

)

,andthuse belongsto R2,whichisacontradiction.

Thisendstheproof.

2

ByProposition 3,thegraphGP

()

representsallthesolutionsofthegreedyalgorithmforSCCS.Wecalledthisgraph,the SuperstringGraph of P ,anddenoteitbyGP.AnexampleofSuperstringGraphforP

:= {

abb

,

bbb

,

bbc

}

isshowninFig. 7.

Tosumup,weknowthat foreach greedysolution w,thereexistsapermutation of

{

1

,

. . . ,

|

P

|}

that canbetranslated intoasetofRed–Bluecircuits,c

(

w

)

= {

c1

(

w

),

. . . ,

cm

(

w

)

}

.RememberthateachRed–BluecircuitismadebychainingRed– Blue paths. Letc be amulti-cycle of G.We saythat c covers GP ifeach arcof GP is includedina single cycleof c.By

Proposition 3,foreachgreedysolution w wehavethatc

(

w

)

coversGP.

Bythefollowingproposition,wehavethateachmulti-cyclethatcoversG comesfromagreedysolution. Proposition4.

{

multi-cycle p

|

p covers GP

}

= {

c

(

w

)

|

w is a greedy solution

}

.

ToproveProposition 4,weﬁrstneedalemma.Acrossing inGP isanode thathasredin- andout-goingarcsandblue in- andout-goingarcs.

Lemma5(Absenceofcrossing).GPdoesnotcontainanycrossing.

Proofof

Lemma 5

. It is enough to see that the only nodes that can have a red out-goingarc and a blue in-going arc correspondtothewordsof P .Butifv

∈

P , v isaleafinT andinL bydeﬁnitionoftheGST,andthus v hasneitherablue arcgoingout,noraredarcgoingin.

2

Fig. 5. Illustration fortheproofofProposition 4.Intheproofof⊆,theargumentistheexistenceofacrossingnodeinanon-greedy solution.Thisﬁgure displaysthefournodes v1, . . . , v4 andthenodes(circles)representingtheiroverlaps.Then,thenoderepresentingov(v3, v4)(greynode)isacrossing, i.e. ithasblueandredin-goingandout-goingarcs.(Forinterpretationofthereferencestocolourinthisﬁgurelegend,thereaderisreferredtotheweb versionofthisarticle.)

Proofof

Proposition 4

. To understandthis equality,it issuﬃcient tonote that there is arelation betweenchoosing the longestoverlapin

greedy

andtheabsenceofcrossinginGP.

Proofof

⊆

.

Letusshow

{

multi-cycle p

|

p covers GP

}

⊆ {

c

(

w

)

|

}

by contraposition.Assumethatthere exists a multi-cycle p coveringGP andtheredoesnotexist agreedysolution w of P suchthat p

=

c

(

w

)

.Byhypothesis, there exist four nodes v1

,

v2

,

v3 and v4 in P that do notsatisfy the greedyproperty. In other words,although

|

ov

(

v1

,

v2

)

| <

(9)

|

ov

(

v3

,

v4

)

|

,both P ath

(

v1

,

v4

)

and P ath

(

v3

,

v2

)

arein p. Hence,the nodeov

(

v3

,

v4

)

isa crossingin GP (see Fig. 5).As theredoesnotexistanycrossinginGP (Lemma 5),weobtainacontradiction,andget

{

multi-cycle p

|

p covers GP

} ⊆ {

c

(

w

)

|

}.

Proofof

⊇

.

