Mental map preserving graph drawing using simulated annealing

20 Pages·2011·1.35 MB·English

InformationSciences181(2011)4253–4272
ContentslistsavailableatScienceDirect
Information Sciences
journal homepage: www.elsevier.com/locate/ins
q
Mental map preserving graph drawing using simulated annealing
Chun-Cheng Lina, Yi-Yi Leeb, Hsu-Chun Yenb,⇑
aDepartmentofIndustrialEngineeringandManagement,NationalChiaoTungUniversity,Hsinchu300,Taiwan,ROC
bDepartmentofElectricalEngineering,NationalTaiwanUniversity,Taipei106,Taiwan,ROC
a r t i c l e i n f o a b s t r a c t
Articlehistory: Visualizinggraphshasbeenstudiedextensivelyinthecommunityofgraphdrawingand
Received26May2009 informationvisualizationovertheyears.Insomeapplications,theuserisrequiredtointer-
Receivedinrevisedform19December2010 actwithagraphbymakingslightchangestotheunderlyinggraphstructure.Tovisualize
Accepted2June2011
graphsinsuchaninteractiveenvironment,itisdesirablethatthedifferencesbetweenthe
Availableonline6June2011
displays of the original and the modified graphs be kept minimal, allowing the user to
comprehendthechangesinthegraphstructurefaster.Asthementalmapconceptrefers
Keywords:
tothepresentationofaperson’smindwhileexploringvisualinformation,thebetterthe
Mentalmap
mentalmapispreserved,theeasierthestructurechangeofagraphisunderstood.Itis
Simulatedannealing
somewhat surprising that preserving the user’s mental map has largely been ignored in
Graphdrawing
thegraphdrawingcommunityinthepast.Weproposeaneffectivemental-map-preserving
graphdrawingalgorithmforstraight-linedrawingsofgeneralundirectedgraphsbasedon
thesimulated-annealingtechnique.Ourexperimentalresultsandquestionnaireanalysis
suggestthisnewapproachtobepromising.
(cid:2)2011ElsevierInc.Allrightsreserved.
1.Introduction
Graphsarewidelyrecognizedasapopularandusefulmodelforcapturingawidevarietyofreal-worldconceptsinappli-
cationareassuchassocialnetworks,databases,discreteeventsystems,biologicalpathways,andmanymore[17].Througha
nicevisualdisplayofagraph,theuserislikelytogainabetterinsightintotheinformationconveyedbythegraph.Asitis
virtuallyinfeasibletomanuallydrawgraphsofhighcomplexityinstructureandsize,manysophisticatedautomaticdrawing
methodsfordisplayingcomplicatedgraphshavebeendevelopedovertheyears(see,e.g.,[9,10,20]).Theinterestedreaderis
referredto[1,21]fornicesurveysongraphdrawing.
Inordertofindnicedrawings,graphdrawingalgorithmsareoftendesignedtomeetcertainaestheticcriteria(see,e.g.,
[31]), which specify the desired graphical structures and drawing properties. Such aesthetic criteria include minimizing
thenumberofedgecrossingsorbends,minimizingthedrawingarea,maximizingthedegreeofsymmetry,andmanymore.
Unfortunately,theproblemofsimultaneouslyoptimizingtwoormorecriteria,inmanycases,isNP-hard.Althoughthecom-
putationalefficiencyaswellasthedrawingquality(i.e.,theextenttowhichtheoutputdrawingsconformtothedesired
aestheticcriteria)remainthemostpopularmeasuresforjudgingtheeffectivenessofadrawingalgorithm,moreandmore
attentionhasbeengiventotheissueofpreservingthementalmap[30]intheframeworkofinteractivegraphdrawing.
In some applications, the user can interact with a graph by applying operations such as adding/deleting one or more
nodes/edgestothegraphonaregularbasis.Tocopewiththistypeofdynamicallychanginggraphs,mostautomaticdrawing
strategiesrelyonusinganexistingstaticgraphdrawingalgorithmtoredraweachmodifiedgraphfromscratch.Although
q ApreliminaryversionofthisworkwaspresentedatAsiaPacificSymposiumonInformationVisualization2006,February1–3,2006,Tokyo,Japan.
⇑
Correspondingauthor.
E-mailaddress:[email protected](H.-C.Yen).
0020-0255/$-seefrontmatter(cid:2)2011ElsevierInc.Allrightsreserved.
doi:10.1016/j.ins.2011.06.005
4254 C.-C.Linetal./InformationSciences181(2011)4253–4272
suchstrategiesareconvenientandsimple,theysufferfromthefollowingdrawback.Sincetheuserisalreadyfamiliarwith
the original drawing, applying an existing static graph drawing algorithm to the modified graph may render the output
drawingill-behavedasfaraspreservingtheuser’s‘‘mentalmap’’isconcerned.Toseethis,considerFig.1.Fig.1(a)isthe
drawingofagraphproducedbythewell-knownspringalgorithm[12].Supposeweaddnode4andconnectittotheremain-
ingnodesofthegraph.Fig.1(b)and(c)aretwopossibledrawingsofthemodifiedgraphproducedbythespringalgorithm
underdifferentsettingsofspringlengths.Clearly,Fig.1(c)isabetteroneasfaraspreservingthementalmapisconcerned,
sinceitinheritsmoreinformationfromtheoriginaldrawing(i.e.,Fig.1(a)).Fig.2(originallygivenin[22])showsamore
dramaticexampleillustratingtheimportanceofpreservingthementalmap.Theinitialgraph(Fig.2(a),excludingthethick
curve)consistsofachainofquadrangles,whicharedisplayednicelybythe‘‘CircularwithRadial’’(CRforshort)layoutalgo-
rithm[36].WhenweconnectthetwooutermostnodesbyasingleedgewhichismarkedbythethickcurveinFig.2(a),all
thequadranglescollapseintooneinthelayoutredrawnbytheCRalgorithm(seeFig.2(b)).NotethatthebasicideaoftheCR
algorithmistoplaceallthenodesonacircleandthenperformacrossingminimizationheuristic,whichstillresultsinat
leastn/4crossings(wherenisthenumberofnodes)inthisexample.Unfortunately,thenewdrawingsuffersfromarather
smallangularresolutionsuchthatitishardtorecognizetheoriginalstructure.AnalternativelayoutisdisplayedinFig.2(c)
whichshowsacompleteringofquadranglesasitisconstructedfromtheinitialdrawing.Theabovetwoexamplessuggest
thatusingastaticdrawingalgorithmtodrawamodifiedgraphfromscratchmayresultinsomethingthatlookscompletely
differentfromtheoriginal,causingtheusertohavetospendaconsiderableamountofextratimetocomprehendthenew
drawing.
