Label Placement in Road Maps

0.64 MB·English

by  Andreas Gemsa

Label Placement in Road Maps
AndreasGemsa,BenjaminNiedermann,andMartinNöllenburg
InstituteofTheoreticalInformatics,KarlsruheInstituteofTechnology,Germany
5
Abstract. Aroadmapcanbeinterpretedasagraphembeddedintheplane,in
1
whicheachvertexcorrespondstoaroadjunctionandeachedgetoaparticular
0
roadsection.Weconsiderthecartographicproblemtoplacenon-overlappingroad
2
labelsalongtheedgessothatasmanyroadsectionsaspossibleareidentifiedby

n theirname,i.e.,coveredbyalabel.WeshowthatthisisNP-hardingeneral,but
a theproblemcanbesolvedinpolynomialtimeiftheroadmapisanembeddedtree.
J

8
2
1 Introduction

]
G
Map labeling is a well-known cartographic problem in computational geometry [11,
C Chapter 58.3.1], [13]. Depending on the type of map features, one can distinguish
. labelingofpoints,lines,andareas.Commoncartographicqualitycriteriaarethatlabels
s
c must be disjoint and clearly identify their respective map features [7]. Most of the
[ previousworkconcernspointlabeling,whilelabelinglineandareafeaturesreceived

1 considerablylessattention.Inthispaperweaddresslabelinglinearfeatures,namely
v roadsinaroadmap.
8 Geometrically,aroadmapistherepresentationofaroadgraphGasanarrangement
8 offatcurvesintheplaneR2.EachroadisaconnectedsubgraphofG(typicallyasimple
1
path) and each edge belongs to exactly one road. Roads may intersect each other in
7
0 junctions,theverticesofG,andwedenoteanedgeconnectingtwojunctionsasaroad
. section.Inroadlabelingthetaskistoplacetheroadnamesinsidethefatcurvessothat
1
theroadsectionsareidentifiedunambiguously,seeFig.1.
0
5 Chirié[1]presentedasetofrulesandqualitycriteriaforlabelplacementinroad
1 mapsbasedoninterviewswithcartographers.Thisincludesthat(C1)labelsareplaced
v: insideandparalleltotheroadshapes,(C2)everyroadsectionbetweentwojunctions
i should be clearly identified, and (C3) no two road labels may intersect. Further, he
X
gaveamathematicaldescriptionforlabelingasingleroadandintroducedaheuristic
r
forsequentiallylabelingallroadsinthemap.Imhof’sfoundationalcartographicwork
a
onlabelpositioninginmapslistsverysimilarqualitycriteria[3].Edmondsonetal.[2]
took an algorithmic perspective on labeling a single linear feature (such as a river).
While Edmondson et al. considered non-bent labels, Wolff et al. [12] introduced an
algorithmforsinglelinearfeaturethatplaceslabelsfollowingthecurvatureofthelinear
feature.Strijk[9]consideredstaticroadlabelingwithembeddedlabelsandpresenteda
heuristicforselectingnon-overlappinglabelsoutofasetoflabelcandidates.Seibertand
Unger[8]consideredgrid-shapedroadnetworks.Theyshowedthatinthosenetworksit
isNP-completeandAPX-hardtodecidewhetherforeveryroadatleastonelabelcan
beplaced.Yet,NeyerandWagner[6]introducedapracticallyefficientalgorithmthat
2 AndreasGemsa,BenjaminNiedermann,andMartinNöllenburg
HammingSt. HammingSt. junction
edge
H
am DijkstraSt.
m junction
in roadsection vertex
(aK)nuthTuSritng.St. gSt.HammingSt. (bK)nuthSt.TuringSt. HammingSt. (c) label rveegrtuelxar
Fig.1. a–b): Two ways to label the same road network. Each road section has its own color.