Reciprocally,assumethatthereexistsagreedysolution w of P andtheredoesnotexistamulti-cycle p thatcoversGP such that p

=

c

(

w

)

.Thus,there existsan arcofc

(

w

)

whichisnot an arcofGP.Among thosearcs,let e

= (

u

,

v

)

bethe one withthedeepest node(i.e. longest word) v in T .Bythe deﬁnitionof GP, e belongs to R (i.e. is ared arc),because it is the deepest ofits kind. Let v1 and v2 be the nodesof P such that e belongs to P ath

(

v1

,

v2

)

. As e is not in GP, foreach u in P such that P ath

(

v1

,

u

)

is in GP, u

=

v2.Thus, there exists v4 in P such that P ath

(

v1

,

v4

)

is in GP and

|

ov

(

v1

,

v2

)

| < |

ov

(

v1

,

v4

)

|

.As w isagreedysolutionof P ,thereexists v3 in P andnotinGP suchthat P ath

(

v3

,

v4

)

isin

c

(

w

)

and

|

ov

(

v1

,

v4

)

| ≤ |

ov

(

v3

,

v4

)

|

.Thus,there existsan arcinc

(

w

)

that isnot inGP witha deeperextremity thane. Thiscontradictsourhypothesisandweget

{

c

(

w

)

|

}

⊆ {

multi-cycle p

|

p covers GP

}

.

2

3.2. CharacterisationoftheSuperstringGraphandaconstructionalgorithm

Now,wearegoingtocharacterisetheSuperstringGraphandgivealineartimealgorithmtobuilditfromasetofwords. Forthisalgorithm,wedeﬁneandcomputeweights(respectivelyn

(.)

andd

(.)

)fortheredandforthebluearcsoftheGST. Finally,theSGwillbecomposedofthosearcshavingastrictlypositiveweight(andofthenodesvisitedbythem).

As

greedy

selectsoverlapsinorderofdecreasinglength,onecanseeitasarecursivealgorithmthatprogressesalong thebranchesoftheGSTof P .OntheGST,

greedy

progresses fromtheleavestowardtheroot,i.e. inorderofdecreasing worddepthofthenodes.Letususethisrecursivesideof

greedy

toexplaintheSuperstringGraph.

Letv be anodeoftheGST;when

greedy

sees theoverlaps oflength

|

v

|

,ithasalreadyconsideredoverlaps oflarger lengths.Wedeﬁnetwoweightsn

()

andd

()

suchthat:

•

n

(

v

)

isthenumberofstringsof P whichhavev assuﬃxandarenotmergedontheirrightwithanotherstringof P withalargeroverlap.

•

d

(

v

)

isthenumberofstringsof P whichhave v aspreﬁx andare notmergedontheir leftwithanotherstringof P withalargeroverlap.

Inother words,n

(

v

)

isthenumberofleavesofthesubtreeofv inL thathavenotyetbeenmergedontheirright,and d

(

v

)

isthenumberofleavesofthesubtreeofv inT thathavenotyetbeenmergedontheirleft.

First,itcanbeseeneasilyseenthatanode v whichhasnootherleafbothinitssubtreesinL andinT ,isastringofP . Thus,forthesenodesweset,n

(

v

)

=

d

(

v

)

:=

1.

Fortheother nodes,weusetherecursionof

greedy

.Infact, wecan determinehowmanystringsof P havenotyet beenmergedwithanotherstringbylookingtheweightsn

(.)

andd

(.)

ofthechildrenofv inL andofitschildreninT (see

Fig. 6).

Leta

(

v

)

bethe difference betweenthesumofthen

(

u

)

ofthe children u of v inL and the sumofthed

(

u

)

ofthe childrenu ofv inT : a

(

v

)

:=

u∈ChildrenL(v) n

(

u

)

−

u∈ChildrenT(v) d

(

u

).

Hence, theabsolutevalue ofa

(

v

)

is thelargest numberofstrings of P that

greedy

can mergewithother strings of P usingtheoverlap v.Witha

(

v

)

,wecancalculaten

(

v

)

andd

(

v

)

:

If a

(

v

)

≥

0

,

we have

n

(

v

)

:=

a d

(

v

)

:=

0

,

if a

(

v

) <

0

,

we have

n

(

v

)

:=

0 d

(

v

)

:= −

a

.