Eades,Lai,Misue,andSugiyama[13]werethefirsttoproposethementalmapconceptofgraphdrawing,whichcanbe
thoughtofasthepresentationofaperson’smindwhileexploringvisualinformationofagraph.Whatmakesthepreserva-
tionofthementalmapimportantingraphdrawingisthattheuserisabletoquicklyrecognizetheredrawnlayoutofthe
modifiedgraph,basedonhis/herunderstandingoftheoriginallayout.Mostoftheexistingalgorithmsfortheproblemof
preservingthementalmapfocusonpreservingtheorthogonalordering(see,e.g.,[25])oraddingconstraintstotheinputgraph
(see,e.g.,[2,19]).Fortwonodesuandvinagraph,thepreservationoftheorthogonalorderingmeanstherelativeabove/
below and left/right positions of the nodes remain unchanged after the graph is redrawn. Preserving the user’s mental
map is sometimes referred to as a stability problem(see, e.g., [26]). Recently, Brandes and Pampel [3] have showed that
itisNP-hard topreservetheorthogonalorderingevenfor thesimplestgraphs(paths).Forspecificdrawingstylestaking
mentalmappreservationintoaccount,thereaderisreferredto[22]forcirculardrawingand[16]foronlinedynamicgraph
drawing.Experimentalstudiesconcerningtheeffectofpreservingthementalmapcanbefoundin[32,33,35].Theimpor-
tance of preserving the mental map can also be found in applications including distributed mobile object environments
[15],onlinelearningenvironments[37],andmore.
Thebasicrequirementofagraphdrawingalgorithmistoproduceanaestheticallypleasinglayoutin2-Dor3-D,taking
theabstractrepresentationofagraphastheinput.InDavidsonandHarel’swork[6],anice-lookinggraphdrawingreferstoa
drawingwhich:(1)distributesnodesevenly,(2)makesedge-lengthsuniform,(3)minimizesedge-crossings,and(4)keeps
nodesfromcomingtooclosetoedges.Todrawgraphsnicely,asimulatedannealing(SA)methodwasusedin[6]withthe
energycostfunctiondesignedtorespecttheaboverequirements.Theimplementationoftheirworkallowstheusertoadjust
therelativeweightsofcriteriaintheenergycostfunctionamongvariousparameters,thoughsomeofthesecriteriamaybe
inconflictwitheachother.TheSAmethodoffersamoreflexiblewayofdrawinggraphsthanmanyotheralgorithms.Our
simulatedannealinggraphdrawingmethodisbasicallybasedonthestrategyreportedin[6].Theinterestedreaderisalso
referredto[7,8]forearlierworkofapplyingtheSAmethodtographdrawing.
Astherearemanyintuitivefactorsthataffectthementalmap,BridgemanandTamassia[5]formallydefinedthoseintu-
itivefactorsfororthogonaldrawingsandpresentedsomestatisticalresultstoevaluatetheireffectsonhumanperceptionof
similarity.Theyusedmanymetricstomeasurethesimilaritybetweendifferentdrawings,andthenaskedrespondentsto
evaluatethebestmeasuresthatreflecttheirperceptionofsimilarityamongasetoforthogonaldrawings.Finally,theyused
astatisticalmethodtofindtherelationshipbetweenthesemeasurementsandusers’feelings,andsuggestedthefollowing
formeasuringthedegreeofpreservationofthementalmap:ranking,orthogonalordering,k-matrix,unweightednearestneigh-
borwithin,averagedistance,weightednearestneighborbetween,shape,andrelativedistance.Previousstudies([32,33,35]),on
theotherhand,oftenapplystudent-basedsurveystoevaluatingtheperformanceofmental-map-preservinggraphdrawing
algorithms.
Inthispaper,weproposeamental-map-preservinggraphdrawingalgorithmforstraight-linedrawingsofgeneralundi-
rectedgraphs,baseduponthesimulatedannealinggraphdrawingapproachof[6]withtheenergycostfunctionincorporating
1 1 4 1
4
2 3 2 3 2 3
(a) (b) (c)
Fig.1. Anexampleofthementalmapproblem.
C.-C.Linetal./InformationSciences181(2011)4253–4272 4255
(a) (b) (c)
Fig.2. Anexampleofthementalmapproblemincirculardrawing[22].
sixcriteriaof[5]toreflecttheuser’smentalmap.Aspointedoutin[24],student-basedsurveysmaynotalwaysbeappropriate
asfarasevaluatingthequalityofamental-mappreservinggraphdrawingalgorithmisconcerned.Hence,inourstudywe
considernotonlyexperimentalevaluationbutalsostudent-basedquestionnaireanalysis,inordertobetterjustifytheclaimed
performanceofourdesign.Theexperimentalresultsshowthattoacertainextent,ourapproachiscapableofpreservingthe
drawingcontourofagraphandtherelativepositionsofnodeswhenapplyingavertex/edgeinsertion/deletionoperationtoa
graph.Similarto[6],ourapproachisveryflexibleinthesensethatitallowstheusertoadjusttherelativeweightsofcriteria
among various parameters. We also see from the experimental evaluation that there exists a trade-off between drawing
graphsnicelyandpreservingthementalmap.Asforthequestionnaireanalysis,ahighpercentageofrespondentschoose
themodifieddrawingproducedbyourapproachasthebestmental-map-preservingmodifieddrawingintheirminds.
Themaincontributionsofthispaperaresummarizedasfollows:
1. Mostofthepreviousworkofpreservingthementalmapfocusedonhowtopreservetheorthogonalordering(see,e.g.,
[25])oraddconstraintstotheinputgraph(see,e.g.,[2,19]).Toourknowledge,therehavenotbeengraphdrawingalgo-
rithmsaimingatpreservingthementalmaponstraight-linedrawingsofgeneralundirectedgraphswhileusingmore
sophisticatedcriteriasimultaneously.