Junctionsaremarkedgray.Fig.b)identifiesallroadsections.c)Illustrationoftheroadgraphand
relevantterms.
findssuchagridlabelingifpossible.MaassandDöllner[5]presentedaheuristicfor
labelingtheroadsofaninteractive3Dmapwithobjects(suchasbuildings).Apartfrom
label-labeloverlaps,theyalsoresolvelabel-objectocclusions.Vaaraniemetal.[10]used
aforce-basedlabelingalgorithmfor2Dand3Dscenesincludingroadlabelplacement.
Contribution. While in grid-shaped road networks it is sufficient to place a single
labelperroadtoclearlyidentifyallitsroadsections,thisisnotthecaseingeneralroad
networks.ConsidertheexampleinFig.1.InFig.1a),itisnotobviouswhethertheorange
roadsectioninthecenterbelongstoKnuthSt.ortoTuringSt.Simplymaximizingthe
numberofplacedlabels,asoftendoneforlabelingpointfeatures,cancauseundesired
effectslikeunnamedroadsorclumsylabelplacements(e.g.,aroundDijkstraSt.and
HammingSt.inFig.1a)).Therefore,incontrasttoSeibertandUnger[8],weaimfor
maximizingthenumberofidentifiedroadsections,i.e.,roadsectionsthatcanbeclearly
assignedtolabels;seeFig.1b).
Basedoncriteria(C1)–(C3)weintroduceanewandversatilemodelforroadlabeling
in Section 2. In Section 3 we show that the problem of maximizing the number of
identifiedroadsectionsisNP-hardforgeneralroadgraphs,evenifeachroadisapath.
Forthespecialcasethattheroadgraphisatree,wepresentapolynomial-timealgorithm
inSection4.Thisspecialcaseisnotonlyoftheoreticalinterest,butouralgorithminfact
providesaveryusefulsubroutineinexactorheuristicalgorithmsforlabelinggeneral
roadgraphs.Ourinitialexperiments,sketchedinSection5,showthatreal-worldroad
networksdecomposeintosmallsubgraphs,alargefractionofwhich(morethan85.1%)
areactuallytrees,andthuscanbelabeledoptimallybyouralgorithm.
2 Preliminaries
Asarguedabove,aroadmapisacollectionoffatcurvesintheplane,eachrepresenting
aparticularpieceofanamedroad.Iftwo(ormore)suchcurvesintersect,theyform
junctions.Aroadlabelisagainafatcurve(theboundingshapeoftheroadname)thatis
containedinandparalleltothefatcurverepresentingitsroad.Weobservethatlabelsof
differentroadscanintersectonlywithinjunctionsandthattheactualwidthofthecurves
LabelPlacementinRoadMaps 3
isirrelevant,exceptfordefiningtheshapeandsizeofthejunctions.Theseobservations
allowustodefinethefollowingmoreabstractbutequivalentroadmapmodel.
Aroadmap isaplanarroadgraphG=(V,E)togetherwithaplanarembedding
M
E(G),whichcanbethoughtofasthegeometricrepresentationoftheroadaxesasthin
curves;seeFig1c).WedenotethenumberofverticesofGbyn,andthenumberof
edgesbym.ObservethatsinceGisplanarm=O(n).Eachedgee Eiseitheraroad
∈
section,whichisnotpartofajunction,orajunctionedge,whichispartofajunction.
Eachvertexv V iseitherajunctionvertexincidentonlytojunctionedges,oraregular
∈
vertexincidenttooneroadsectionandatmostonejunctionedge,whichimpliesthat
eachregularvertexhasdegreeatmosttwo.Ajunctionvertexvanditsincidentedges
are denoted as a junction. The edge set E decomposes into a set of edge-disjoint
R
roads,whereeachroadR inducesaconnectedsubgraphofG.Withoutlossof
∈ R
generalityweassumenotworoadsectionsGareincidenttothesamevertex.Thus,a
roaddecomposesintoroadsections,separatedbyjunctionverticesandtheirincident
junctionedges.Inrealisticroadnetworksthenumberofroadsconnectedpassingthrough
ajunctionissmallanddoesnotdependonthesizeoftheroadnetwork.Wetherefore
assumethateachvertexinGhasconstantdegree.WeassumethateachroadR has
∈R
anamewhoselengthwedenotebyλ(R).