Withthisrecursive deﬁnitionoftheweights n

()

andd

()

, wecan constructtheSuperstring GraphfromtheGST of P . WhenevertheweightofanarcoftheGSTisequalto0,thisarcdoesnotbelongtotheSuperstringGraph.Similarly,iffor anode,sayv,allofitsin-goingandout-goingarcshaveazeroweight, v isisolatedandisnotincludedintheSuperstring Graph–thiscorrespondstothesubsetU inDeﬁnition 1(whichstatesanotherdeﬁnitionoftheSuperstringGraph).

Deﬁnition1.TheSuperstringGraphGP

= (

V

,

R

,

B

)

where V

=

VT

\

U

(10)

Fig. 6. General schemeofthein- andout-goingarcsofanodev inGST.TheredarcsbelongtoR andthebluearcstoB.Ifaweightiszero,thearcis notincludedintheSuperstringGraph.(Forinterpretationofthereferencestocolourinthisﬁgurelegend,thereaderisreferredtothewebversionofthis article.)

B

= {(

t

,

v

)

d(v)

|

v

∈

VT

,

t the parent of v in T

}

andU

= {

v

∈

VT

|

vis not an extremity of an arc of R

∪

B

}

wheren

()

andd

()

aretheweightsonarcsasdeﬁnedabove. Now thatwecanbuild theSuperstringGraphforaninput P by traversingthegeneralisedsuﬃxtreeof P ,weprovide analgorithm(Algorithm 2)thatusesthisgraphtosolveSCCSforP .

Theorem6.

Algorithm 2

solvesSCCSinlineartimeandspace.

Proof. Correctness.ByProposition 4,theset w ofcyclicwordscomputedbyAlgorithm 2isasolutionof

greedy

forSCCS. Hence,eachwordof P appearsasasubstringofacyclicwordinw,andbyTheorem 1,thenormof w isminimal.

Complexity.TheSuperstringGraphof P isembeddedinthegeneralisedsuﬃxtreeof P ,andthusbuiltinatime linear in

P

.

Firstthecyclicconcatenationofthewordsin P islongerthantheSCCSobtainedthroughanycyclicpermutationof P . ByProposition 2,thelengthofasolutionof

greedy

isboundedby

P

;thus,thelengthofamulti-cycleofGP isbounded by2

×

P

.Consequently,onecanﬁndamulti-cyclealsoin O

(

P

)

timeandspace.Finally,traversingallitscyclestakesa timeproportionalinthesizeoftheSuperstringGraph.

2

NotethatwithAlgorithm 2,oneobtainsacycliccover.

Algorithm 2: TheSuperstringGraphalgorithmforSCCS.

1 Input: P asetoflinearwords. Output:w agreedysolutionforSCCS;

2 buildGPtheSuperstringGraphofP ;

3 computeacycliccoverc= (c1,. . . ,cn)ofGP;

4 for i ∈ [1, n]do

5 traverseci:listthewordsofP whosenodeareinciandcreatewibyconcatenatingcyclicallypr(sj, sk)wheneverskisthesuccessorofsj;

6 insertthecyclicstringwiinw;

7 return w

4. ImplicationsoftheSuperstringGraph

4.1. SCCSwithacardinalityconstraintongreedysolutions

We cansplitup GP intoconnectedcomponents: GP

= {

C1

,

. . . ,

Cg

}

.ByProposition 4,eachmulti-cycle c ofGP can be partitionedinasubsetofcyclesc

=

S1

∪ . . . ∪

Sg,whereeachsetofcyclesSi coversaconnectedcomponentCiofGP.

Algorithm 2, on its third line (in blue),1 _computes _a _cyclic _cover _of _G

P. This requires to perform a random choice when facing two alternative out-goingarcs.Clearly, the combinationof all possiblesuccessive choicesallows to produce all possiblecycliccovers ofGP.Certain choiceswill generateasolution containingseveralcycles tocoverone connected component(seeFig. 7b),althoughitisalwayspossibletocoveroneconnectedcomponentwithasinglecycle(seeFig. 7c). Thus, anytwocycliccoversmaydifferintheirnumberofcycles,i.e. intheircardinality.Thisgivesrisenaturallytoanew question,toanewvariantofSCCS,whichwedeﬁnebelow.