2. Our approach allows the user to adjust the relative weights of various criteria, making our drawing algorithm very
flexible.
3. Bytakingsamplesamongagroupofstudentswithbasicknowledgeofgraphsandgraphdrawing,BridgemanandTam-
assia[5]suggestedanumberofgoodcriteriasuitableforcapturingthenotionofthementalmap.Ourgraphdrawing
approachincorporatesthecriteriaof[5]intotheenergycostfunctionofasimulatedannealinggraphdrawingalgorithm
topreservethedrawingcontourofagraphandtherelativepositionsofnodeswhenmodifyingthegraph.
4. Ourapproachobtainsthelocationofeachnodeineachiteration,resultinginasmoothprocessfromtheinitialdrawingto
thefinaldrawing–adesirablepropertyininformationvisualization.
Therestofthispaperisorganizedasfollows.InSection2,weintroducethesimulatedannealinggraphdrawingmethod,
uponwhichourgraphdrawingalgorithmisbased.Section3involvesthedesignofanenergycostfunctiontopreservethe
mentalmap.Section4showstheexperimentalresultsofourapproach.Section5givesaquestionnaireanalysisofourap-
proach.AconclusionisgiveninSection6.
2.Preliminaries
Asmentionedintheprevioussection,thefourrequirementsofa‘‘nicedrawing’’includedistributingnodesevenly,making
edge-lengthsuniform,minimizingedge-crossings,andkeepingnodesfrombeingtooclosetoedges.Inthissection,wefirst
introducethe generalframeworkof simulatedannealing,and then showhow todraw graphsnicelyusingthe simulated
annealinggraphdrawingalgorithmproposedbyDavidsonandHarel(DHforshort)[6].
2.1.Introductiontosimulatedannealing
Simulatedannealing(SA)[23]isageneralizationoftheMonteCarlomethodforexaminingtheequationsofstatesand
frozenstatesofn-bodysystems.SAisflexibleandhassuccessfullybeenappliedtosolvingvariouscombinationaloptimiza-
tionproblems,includingsolvingthemaximumcliqueproblem[18],determiningthethicknessofagraph[28],etc.Itallows
‘‘uphill’’moves–movesthatspoil,ratherthanimprove,thetemporarysolution.ThisuniquefeatureallowsSAtoescapefrom
alocalminimalsolution,thoughitisnotguaranteedthataglobalminimumcanbereachedeventually.
ThebasicstepsofSAaregivenasfollows.ThefirststepofSAistogenerateaninitialconfiguration(eitherrandomlyor
heuristicallyconstructed)andinitializetheso-calledtemperatureparameterT.Thenthefollowingisrepeateduntiltheter-
minationconditionissatisfied.Anewsolutionfromtheneighborhoodofthecurrentsolutionisselectedrandomly.Theen-
ergycostofthenewsolutioniscalculatedovertheobjectivefunctionandTisdecremented.Let/and/0denotethecurrent
andnewenergycosts,respectively.If/0</,thenewsolutionwillbeaccepted.If/0>/,thissolutionwillbeacceptedwitha
probabilityfromaBoltzmanndistributionfunctione(cid:2)ð/0(cid:2)/Þ=kT,wherekistheBoltzmannconstant.Weassumek=1.Thiskind
of‘‘uphill’’moveswillbefewerasSAproceeds,becauseTdecreasesaftereachiteration.
4256 C.-C.Linetal./InformationSciences181(2011)4253–4272
2.2.Thegraphdrawingalgorithmusingsimulatedannealing
TheinputoftheDHalgorithmisagraphG=(V,E),whereVisthesetofnodesandEisthesetofedges.EachnodeinVhas
auniqueindex.ThekeydesigninSAistheenergycostfunction,onwhichtheefficiencyofthealgorithmhighlydepends.A
goodenergycostfunctionshouldreflectthepropertiesofnice-lookingdrawings.ThecriteriausedbytheDHalgorithminthe
designoftheenergycostfunctionaregivenasfollows,inwhichk ,...,k arenormalizingfactorsthatdefinetherelative
1 5
importanceofeachcriterioncomparedtotheothers.
1. Nodedistribution:"(i,j)2V(cid:3)V,a ¼k =d2 isaddedsothatnodesarespreadevenly,whered isthedistancebetween
i;j 1 i;j i,j
nodesiandj.
2. Borderlines:"i2V, m ¼k ð1=r2þ1=l2þ1=t2þ1=b2Þisadded so thatnodesarepreventedfrombeingtooclosetothe
i 2 i i i i
borderlines,wherer,l,t,b standforthedistancesbetweennodeiandtheright,left,topandbottomsidesofthedrawing
i i i i
frame,respectively.
3. Edgelengths:TheDHalgorithmusesc ¼k d2,whered isthelengthofedgek.Instead,ouralgorithmusesr=k s2,where
k 3 k k 3
s is the standard deviationof edge lengths, becausethe utilizationof c tends to producedrawingswith shorteredge
k
lengthswhichmaynotbedesirableinsomecasesofpreservingthementalmap.
4. Edgecrossings:"(k,l)2E(cid:3)E,wesimplyaddtheterme whichequalsk f f ifedgeskandlcross;e =0otherwise,where
k,l 4kl k,l
f (resp.,f)istheweightattributedtoedgek(resp.,l),representingtheimportanceofedgek(resp.,l)notintersecting
k l
otheredges.Notethatk ¼k =g2 whereg isthegivenminimumdistancebetweennodesandedges.