Forsimplicity,weidentifytheembeddingE(G)withthepointsintheplanecovered
byE(G),i.e.E(G) R2.WealsouseE(v),E(e),andE(R)todenotetheembeddings
⊆
ofavertexv,anedgee,andaroadR.
Wemodelalabelasasimpleopencurve(cid:96): [0,1] E(G)inE(G).Unlessmen-
→
tioned otherwise, we consider a curve (cid:96) always to be simple and open, i.e., (cid:96) has no
self-intersectionsanditsendpointsdonotcoincide.Inordertoeasethedescription,we
identifyacurve(cid:96)inE(G)withitsimage,i.e.,(cid:96)denotestheset (cid:96)(t) E(G) t [0,1] .
{ ∈ | ∈ }
Thestartpointof(cid:96)isdenotedasthehead h((cid:96))andtheendpointasthetailt((cid:96)).The
lengthof(cid:96)isdenotedbylen((cid:96)).Thecurve(cid:96)identifiesaroadsectionrif(cid:96) E(r)= .
∩ (cid:54) ∅
Foraset ofcurvesω( )isthenumberofroadsectionsthatareidentifiedbythecurves
L L
in .Forasinglecurve(cid:96)weuseω((cid:96))insteadofω( (cid:96) ).Fortwocurves(cid:96) and(cid:96) itis
1 2
L { }
notnecessarilytruethatω( (cid:96) ,(cid:96) ) = ω((cid:96) )+ω((cid:96) ),becausetheymayidentifythe
1 2 1 2
{ }
sameroadsectiontwice.
Alabel(cid:96)foraroadRisacurve(cid:96) E(R)oflengthλ(R)whoseendpointsmustlie
⊆
onroadsectionsandnotonjunctionedgesorjunctionvertices.Requiringthatlabelsend
onroadsectionsavoidsambiguousplacementoflabelsinjunctionswhereitisunclear
howtheroadpassesthroughit.Alabeling foraroadmapwithroadset isasetof
L R
mutuallynon-overlappinglabels,wherewesaythattwolabels(cid:96)and(cid:96) overlapifthey
(cid:48)
intersectinapointthatisnottheirrespectiveheadortail.
Followingthecartographicqualitycriteria(C1)–(C3),ourgoalistofindalabeling
L
thatmaximizesthenumberofidentifiedroadsections,i.e.,foranylabeling wehave
(cid:48)
L
ω( (cid:48)) ω( ).WecallthisproblemMAXIDENTIFIEDROADS.
L ≤ L
Note that assuming the road graph G to be planar is not a restriction in practice.
Considerforexamplearoadsectionrthatoverpassesanotherroadsectionr ,i.e.,risa
(cid:48)
bridgeoverr ,orr isatunnelunderneathr.Inordertoavoidoverlapsbetweenlabels
(cid:48) (cid:48)
placedonrandr ,weeithercanmodeltheintersectionofrandr asaregularcrossing
(cid:48) (cid:48)
oftworoadsorwesplitr insmallerroadsectionsthatdonotcrossr.Inbothcases
(cid:48)
4 AndreasGemsa,BenjaminNiedermann,andMartinNöllenburg
truefalse
x2∨x4∨x3 x4∨x1∨x5
true
x2 x3 F F x4 x5 x1 false false (c)Chain
F
x¯2∨x¯1∨x¯3 x¯3∨x¯5∨x¯4
false (d)Fork
(a)SketchofMϕ (b)Clause
Fig.2.IllustrationofNP-hardnessproof.(a)3-Satformulaϕ=(x x x ) (x x
4 1 5 2 4
∨ ∨ ∧ ∨ ∨
x ) (x¯ x¯ x¯ ) (x¯ x¯ x¯ )representedasroadgraph .Truthassignmentis
3 2 1 3 3 5 4 ϕ
∧ ∨ ∨ ∧ ∨ ∨ M
x =true,x =true,x =false,x =falseandx =false.(b)Clausegadgetintwostates.