Problem2(SCCSwithacardinalityconstrainton

greedy

solutions). Input: Asetoflinearstrings P .

Output: A setofcyclicstringsc,withtheleastnumberofelements,thatisasolutionofthegreedyalgorithmonShortest CyclicCoverofStrings.

(11)

Fig. 7. A SuperstringGraphandSCCSsolutions.(a)ThegeneralisedsuﬃxtreeandtheSuperstringGraph(SG)forP:= {abb, bbb, bbc}.SquarenodesofT

representstringsthataresuffixesofsomewordofP :theintegerinthesquarenodeisastartingpositionofitssuffix.AwordofP isidentifiedbyits colour.Figure (b)and(c)showthetwomulti-cyclesthatcovertheSGandtheirrespectivesetsofcyclicstrings.Thisexampleiswellknownforshowing thelowerboundoftheapproximationratioofthegreedyalgorithmforSLiS.TheSGforthisinstanceisconnectedandthusitalsogivesanoptimalcyclic superstring(asolutionofSCyS).AstheEuleriancircuitvisitstherootoftheGST,thisoptimalcyclicsuperstringcanbecutattheroottoobtainanoptimal linearsuperstring.

Algorithm 3: TheSuperstringGraphalgorithmforSCCSwithacardinalityconstraint.

1 Input: P asetoflinearwords. Output:w agreedysolutionforSCCS;

2 buildGP theSuperstringGraphofP ;

3 computeaEulerianmulti-cyclec= (c1,. . . ,cn)ofGP;

4 for i ∈ [1, n]do

5 traverseci:listthewordsofP whosenodeareinciandcreatewibyconcatenatingcyclicallypr(sj, sk)wheneverskisthesuccessorofsj;

6 insertthecyclicstringwiinw;

7 return w

Algorithm 3differsfromAlgorithm 2onlyon thethirdline:insteadofcomputinganycycliccoverof GP,it computes aEulerianmulti-cycleof GP.ThisispossiblesinceeachconnectedcomponentoftheSuperstringGraphisEulerianandit canbedoneinlineartimeinthenormof P .AEulerianmulti-cycleofGP coverseachconnectedcomponentwithasingle cycle, henceitminimisesthecardinalityofthe cycliccover, andasitisa greedysolutionitsolves Problem 2.Weobtain

Theorem 7.TheminimisationofthenumberofcyclicstringsbyAlgorithm 3isillustratedinFig. 7.

Theorem7.

Algorithm 3

solves

Problem 2

inlineartime.

Proof. AstheconnectedcomponentsoftheSuperstringGraphareEulerian,thisisdoneinlineartimeandspaceinthesize ofGP.ByProposition 4,alabelofaEulerianmulti-cycleofGP isanoptimalsolutionfortheProblem 2.

2

Remark.AlthoughforSCCS,allsolutionsof

greedy

areoptimal,someoptimalsolutionsarenotoutputby

greedy

.Thus, thequestionofﬁndingashortestcycliccoverwithaminimumnumberofelementsisnotsolvedbytheAlgorithm 3(seea counterexampleinFig. 8).

4.2. ConsequencesofthetopologyoftheSuperstringGraph

Thecharacterisation oftheSGcanalsobe appliedtosolveorapproximatethe“classical”shortestsuperstringproblem (SLiS–forShortestLinearSuperstring),inwhichthesolutionisalinearsuperstring,aswellasitsvariantwherethesolution isasinglecyclicword(SCyS-=forShortestCyclicSuperstring).LetwO P T(S Li S),respectively wO P T(SC y S),denoteanoptimal solutionforSLiS,resp.forSCyS.Let w agreedysolutionforSCCS,wehave:

|

w

| ≤

wO P T(SC y S)

≤

wO P T(S Li S)

.IftheSGis connected,then w is acyclicsuperstringof P ,i.e. asolution ofSCyS.Bytheabove relation,itis anoptimalsolution for SCyS.