4 5 min min
5. Node-edgedistances:Fornodeiandedgekwithdistanceg ,thetermh ¼k =g2 isadded.
i,k i;k 5 i;k
Inlightoftheabove,theenergycost/ofadrawingofgraphG=(V,E)iscomputedasfollows:
/¼ X a þXm þrþ X e þ X h ;
i;j i k;l i;k
ði;jÞ2V(cid:3)V;i–j i2V ðk;lÞ2E(cid:3)E;k–l;kRNðlÞ ði;kÞ2V(cid:3)E;kRNðiÞ
whereweslightlyabusethenotationN((cid:4))withN(l)denotingthesetofneighboringedgesofedgel,andN(i)representingthe
setoftheedgesincidenttonodei.Infact,onlyanodemovesfromtheoriginalconfigurationtothenewconfigurationineach
stepofthealgorithm.Fornotationalconvenience,thenodeintheoriginal(resp.,new)configurationiscalledp(resp.,p0).If
thecalculationofacomponenttakesp(resp.,p0)intoaccount,itimpliesthatthecomponentbelongstotheenergycostfunc-
tionoftheoriginal(resp.,new)configuration.Therefore,thedifferenceoftheenergycostsbetweentheoriginalandthenew
configurations can be computed more efficiently as follows: /0(cid:2)/¼Pðp;jÞ2NðpÞðap0;j(cid:2)ap;jÞþðmp0(cid:2)mpÞþðr0(cid:2)rÞþ
ðPðk;lÞ2Nðp0Þ(cid:3)E;k–lek;l(cid:2)Pðk;lÞ2NðpÞ(cid:3)E;k–lek;lÞþPk2E;kRNðpÞðhp0;k(cid:2)hp;kÞ, where r0 is the variance of the edge lengths in the new
configuration.
3.Ourmental-map-preservinggraphdrawingalgorithm
TheworkofBridgemanandTamassia[5]canberegardedasthemostcompleteworkonmentalmapmetrics.Theypre-
sented the results of a user study based on several similarity measures between orthogonal drawings, which reflect the
user’smentalmapaccordingtotheuser’sdirectresponses.Asaresult,ourapproachtothementalmappreservationprob-
lemistoincorporatesomeofthebestmeasuresusedin[5]forstraight-linedrawingsintotheDHalgorithm.Inwhatfollows,
weintroducethemaincomponentsofouralgorithm.
3.1.Configuration
GivenagraphG=(V,E),aconfigurationspecifiestheplacementofnodesinVonaplane,assumingthatedgesaredrawnas
straight-linesegments,andGisdrawnwithinagivenrectangularframe.Ifthereisnoaprioriinformationaboutthenodes,
weplacethenodesrandomlywithinthedrawingframe.TheneighborhoodofaconfigurationCisaconfigurationthatisob-
tainedfromCbychangingthelocationofacertainnode.Ineachiteration,wechooseanodearbitrarilyandthenassignita
newrandomlocationwithinacircleofdecreasingradiusarounditsoriginallocation.Thesettingofadecreasingradiusmay
acceleratetheconvergenceofthealgorithm[6].
3.2.Thementalmapcostfunction
Inordertopreservethementalmap,wedefinethementalmapcostMbetweentheoriginaldrawingDandthemodified
drawingD0toreflectthesimilarityofdrawingsbasedoncertain’’components’’usedintheexperimentsreportedin[5].Note
thatthevalueofeachcomponentinMisnonnegative,andhenceM=0impliesthatthetwodrawingsareviewedasthesame
drawings.AlargermagnitudeofMmeansahigherdegreeofdifferencebetweenthetwodrawings.
Notethatallthemeasuresproposedin[5]exceptthe‘‘Shape’’measurearedefinedintermsofpointsets,whicharede-
rivedfromtheedgesandnodesofthegraphinsteadoftheedgesandnodesthemselves.Eachpointintheoriginaldrawing
hasacorrespondingpointinthemodifieddrawing.Althoughthereareavarietyofwaystoselectthepointsetsasindicated
C.-C.Linetal./InformationSciences181(2011)4253–4272 4257
in[5,27],inthispaperwechoosethepointsetincludingallthenodesofthegraph,becausethereisatmostanodemovingin
eachtrialofourSAalgorithm–itsufficestocomputeonlythevaluesrelatedtothemovednode.
Weadoptthenotationsoriginallyusedin[5]inoursubsequentdiscussion.PandP0alwaysrefertothepointsetsofdraw-
ingsDandD0,respectively.SinceonlyonepointmovesineachtrialofourSAalgorithm,weletpdenotethemovedpoint,and
p02P0isthecorrespondingpointofp(andviceversa).ThenumberofnodesinVisn.Recallthatd denotestheEuclidean
p,q
distancebetweenpointspandq.
Inthefollowing,wedefinecomponentsusedinM,inwhichg ,...,g arethenormalizingfactorsthatdefinetherelative
1 6
importanceofacriterioncomparedtotheothers.
1. Ranking:Consideringthattherelativehorizontalandverticalpositionsofanodeshouldnotvarytoomuch,weaddtoM
thecomponent
g
rankðP;P0Þ¼ 1 minfjrightðpÞ(cid:2)rightðp0ÞjþjaboveðpÞ(cid:2)aboveðp0Þj;UBg;
UB
whereUB=1.5(jPj(cid:2)1);right(p)andabove(p)arethenumbersofpointstotherightandaboveofp,respectively.Notethat
theupperboundUBissetto1.5(jPj(cid:2)1)ratherthan2(jPj(cid:2)1)becausetheprobabilityunderwhichtheactualmaximum
valuehappensisverysmall.
2. k-matrix: Two point sets P and P0 have the same order type if, for every triple of points (p,q,r), they are oriented
counterclockwiseifandonlyif(p0,q0,r0)arealsoorientedcounterclockwise.Thek-matrixisdesignedtopreserveanorder
typeofpoints.Hereweuseasimplifiedversion,whichcalculatesthenumberofnodestotheleftoftwodesignatednodes.
Letk(p,q)bethenumberofpointsinPtotheleftofthedirectedlinefromptoq.WeaddtoMtheterm
g X
lambdaðP;P0Þ¼ 2 jkðp;qÞ(cid:2)kðp0;qÞj;
UB
q2P;q–p
whereUB¼njðn(cid:2)1Þ2k.
2
3. Unweighted nearestneighbor within: This componentcapturesthe intuitive idea thata node’s nearest neighborshould
remainitsclosestneighborafterthegraphisredrawn.Letnn(p)bethenearestneighborofpinPandnn(p)0bethecor-
respondingnodeinP0 tonn(p).Thecomponentisdefinedas:
nnwðP;P0Þ¼ g3 XminfjnearerðpÞj;1g;
UB
p2P
whereUB¼jPj;nearerðpÞ¼fqjdp0;q<dp0;nnðpÞ0;q2P;q–p;q–nnðpÞg.