1 2 3 4 5
(c)Thechainisthebasicbuildingblockfortheproof.(d)Schematizedforkgadget.
thecorrespondingroadgraphbecomesplanar.Inthelattercasewemayobtainmore
independentroadscreatedbychoppingr intosmallerpieces.
(cid:48)
3 ComputationalComplexity
WefirststudythecomputationalcomplexityofroadlabelingandproveNP-hardnessof
MAXIDENTIFIEDROADSinthefollowingsense.
Theorem1. Foragivenroadmap andanintegerK itisNP-hardtodecideifin
M
totalatleastK roadsectionscanbeidentified.
Proof. We perform a reduction from the NP-complete PLANAR MONOTONE 3-SAT
problem[4].AninstanceofPLANARMONOTONE3-SATisaBooleanformulaϕwithn
variablesandmclauses(disjunctionsofatmostthreeliterals)thatsatisfiesthefollowing
additionalrequirements:(i)ϕismonotone,i.e.,everyclausecontainseitheronlypositive
literalsoronlynegativeliteralsand(ii)theinducedvariable-clausegraphH ofϕis
ϕ
planarandcanbeembeddedintheplanewithallvariableverticesonahorizontalline,
allpositiveclauseverticesononesideoftheline,allnegativeclausesontheotherside
oftheline,andtheedgesdrawnasrectilinearcurvesconnectingclausesandcontained
variablesontheirrespectivesideoftheline.Weconstructaroadmap thatmimics
ϕ
M
theshapeoftheaboveembeddingofH bydefiningvariableandclausegadgets,which
ϕ
simulatetheassignmentoftruthvaluestovariablesandtheevaluationoftheclauses.
WerefertoFig.2forasketchoftheconstruction.
ChainGadget.Thebasicbuildingblockisthechaingadget,whichconsistsofan
alternatingsequenceofequallylonghorizontalandverticalroadswithidenticallabel
lengthsthatintersecttheirrespectiveneighborsinthesequenceandformjunctionswith
LabelPlacementinRoadMaps 5
r r
1 2
g g
1 2
r
r4 false false true true
5
r
6
r
3
g false true
3
(a) (b) (c)
Fig.3.Illustrationoftheforkgadget.(a)Structureoftheforkgadget.(c)Configurationtransmitting
thevaluefalse.(b)Configurationtransmittingthevaluetrue.
them as indicated in Fig. 2c). Assume that the chain consists of k 3 roads. Then
≥
eachroadexceptthefirstandlastonedecomposesintothreeroadsectionssplitbytwo
junctions, a longer central section and two short end sections; the first and last road
consistofonlytworoadsections,ashortoneandalongone,separatedbyonejunction.
(Thesetworoadswilllaterbeconnectedtoothergadgets;indicatedbydottedsquaresin
Fig.2c).)Thelabellengthanddistancebetweenjunctionsischosensothatforeachroad
eitherthecentralandoneendsectionisidentified,ornosectionatallisidentified.For
thefirstandlastroad,bothsectionsareidentifiedifthejunctioniscoveredandotherwise
onlythelongsectioncanbeidentified.Wehavekroadsandk 1junctions.Eachlabel
−
mustblockajunction,ifitidentifiestwosections.Sothebestpossibleconfiguration
blocksalljunctionsandidentifies2(k 1)+1=2k 1roadsections.
− −
Thechaingadgethasexactlytwostates,inwhich2k 1roadsectionsareidentified.
−
Eitherthelabelofthefirstroaddoesnotblockajunctionandidentifiesasinglesection
andallsubsequentroadshavetheirlabelcoverthejunctionwiththeprecedingroadin
thesequence,orthelabelofthelastroaddoesnotblockajunctionandallotherroads
havetheirlabelcoverthejunctionwiththesuccessiveroadinthesequence.Inanyother
configurationthereisatleastoneroadwithoutanyidentifiedsectionandthusatmost
2k 2sectionsareidentified.Weusethetwooptimalstatesofthegadgettorepresent
−
andtransmitthevaluestrueandfalsefromoneendtotheother.