(12)

Fig. 8. An instance with an optimal solution that is nota solution of greedy for SCCS (however, the solution ofthe SG is also optimal). Let

P= {abec, bed, c f abe, dgab}.(Top)thegeneralised suﬃxtree,(middle)theSG,and(bottom)anoptimalsolutionofSCCSthatconsistsinsinglecycle, whiletheSGhastwocomponents.

Now,fromthiscyclicsuperstring,wecaneasilyderivealinearsuperstringof P .Basically,wecutthecyclicstringatthe startofasuﬃxofawordofP andpre-catenatethecorrespondingpreﬁx.Letv anodeoftheSGsuchthatn

(

v

)

=

d

(

v

)

=

0, thenv isapreﬁxofsomewordofP .Then v w makesupalinearsuperstringofP .Weobtain

|

w

| ≤

wO P T(S Li S)

≤|

v

|+|

w

|

. As

|

v

| ≤

wO P T(S Li S)

, we get

|

v

| + |

w

| ≤

2

wO P T(S Li S)

. In other words, forinstances P that have a connected SG,we obtain a 2-approximation for the shortest linear superstring. Furthermore, if the Eulerian cycle visits the root of the suﬃx tree, then v isthe empty word andthen

|

w

| ≤

wO P T(S Li S)

≤ |

w

|

, meaning that w is an optimal linear super-string.

Notethatthereexistnon-trivial instances,withanarbitrarylargenorm,whoseSGisconnected:anyinstanceinwhich allwordsarecyclicshiftsofanotherbelongtothisclass.AstheSGcanbebuiltandexploredinlineartimein

P

,onecan usethisprocedureasapredicatetocheck whetherSCySorSLiSaresolvableinlineartime.Ifnot,onecanlaunchamore complexapproximationalgorithm[15].

5. Conclusion

WeconsidertheproblemShortestCyclicCoverofStrings,whichissolvedbyAlgorithm

greedy

.Thesetofsolutionsof

greedy

canbeexponential.Nevertheless,withtheSG,weexhibitacharacterisationofthissetinagraphthatisEulerian and linear inthe input size. This enablesus to solve SCCSin linear time, butalso to ﬁnd a

greedy

solutionwith the leastnumberofcyclicstrings.Moreover,thisapproachcansolveorapproximatesome non-trivial instancesoftheshortest superstringproblems.OurapproachcanbeadaptedtoavariantofSCCSwherethemultiplicityofeachinputwordisgiven asinput.

Currently,approximationalgorithmsforshortestsuperstringproblemsuseaprocedureforsolvingSCCSthattakescubic time (i.e. O

(

P

+ |

P

|

3

)

time) andquadratic spacein the numberof strings. Thisstep represents their time complexity bottlenecksinceotherstepscanbedonein O

(

P

)

time.Usingouralgorithmmakestheoveralltimecomplexitylinearin theinputsize.

(13)

Acknowledgements

This work is supported by ANR Colib’read (ANR-12-BS02-0008), by ANR Institut de Biologie Computationnelle (ANR-11-BINF-0002),andbyCNRS(DéﬁMASTODONS).

References

[1]A.Blum,T.Jiang,M.Li,J.Tromp,M.Yannakakis,Linearapproximationofshortestsuperstrings,in:ACMSymposiumontheTheoryofComputing,1991, pp. 328–336.

[2]D.Breslauer,G.F.Italiano,Onsuﬃxextensionsinsuﬃxtrees,Theor.Comput.Sci.457(2012)27–34.

[3]D.Breslauer,T.Jiang,Z.Jiang,Rotationsofperiodicstringsandshortsuperstrings,J.Algorithms24 (2)(1997)340–353.