4. Weightednearestneighborbetween:Thisconceptisbasedontheassumptionthatanodeshouldnotmovetoofarfromthe
originalposition,andhenceitsnewpositionshouldbeasclosetoitsoriginalpositionaspossible.Asthenewdrawingis
presented,theuseroughttobeabletofindacertainnodewhichisrecordedinhis/hermentalmap.Thecomponentis
definedas:
g X
nnbðP;P0Þ¼ 4 jnearerðpÞj;
UB
p2P
whereUB¼jPjðjPj(cid:2)1Þ;nearerðpÞ¼fqjdp;q<dp;p0;q2P;q–pg.
5. Shape:Thiscomponentrecordstheedgeorientationofthegraph.Notethattheoriginalmeasurein[5]wasdesignedfor
orthogonaldrawings.Tocopewithstraight-linedrawings,themeasureismodifiedasfollows:
g X
shape¼ 5 editsðe;e0Þ;
UB
e2E
whereUB=jEjand
(cid:2)1; if theincludedanglebetweene and e0 issmall;
editsðe;e0Þ¼
0; otherwise;
inwhiche0isthecorrespondingedgeofeinD0.Notethatwhethertheincludedanglebetweentwoedgesissmallisdeter-
minedbyusers.
6. Average relative distance: The average relative distance is the average of the distances from p to the remaining nodes.
Therefore,weaddtheterm
rdistðP;P0Þ¼nðng(cid:2)6 1Þ Xjdp;q(cid:2)dp0;qj:
p;q2P
Althougheightbestsimilaritymeasuresfororthogonaldrawingsweresuggestedin[5],onlysixofthemareincludedin
ourmentalmapcostfunction.Thetwomeasuresdesignedforpreservingtheorthogonalordering(calledOrthogonalOrdering
4258 C.-C.Linetal./InformationSciences181(2011)4253–4272
in[5])ofnodesandminimizingthemovementdistanceoftheselectednode(calledAverageDistancein[5])areexcludedin
ourworkforthereasonsgivenbelow.First,ifsomenewnodesandedgesarearbitrarilyaddedtoadrawingcausingnew
edgecrossings,thenthosenewedgecrossingscannotbeeliminatedbyanalgorithmwhichpreservestheorthogonalorder-
ingofnodes.Also,minimizingthenumberofedgecrossingsisusuallyregardedasthemostimportantaestheticcriterionin
graphdrawing[29].Therefore,weexcludethemeasureofpreservingtheorthogonalorderingofnodes.Asourexperimental
resultsindicate,ahighdegreeoforthogonalorderofnodesisstillpreservedwithouttakingthismeasureintoaccount.On
theotherhand,themeasureofminimizingthemovementdistanceoftheselectednodeisexcludedbecauseithasbeenim-
pliedbytheselectionoftheneighborhoodofaconfigurationintheSAalgorithm.Inaddition,itcanalsobeviewedasbeing
achievedinpartbytheedgelengthcomponentintheenergycostfunction.Finally,theexclusionofthetwomeasuresresults
inamoreefficientSAalgorithm,asexpected.
3.3.Ouralgorithm
Ouralgorithm,describedinAlgorithm1,isalmostthesameastheDHalgorithmexcepttheifconditionatstep2(b)is
replaced by the conjunction of the original condition and M0< (cid:2) , where M0 is the value of the mental map function at
M
thenewdrawingD0 and(cid:2) isthemaximaltoleranceofthementalmapcostthatacceptsthenewdrawingD0.
M
Algorithm1SA_MENTAL_MAP
1:denotetheinputdrawingasDandsetaninitialtemperatureT ;
0
2:fori-thstage,denotethetemperatureasT,andrepeatthefollowingfor30jVjtrials:
i
(a) randomlyselectapointp2P(notethatPisthepointsetofD);selectanewpositionp0 inadecreasing-radius
circlearoundp;letD0 beanewdrawingwhichisalmostthesameasDexceptpisreplacedbyp0;
(b) let/and/0bethevaluesoftheenergycostfunctionatDandD0respectively;letM0bethevalueofthemental
mapfunctionatD0;if(/0(cid:2)/<0orrandom<e(cid:2)ð/0(cid:2)/Þ=Ti)and(M0<(cid:2)M),thensetD D0;
3:T =cT;
i+1 i
4:iftheterminationconditionissatisfied,stop;otherwisegotostep2.
TheenergycostandthementalmapcostareconsideredindependentlyinAlgorithm1,becausetheyservefordifferent
purposes.Theformerisusedfordrawinggraphnicely,whereasthelatterisusedforpreservingthementalmap.Eachcom-
ponentinthementalmapcostcanbenormalizedwiththeupperboundofvalue1exceptforthe‘Averagerelativedistance’
component(whichisofsmallvaluewhilepreservingthementalmap),andhence,itiseasytocontroltherelativeweightsof
thesecomponentstopreservethementalmap.However,thevaluesofthecomponentsintheenergycostaresodiversethat
itisdifficulttoadjusttheirweightstogenerateanicedrawingwhenthealgorithmconverges.Consolidatingthetwocosts
withdifferentscalesintoonedoesnothelp,sinceitbecomesmoredifficulttoadjusttheweightstoreachconvergence,and
moreimportantly,thegoalsassociatedwiththetwocostsarelikelytobeinconflictwitheachother.Asaresult,theenergy
costandthementalmapcostareconsideredindependentlyinAlgorithm1.
AlsonoteinAlgorithm1thattheSAschemeisonlyappliedtotheenergycost,becauseitwouldbedifficulttoachieve
convergence if applyingto the two costs simultaneously. In addition,we believe that in most cases a user prefers a nice
drawingwithareasonablepreservationofthementalmapthanabaddrawingwithawell-preservedmentalmap.Conse-
quentlyweapplytheSAschemeonlytotheenergycost.
ThecoolingscheduleusedinAlgorithm1followsthebasicgeometricscheduledescribedin[23].