ForkGadget.Theforkgadgetallowstosplitthevaluerepresentedinonechaininto
twochains,whichisneededtotransmitthetruthvalueofavariableintomultipleclauses.
Tothatenditconnectstoanendroadofthreechaingadgetsbysharingjunctions.
The core of the fork consists of six roads r ,...,r , whereas r , r , and r are
1 6 1 2 3
verticallinesegmentsandr ,r andr arehorizontallinesegments;seeFig.3.We
4 5 6
arrangethoseroadssuchthatr andr haveeachonejunctionwithr andonejunction
1 2 4
withr .Further,r hasonejunctionwithr ,onewithr andonewithr .Thelabel
5 3 4 5 6
length of those roads is chosen so that it is exactly the length of the roads. Hence, a
placedlabelidenfiesallroadsectionsoftheroads.
Further,therearethreeroadsg ,g ,g suchthatg hasonejunctionwithr ,g has
1 2 3 1 1 2
onejunctionwithr andg hasonejunctionwithr .Inallthreecasesweplacethe
2 3 6
junctionsothatitsplitstheroadinashortroadsectionthatisshorterthantheroad’s
labellengthandalongroadsectionthathasexactlytheroad’slabellength.Wecallg ,
1
6 AndreasGemsa,BenjaminNiedermann,andMartinNöllenburg
g andg gates,becauselatertheseroadswillbeconnectedtotheendroadsofchains
2 3
byjunctions.Tothatendthoseconnectingjunctionswillbeplacedonthelongroad
sectionsofthegates;seevioletdottedareasinFig.3.
Theforkgadgethasexactlytwostates,inwhich16roadsectionsareidentified.In
thefirststatethelabelsofr ,r andr areplaced;seeFig3(b).Hence,thelabelsofg
1 2 3 1
andg identifyonlythelongroadsectionsofg andg ,butnottheshortones.Thelabel
2 1 2
ofg idenfiesboththelongandshortroadsectionofg .Inthesecondstatethelabelsof
3 3
r ,r ,r areplaced;seeFig3(c).Hence,thelabelsofg andg identifythelongand
4 5 6 1 2
shortroadsectionsofg andg ,whileonlythelongroadsectionofg isidentifiedbya
1 2 3
label.Inanyotherconfigurationfewerroadsectionsareidentifiedbylabels.Weusethe
twooptimalstatesofthegadgettorepresentandtransmitthevaluestrueandfalsefrom
onegatetotheothertwogates.Morespecificallythegatesg andg areconnectedwith
1 2
chainsthatleadtothesameliteral,whileg isconnectedwithachainthatleadstothe
3
complementaryliteral.
VariableGadget.Wedefinethevariablegadgetssimplybyconnectingchainandfork
gadgetsintoaconnectedcomponentofintersectingroads.Thisconstructionalreadyhas
thefunctionalityofavariablegadget:itrepresents(inalabelingidentifyingthemaximum
numberofroadsections)thesametruthvalueinallofitsbranches,synchronizedbythe
forkgadgets,seethebluechainsandyellowforksinFig.2a).Moreprecisely,weplacea
sequenceofchainslinkedbyforkgadgetsalongthehorizontallineonwhichthevariable
verticesareplacedinthedrawingH .Eachforkcreatesabranchofthevariablegadget
ϕ
eitheraboveorbelowtheline.Wecreateasmanybranchesabove(below)thelineas
thevariablehasoccurrencesinpositive(negative)clausesinϕ.Thefirstandlastchain
onthelinealsoserveasbranches.Thesynchronizationofthedifferentbranchesviathe
forksissuchthateitheralltopbrancheshavetheirroadlabelspushedawayfromthe
lineandallbottombranchespulledtowardsthelineorviceversa.Inthefirstcase,we
saythatthevariableisinthestatefalseandinthelattercasethatitisinthestatetrue.
TheexampleinFig.2hastwovariablessettotrueandthreevariablessettofalse.