[4] B. Cazaux, E. Rivals, Approximation of greedy algorithms for Max-ATSP, Maximal Compression, Maximal Cycle Cover, and Shortest Cyclic Cover of Strings, in: Proc. of Prague Stringology Conference, PSC, Czech Technical Univ. Prague, 2014, pp. 148–161, http://www.stringology. org/event/2014/p14.html.

[5] B.Cazaux, E. Rivals, Reverseengineering of compactsuﬃx trees and links: a novel algorithm, J. Discret. Algorithms 28 (2014) 9–22, http:// dx.doi.org/10.1016/j.jda.2014.07.002.

[6] B.Cazaux,E.Rivals, The powerofgreedy algorithms for approximating Max-ATSP, CyclicCover,and superstrings,Discrete Appl.Math. (2015),

http://dx.doi.org/10.1016/j.dam.2015.06.003.

[7]J.Gallant,D.Maier,J.A.Storer,Onﬁndingminimallengthsuperstrings,J.Comput.Syst.Sci.20(1980)50–58.

[8]A.Golovnev,A.Kulikov,I.Mihajlin,ApproximatingshortestsuperstringproblemusingdeBruijnGraphs,in:CombinatorialPatternMatching,in:Lect. NotesComput.Sci.,vol. 7922,SpringerVerlag,2013,pp. 120–129.

[9]D.Gusﬁeld,Fasterimplementationofashortestsuperstringapproximation,Inf.Process.Lett.51 (5)(1994)271–274.

[10]D.Gusﬁeld,AlgorithmsonStrings,TreesandSequences,CambridgeUniv.Press,1997.

[11]D.Gusfield,G.M.Landau,B.Schieber,Anefficientalgorithmfortheallpairssuffix–prefixproblem,Inf.Process.Lett.41 (4)(1992)181–185.

[12]S.R.Kosaraju,A.L.Delcher,Large-scaleassemblyofDNA stringsandspace-eﬃcientconstructionofsuﬃxtrees,in:ProceedingsoftheTwenty-Seventh AnnualACMSymposiumonTheoryofComputing,STOC ’95,1995,pp. 169–177.

[13]S.R.Kosaraju,A.L.Delcher,Large-scaleassemblyofDNA stringsandspace-eﬃcientconstructionofsuﬃxtrees,in:ProceedingsoftheTwenty-Eighth AnnualACMSymposiumonTheoryofComputing,STOC ’96,ACM,NewYork,NY,USA,1996,p. 659.

[14]M.Mucha,Lyndonwordsandshortsuperstrings,in:ProceedingsoftheTwenty-FourthAnnualACM–SIAMSymposiumonDiscreteAlgorithms,SODA, 2013,pp. 958–972.

[15]K.E.Paluch,Betterapproximationalgorithmsformaximumasymmetrictravelingsalesmanandshortestsuperstring,CoRRarXiv:1401.3670[abs],2014.

[16]C.H.Papadimitriou,K.Steiglitz,CombinatorialOptimization:AlgorithmsandComplexity,2ndedition,DoverPublications,Inc.,1998,496pp.

[17]J.Tarhio,E.Ukkonen,Agreedyapproximationalgorithmforconstructingshortestcommonsuperstrings,Theor.Comp.Sci.57(1988)131–145.

[18]S.Teng,F.F.Yao,Approximatingshortestsuperstrings,in:34thAnnualSymposiumonFoundationsofComputerScience,1993,pp. 158–165.

[19]E.Ukkonen,Alinear-timealgorithmforﬁndingapproximateshortestcommonsuperstrings,Algorithmica5 (1–4)(June1990)313–323.

[20]V.Vassilevska,Explicitinapproximabilityboundsfortheshortestsuperstringproblem,in:MathematicalFoundationsofComputerScience2005,30th InternationalSymposium,Proceedings,MFCS2005,Gdansk,Poland,August29–September2,2005,in:Lect.NotesComput.Sci.,vol. 3618,Springer, 2005,pp. 793–800.