1. Initialtemperature:Wesettheinitialtemperaturehighenoughinorderforanyconfigurationtoworkasthestartingpoint
ofthealgorithm.Alowerinitialtemperaturecanbechoseninthebeginning,iftheinitialconfigurationisknowntobear
someresemblancetothedesiredfinalresult.
2. Temperaturereduction:Thetemperaturereductionstagedetermineswhentochangethetemperatureandhowtochange
it.Thisstageaffectsthenumberofmovesateachtemperatureandtherangeofeachmove.Typicalapplicationssetthe
numberoftrialsateachtemperaturetobepolynomialinthesizeoftheinput.Inoursetting,thenumberofmovesateach
temperatureissetto30jVj,suggestedbytheDHalgorithm.Thecoolingscheduleruleisgeometric.IfT denotesthetem-
i
peratureatthei-thstage,thenthetemperatureatthenextstageisT =cT,wherecisarealnumberbetween0.6and
i+1 i
0.95.SimilartotheDHalgorithm,weletc=0.75.Therangeofeachmovedecreaseswhenthetemperaturereduces.Ifthe
rangeofmovementatthei-thstageisR,thentherangeofmovementservingforthenexttemperatureiscR,whichaccel-
eratestheconvergenceofthealgorithm.
3. Termination condition: This is to decide when to stop the algorithm. Commonly used termination conditions include:
stopping the algorithm when the solution is not modified after several consecutive stages, or fixing the number of
stages to be a constant. To carry out a fair comparison between our algorithm and the DH algorithm, the number
ofstagesinourexperimentsissettoten,whichisidenticaltothatusedin[6].Experimentalresultshaveshownthat,
in many cases, substantial changes only happen in the early stages; subsequent stages yield only marginal
improvement.
C.-C.Linetal./InformationSciences181(2011)4253–4272 4259
Nowweturnourattentiontothetimecomplexityofouralgorithm.AsouralgorithmisbuiltupontheDHalgorithm,it
sufficestoconsiderthetimecomplexityoftheDHalgorithm.Asindicatedin[6],theDHalgorithmrunsinO(jVj2jEj)time
becausethenumberofstagesisaconstant,thenumberoftrialsineachstageis30jVj,andeverytermintheenergycost
functioncanbecomputedinO(jVjjEj)time.Also,itisobviousthateverycomponentofthementalmapcostcanbecomputed
intimenomorethanO(jVjjEj).Hence,Algorithm1alsorunsinO(jVj2jEj)time.
4.Experimentalresults
Thissectionshowsfourexperimentalresultsforevaluatingtheperformanceofourmental-map-preservinggraphdraw-
ingalgorithm.Ineachexperiment,theinitialdrawingoftheinputgraphisarandomdrawing,andthenanicedrawingis
producedbytheDHalgorithm.Subsequently,liketheexperimentdesignedin[5],thegraphismodifiedbyaddingtwonodes
with two and four incident edges, respectively. The locations of the two nodes are decided randomly, and the other end
pointsofthesixedgesareconnectedtoarbitrarynodesintheoriginalgraph.Finally,wecomparethedrawingsofthemod-
ifiedgraphproducedbyouralgorithmandtheDHalgorithm.
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runningtimeoftheDHalgorithmandouralgorithmwithrespecttofourexamplesaregiveninTable1.Ouralgorithmrunsa
bitslowerthantheDHalgorithmsincemorecriteriaaretakenintoaccount.Liketheconclusiongivenin[6]regardingtheDH
algorithm,ouralgorithm,fromapracticalviewpoint,isonlysuitablefordrawingdynamicgraphsofsmall-size.
Inordertoseehowtherunningtimeofouralgorithmincreaseswhenthesizeoftheinputgraphgrows,weconsidersev-
eralgraphstakenfromthewell-knowngraphdataset–RomeGraphs,1whichareoriginatedfrompracticalapplicationsranging
fromdataflowdiagramstoPetrinets.ThelargestgraphintheRomeGraphdatasethas110nodes.Weconductexperimentson
thefirst100graphsintheRomeGraphset,andrecordtherunningtimeofredrawingthemodifiedgraphofeachofthe100
graphs.Tothisend,weaddtwonodes,onewithtwoedgesandtheotherwithfouredges,tothenicedrawingofeachgraph.
TheaveragerunningtimeversusthenumberofnodesisplottedinFig.3.Itturnsoutthatittakesabout25storedrawa100-
nodegraph,whichissomewhatunacceptableforapplicationsthatrequirereal-timeresponses.
ConsiderthefirstsimpleexperimentshowninFig.4.AninitialrandomdrawingisgiveninFig.4(a).AfterapplyingtheDH
algorithmunderdifferentsettingsofparameterk ,threepossiblenice-lookingdrawingsaredisplayedinFig.4(b),inwhich
4
(i)–(iii)illustratehowlesspenaltiesonedgecrossingsaffecttheoutcomesofthealgorithm.Nowwearbitrarilyaddtwonew
nodes7and8toeachofthethreedrawings,wherenode7isconnectedtonodes4and5,andnode8isconnectedtonodes1,
3,4,and5(seeFig.4(c)).Themodifiedgraphsareredrawnbyouralgorithm,asshowninFig.4(d).Forthesakeofcompar-
ison,considerFig.4(e)and(f).Fig.4(e)showsthethreedrawingsobtainedbyapplyingtheDHalgorithmtodrawingsin(i)–
(iii)inFig.4(c)directly.Ontheotherhand,Fig.4(f)displaystheoutcomesofapplyingtheDHalgorithmtorandomdrawings
ofgraphsassociatedwith(i)–(iii)inFig.4(c)underthesamesettingofparameters.
InviewofFig.4(c)and(d),itisobviousthatouralgorithmiscapableofgeneratingnicedrawings.Itisalsonothardtosee
fromFig.4(b)and(d)thatthecontoursofdrawingsandtherelativepositionsofnodesareverysimilarinthetwosetsof
drawings,suggestingthatouralgorithmpreservesahighdegreeofthementalmap.Incontrast,thedrawingsinFig.4(e)
and(f)producedbytheDHalgorithmsharelittlesimilaritywiththoseinFig.4(b).