Clause Gadget. Finally, we need to create the clause gadget, which links three
branches of different variables. The core of the gadget is a single road that consists
ofthreesub-pathsmeetinginonejunction.Eachsub-pathofthatroadsharesanother
junctionwithoneofthethreeincomingvariablebranches.Beyondeachofthesethree
junctionsthefinalroadsectionsarejustlongenoughsothatalabelcanbeplacedon
thesection.However,thesectionbetweenthecentraljunctionoftheclauseroadandthe
junctionswiththeliteralroadsisshorterthanthelabellength.Theroadoftheclause
gadgethassixsectionsintotalandwearguethatthesixsectionscanonlybeidentified
ifatleastoneincomingliteralevaluatestotrue.Otherwiseatmostfivesectionscanbe
identified.Byconstruction,eachroadinthechainofafalseliteralhasitslabelpushed
towardstheclause,i.e.,itblocksthejunctionwiththeclauseroad.Aslongasatleast
oneofthesethreejunctionsisnotblocked,allsectionscanbeidentified;seeFig.2b).
Butifallthreejunctionsareblocked,thenonlytwoofthethreeinnersectionsofthe
clauseroadcanbeidentifiedandthethirdoneremainsunlabeled;seeFig.2b).
Reduction.Obviously,thesizeoftheinstance ispolynomialinnandm.Ifwe
ϕ
M
haveasatisfyingvariableassignmentforϕ,wecanconstructthecorrespondingroad
labelingandthenumberofidentifiedroadsectionsissixperclauseandafixedconstant
LabelPlacementinRoadMaps 7
numberK ofsectionsinthevariablegadgets,i.e.,atleastK =K +6m.Ontheother
(cid:48) (cid:48)
hand,ifwehavearoadlabelingwithatleastK identifiedsections,eachvariablegadget
isinoneofitstwomaximumconfigurationsandeachclauseroadhasatleastonelabel
thatcoversajunctionwithaliteralroad,meaningthatthecorrespondingtruthvalue
assignmentofthevariablesisindeedasatisfyingone.Thisconcludesthereduction.
SinceMAXIDENTIFIEDROADSisanoptimizationproblem,weonlypresenttheNP-
hardness proof. Still, one can argue that the corresponding decision problem is NP-
completebyguessingwhichjunctionsarecoveredbywhichlabelandthenusinglinear
programmingforcomputingthelabelpositions.Weomitthetechnicaldetails.Further,
mostroadsinthereductionarepaths,exceptforthecentralroadineachclausegadget,
whichisadegree-3star.Infact,wecanstrengthenTheorem1byusingamorecomplex
clausegadgetinsteadthatusesonlypaths;seeAppendixA.1.
4 AnEfficientAlgorithmforTree-ShapedRoadMaps
In this section we assume that the underlying road graph of the road map is a tree
T =(V,E).InSection4.1wepresentapolynomial-timealgorithmtooptimallysolve
MAXIDENTIFIEDROADSfortrees;Section4.2shows
howtoimproveitsrunningtimeandspaceconsump-
tion. Our approach uses the basic idea that remov-
ing the vertices, whose embeddings lie in a curve
horizontal c E(T),splitsthetreeintoindependentparts.In
curve ‘ = pa⊆rticularthisistrueforlabels.WeassumethatT is
child of ‘
ρ rooted at an arbitrary leaf ρ and that its edges are
directed away from ρ; see Fig. 4. For two points
p,q E(T) we define d(p,q) as the length of the
∈
head of ‘ shortestcurveinE(T)thatconnectspandq.Fortwo
verticesuandv ofT wealsowrited(u,v)instead
vertical
of d(E(u),E(v)). For a point p E(T) we abbre-
curve ‘
∈
viatethedistanced(p,ρ)totherootρbyd .Fora
tail of ‘ ρ p
curve(cid:96)inE(T),wecallp (cid:96)thelowestpointof(cid:96)
∈
ifd d ,foranyq (cid:96).AsT isatree,pisunique.