Ifweremovetheadditionalnodes7and8andtheiradjacentedgesfromthedrawingsinFig.4(c)–(f),itbecomesclearer
whythedrawingsinFig.4(d)preservebettermentalmaps.SeeFig.5.Sincethementalmapcriteriaandtheaestheticcriteria
areofteninconflictwithoneanother,thereisatrade-offbetweenpreservingahighdegreeofthementalmapandachieving
someaestheticcriteriasuchasminimizingthenumberofedgecrossings.Fortunately,ouralgorithmallowstheusertoadjust
the weights between different criteria, in order to fit his/her drawing requirements. For example, node 5 in Fig. 5(b) (i)
(whichisproducedbyouralgorithm)doesnotpreserveitsoriginalrelativeposition(i.e.,notsatisfiableasfaraspreserving
thementalmapisconcerned).ConsiderFig.4(c)(i).Tokeepnode5inthesamerelativeposition(bychangingthesettingsof
parameters),node8cannotmoveintothequadrangleboundedbynodes1,2,3,and6,whichinturn,inducesadditionaledge
crossings,thoughnode8,inthiscase,keepsitsoriginalrelativepositionintact.
Inthesecondexperiment,theinitialgraphFig.6(b)isacompletedegree-fourtreewith21nodesand20edges.Takingthe
initialrandomdrawingshowninFig.6(a)astheinput,theDHalgorithmoutputsthedrawinginFig.6(b).Fig.6(c)showsthe
graphtransformedbyinsertingnodes22and23(togetherwithsixedges)intothetree.Fig.6(d)isthedrawingproducedby
applyingouralgorithmtothedrawinginFig.6(c).WecanobservethattherearenodrasticdifferencesbetweenFig.6(b)and
(d).Fig.6(e)(resp.,Fig.6(f))isthedrawingproducedbyapplyingtheDHalgorithmtothedrawinginFig.6(c)(resp.,aran-
domdrawingofthemodifiedgraph).AlthoughthedrawinginFig.6(f)almosthasnocommonpointswiththedrawingin
Fig.6(b),thereareonlytwoedgecrossingsinthedrawinginFig.6(f),whichachievestheminimumnumberofedgecrossings
amongallpossibledrawingsofthemodifiedgraph.
Fig.7(a)–(d)simplydisplaythedrawingsafterremovingtheadditionalnodes22and23fromFig.6(c)–(f),respectively.By
comparingFig.7(b)withFig.7(a),itcanbeobservedthatthedrawinginFig.7(b)almostpreservestheshapeandstructureof
thedrawinginFig.7(a)exceptthattheorderingsofnodepairs(6,7)and(15,16)aredifferent.Asmentionedearlier,preserv-
ingthementalmapandoptimizingsomeaestheticcriteriamaynotbeachievedsimultaneously.Thereasonwhytheorder-
1 http://www.dia.uniroma3.it/(cid:5)gdt/.
4260 C.-C.Linetal./InformationSciences181(2011)4253–4272
Table1
ThestatisticsonthetotalrunningtimeoftheDHandouralgorithms.
Problem jVj jEj DH our
Fig.4(c) 8 15 0.1453s 0.1781s
Fig.6(c) 23 26 0.7206s 0.9837s
Fig.8(c) 65 68 5.7500s 8.5030s
Fig.10(c) 102 130 19.3353s 28.1252s
35
ec.) 30
me (s 25
ng ti 20
ni
un 15
e r
ag 10
er
av 5
0
0 20 40 60 80 100 120
number of nodes
Fig.3. Plotofaveragerunningtimeversusnumberofnodes.
ingsofthetwopairsofnodesaredifferentisthatthedrawinginFig.7(b)correspondstothedrawinginFig.6(d)whichin-
ducesfeweredgecrossingsthanthecasewhentheorderingsarenotmodified.IfwecompareFig.7(c)withFig.7(a)assum-
ingnode1tobetherootofthetree,thesubtreesrootedatnodes2,3,and4lookdifferentalthoughthecounterclockwise
orderingofthechildrenofnode1ispreserved.BycomparingFig.7(d)withFig.7(a),itisobviousthatthementalmapisnot
preserved.
Wealsocarryoutanexperimentonacompletebinarytreewith63nodesand62edges.Similartothepreviousexper-
iments,Fig.8(a)–(f)showarandomdrawingoftheinitialgraph,thenicedrawingproducedbytheDHalgorithm,thedraw-
ing ofthe modifiedgraphby addingtwo nodes, themodifieddrawingproducedby our algorithm,themodifieddrawing
producedbytheDHalgorithm,andthedrawingproducedbyapplyingtheDHalgorithmtoarandomdrawingofthemod-
ified graph, respectively. Fig. 9(a)–(d) show the drawings after the two added nodes are removed from Fig. 8(c)–(f),
respectively.
IfwelookatFig.8(b)and(d),thereareonlyslightmodificationsbetweenthetwodrawings,asonlynodepairs(7,15),
(32,33),(36,37),(40,41),(58,59),and(60,61)areofdifferentorderings.Thisismainlycausedbythefactthatthemeasure
ofpreservingtheorthogonalorderingofnodesisexcludedinourmentalmapcostfunction.If,ontheotherhand,sucha
measureisincluded,thenadditionaledgecrossingsbecomeinevitable,causingtheoutputtobelessattractivefromtheaes-
theticviewpoint.Incontrast,althoughthedrawinginFig.8(e)hasfeweredgecrossingsthanthedrawingproducedbyour
algorithm,itdiffersalotincomparisonwiththeoriginaldrawingillustratedinFig.8(b),especially,inthelefthalfofthe
drawing.Assuming node 1tobe theroot of thetree, the subtreesrootedat nodes8, 9,14, 20, and 26in thedrawing of
Fig.8(a)aretotallydifferentfromthoseinthedrawingofFig.8(e).AsforFig.8(f),thedrawinghasonlyoneedgecrossing,
whichachievestheminimumnumberofedgecrossingsamongallthepossibledrawingsofthemodifiedgraph.FromFig.9,
theadvantageofourdrawingalgorithmbecomesapparentwhentheprimaryconcernisplacedonthepreservationofthe
mentalmap.
Finally,wepresentanexperimentcarriedoutonagraphwith100nodesand124edgeschosenfromtheRomeGraphdata
set.Similartothepreviousexperiments,Fig.10(a)–(f)showarandomdrawingoftheinputgraph,thenicedrawingproduced
bytheDHalgorithm,thedrawingofthemodifiedgraphbyaddingtwonodesandsixedges,themodifieddrawingproduced
byouralgorithm,themodifieddrawingproducedbytheDHalgorithm,andthedrawingproducedbyapplyingtheDHalgo-
rithmtoarandomdrawingofthemodifiedgraph,respectively.
AlthoughthedrawingsinFig.10aremorecomplicated,thereisacommonfeaturein(b)–(e)thatmostnodesarecentral-
izedinthebottomrightpartofthewholearea.BycomparingFig.10(b)withFig.10(d)and(e),itisclearthatdrawing(d)by
ouralgorithmpreservesthecontouroftheoriginaldrawing(b),butdrawing(e)bytheDHalgorithmdoesnothavethis
property.
Figs.11and12showhowourSAalgorithmiscapableofescapingfromlocalminima.Fig.11givestheplotoftheenergy
costversusthenumberoftrialsintheprocessofapplyingtheDHalgorithmtoFig.8(c)toyieldFig.8(e).RecallfromAlgo-
rithm1thatthenumberoftrialsisequalto30jVjmultipliedbythenumberofexecutedstagesplusthenumberoftrials
executedatthecurrentstage.Fig.11illustratesthatsome‘uphill’movescansuccessfullyescapefromlocalminimabyusing
theDHalgorithm.Toseethis,considertheenergycostsinthewindowbetweenthe16500-thtrialandthe18000-thtrialin
C.-C.Linetal./InformationSciences181(2011)4253–4272 4261
2 3
2
1 1 2 6 1
4
4 2 6
5
6 5 4
6 3 1 3 4
3
(i) (ii) (iii)
5 5
(a) the initial drawing.(b) applying the DH algorithm to (a) under different settings of parameters.
3
8 8
8
2 1 2
6
1 4
7 2 7 67
5
4
6 5 3 1 4
3
(i) (ii) (iii)
5
(c) adding nodes 7 and 8 to each of the three drawings in (b).
3
8
8
1 2 6 2
7
4 2 1 7
5 4 5
6
8
1
6 3 3
5 7 4
(i) (ii) (iii)
(d) applying our algorithm to each of the three drawings in (c).
3 7
1 2 6 4 1
6
7 2
8 5
4 5 1 4 6
2
3 5
8
8
7
(i) (ii) (iii) 3
(e) applying the DH algorithm to each of the three drawings in (c).
6
4
3 4 8 1
3 7
5 8 5 2 3
2
7 1
5 6
2
1 8 4
6 7
(i) (ii) (iii)
(f) applying the DH algorithm to a random drawing of the graph in (c).
Fig.4. Theexperimentforaninitialcubicgraphwithsixnodes.
Fig.11.Thecurveoftheenergycostbetweenthe16500-thtrialandthe17000-thtrial(‘plain’-moving)isalmostflat.How-
ever,‘downhill’-movingfollowedby‘uphill’-movingnearthe17000-thtrialyieldsabettersolution,whichillustratesthe
advantageoftheSAmethod.
4262 C.-C.Linetal./InformationSciences181(2011)4253–4272
3
2 1 2
6
1 4
2
6
5
4
6 5 3 1 4
3
(i) (ii) (iii)
5
(a) Figure 3(c) without nodes 7 and 8.
3
1 2 2
6
4 1
2
5 5
6
4
6 3 1 3
4
(i) (ii) (iii)
5
(b) Figure 3(d) without nodes 7 and 8.
1
3
1 2 4 2
6
6 5
6
4
4
5 1
2
3 5
(i) (ii) (iii) 3
(c) Figure 3(e) without nodes 7 and 8.
4
3 6 1
5 4
2 3
3
5
5 6
2 2
1 1 4
6
(i) (ii) (iii)
(d) Figure 3(f) without nodes 7 and 8.
Fig.5. Removingnodes7and8fromFig.4(c)(d)and(f).
Fig.12illustratesa‘‘costversustrialnumber’’plotintheprocessofyieldingFig.8(d)fromFig.8(c)usingouralgorithm.
Fig.12(a)and(b)givetheplotsoftheenergycost/andthementalmapcostM,respectively,againstthenumberoftrials.
Fig.12(c)and(d)givedetailedviewsofthedotted-lineboxesin(a)and(b),respectively.Notethatthetolerance(cid:2) ofMis
M
setto0.3inthisexperiment,causingtheconfigurationwiththevalueofMlargerthan0.3toberejectedinFig.12(b)and(d),
whichinturnyieldsaconsiderableamountof‘plain-moving’inFig.12(a).FromFig.12(a)and(c),weobservethatouralgo-
rithm,anSA-basedmethod,alsohastheabilitytoescapefromlocalminimainmanycases.Itshouldbenotedthatthemax-
imumenergycostinFig.11(undertheDHalgorithm)islargerthanthatinFig.12(a)(underouralgorithm)byapproximately
4billion.Suchadisparityisduetothefactthattheweightofthemeasureofpenalizingthenumberofedgecrossings(oneof
themostimportantmeasuresinproducinganicedrawing)issettoahugevalue.Inordertopreservethementalmap,our
algorithmproducesthedrawinginFig.8(d)withthreemoreedgecrossingsthanthedrawinginFig.8(e),causingtheextra4
billioninthecorrespondingenergycost.
Asouralgorithmisbasedonthestochasticsimulatedannealing(SSA)strategy,itisnotsurprisingthatouralgorithmis
somewhatslowwhenapplyingtolargegraphs.Apossiblewaytoimprovetheperformanceofouralgorithmistoconsider
usingthedeterministicsimulatedannealing(DSA)technique[11]instead,whichrunsfasterinmanyreal-worldproblemsof
largescalewithoutsacrificingtheaccuracyofthesolution.ThebasicideaofDSAistouseahyperbolictangentfunctionto
transformdigitalvaluesoftheenergycostinSSA(seeFig.11)intoanalogvalues.DesigningaDSA-basedmental-map-pre-