Fig.4.Basicdefinitions. p ≤ q ∈
WedistinguishtwotypesofcurvesinE(T).Acurve(cid:96)
isverticalifh((cid:96))ort((cid:96))isthelowestpointof(cid:96);otherwisewecall(cid:96)horizontal(see
Fig. 4). Without loss of generality we assume that the lowest point of each vertical
curve(cid:96)isitstailt((cid:96)).Sincelabelsaremodeledascurves,theyarealsoeitherverticalor
horizontal.Foravertexu V letT denotethesubtreerootedatu.
u
∈
4.1 BasicApproach
Wefirstdetermineafinitesetofcandidatepositionsfortheheadsandtailsoflabels,
andtransformT intoatreeT =(V ,E )bysubdividingsomeofT’sedgessothatit
(cid:48) (cid:48) (cid:48)
containsavertexforeverycandidateposition.Tothatendweconstructforeachregular
vertex v V a chain of tightly packed vertical labels that starts at E(v), is directed
∈
8 AndreasGemsa,BenjaminNiedermann,andMartinNöllenburg
towardsρ,andendswheneithertheroadends,oraddingthenextlabeldoesnotincrease
thenumberofidentifiedroadsections.Morespecifically,weplaceafirstverticallabel(cid:96)
1
suchthath((cid:96) ) = E(v).Fori = 2,3,... weaddanewverticallabel(cid:96) withh((cid:96) )=
1 i i
t((cid:96) ),aslongash((cid:96) )andt((cid:96) )donotlieonthesameroadsectionandnoneof(cid:96) ’s
i 1 i i i
end−pointslieonajunctionedge.Weusethetailsofallthoselabelstosubdividethe
treeT.DoingthisforallregularverticesofT weobtainthetreeT ,whichwecallthe
(cid:48)
subdivisiontreeofT.TheverticesinV V areneitherjunctionverticesnorregular
(cid:48)
\
vertices.SinceeachchainconsistsofO(n)labelsthecardinalityofV isO(n2).We
(cid:48)
call an optimal labeling of T an canonical labeling if for each label (cid:96) there
(cid:48)
L ∈ L
existsavertexvin T withE(v)=h((cid:96))orE(v)=t((cid:96)).Thenextlemmaprovesthatis
(cid:48)
sufficienttoconsidercanonicallabelings.
Lemma1. ForanyroadgraphT thatisatree,thereexistsacanonicallabeling .
L
Proof. Let beanoptimallabelingofT.Wepushthelabelsof asfaraspossible
L L
towardstheleavesofT withoutchangingtheidentifiedroadsections;seeFig.5.More
specifically,startingwiththelabelsclosesttotheleaves,wemoveeachlabelawayfrom
therootasfaraspossiblewhileitsheadandtailmustremainontheirrespectiveroad
sections.Foraverticallabelthisdirectionisunique,whileforhorizontallabelswecan
chooseanyofthetwo.Then,foreachlabelitsheadortaileithercoincideswithaleafof
T,withsomeinternalregularvertex,orwiththeheadofanotherlabel.Consequently,
eachverticallabelbelongstoachainoftightlypackedverticallabelsstartingataregular
vertexv V.Further,theheadortailofeachhorizontallabelcoincideswiththeend
∈
ofachainoftightlypackedverticallabelsoraregularvertexofT,whichprovesthe
claim.
(cid:116)(cid:117)
We now explain how to construct such a canonical labeling. To that end we first
introducesomenotations.Foravertexu V let (u)denotealabelingthatidentifies
(cid:48)
∈ L
amaximumnumberofroadsectionsinT onlyusingvalidlabels
(a) inE(T ),whereT denotesthesubtreeofT rootedatu.Note
u(cid:48) u(cid:48) (cid:48)
thatthoselabelsalsomayidentifytheincomingroadsectionof
u,e.g.,label(cid:96)inFig.5b)identifiestheedgee.
(cid:48)
Further,thechildrenofavertexu V aredenotedbythe
(cid:48)
∈
set N(u); we explicitly exclude the parent of u from N(u).
Further, consider an arbitrary curve (cid:96) in E(T) and let (cid:96) =
(cid:48)
(cid:96) t((cid:96)),h((cid:96)) . We observe that removing all vertices of T
(cid:48)
\ { }
containedin(cid:96) togetherwiththeirincidentoutgoingedgescreates
(cid:48)
(b) severalindependentsubtrees.Wecalltherootsofthesesubtrees
(excepttheonecontainingρ)childrenof(cid:96)(seeFig.4).Ifnovertex
T chain
u0 ofT liesin(cid:96),thecurveiscontainedinasingleedge(u,v) E .
(cid:48) (cid:48) (cid:48)
‘ ∈
In that case v is the only child of (cid:96). We denote the set of all
e
u e0 childrenof(cid:96)asN((cid:96)).
added ForeachvertexuinT(cid:48)weintroduceasetC(u)ofcandidates,
vertex which model potential labels with lowest point E(u). If u is a
regular vertex of T or u V V, the set C(u) contains all
(cid:48)
∈ \
verticallabels(cid:96)withlowestpointE(u).Ifuisajunctionvertex,
Fig.5. Canonical la-
beling. C(u)containsallhorizontallabelsthatstartorendatavertexof
LabelPlacementinRoadMaps 9
T andwhoselowestpointisE(u).InbothcasesweassumethatC(u)alsocontainsthe
(cid:48)
degeneratedcurve =E(u),whichisthedummylabelofu.WesetN( )=N(u)
u u
⊥ ⊥
andω( )=0.
u
⊥ (cid:83)
For a curve (cid:96) we define ((cid:96)) = (v) (cid:96) . Thus, ((cid:96)) is a labeling
L v N((cid:96))L ∪ { } L
∈
comprising(cid:96)andthelabelsofitschildren’soptimallabelings.Wecallalabel(cid:96) C(u)
∈
with(cid:96)=argmax ω( ((cid:96))) (cid:96) C(u) anoptimalcandidateofu.Next,weprovethat
{ L | ∈ }
itissufficienttoconsideroptimalcandidatestoconstructacanonicallabeling.
Lemma2. GivenavertexuofT andanoptimallabeling (u)andlet(cid:96)beanoptimal
(cid:48)
L
candidateofu,thenitistruethatω( (u))=ω( ((cid:96))).
L L
Proof. Firstnotethatω( (u)) ω( ((cid:96)))becausebothlabelings (u)and ((cid:96))only
L ≥ L L L
containlabelsthatareembeddedinE(T ).ByLemma1wecanassumewithoutlossof
u(cid:48)
generalitythat (u)isacanonicallabeling.Let(cid:96)bethelabelof (u)withE(u)asthe
L L
lowestpointof(cid:96)(ifitexists).
If(cid:96)exists,thentheverticesinN((cid:96))arerootsofindependentsubtrees,whichdirectly
yieldsω( (u))=ω( ((cid:96))).ByconstructionofC(u)wefurtherknowthat(cid:96)iscontained
L L
inC(u).Hence,(cid:96)isanoptimalcandidateofu,whichimpliesω((cid:96))=ω((cid:96)).
If (cid:96)doesnotexist,thenwehave
(cid:91) (1) (cid:91)
ω( (u))=ω( (v)) = ω( (v) )=ω( ( )).
u u
L L L ∪{⊥ } L ⊥
v∈N(u) v∈N(⊥u)
Equality(1)followsfromN( )=N(u)andthedefinitionthat doesnotidentify
u u
⊥ ⊥
anyroadsection.Since iscontainedinC(u),thedummylabel istheoptimal
u u
⊥ ⊥
candidate(cid:96).
(cid:116)(cid:117)
Algorithm 1 first constructs the subdivision tree T = (V ,E ) from T. Then
(cid:48) (cid:48) (cid:48)
starting with the leaves of T and going to the root ρ of T , it computes an optimal
(cid:48) (cid:48)
candidate(cid:96)=OptCandidate(u)foreachvertexu V inabottom-upfashion.By
(cid:48)
∈
Lemma2thelabeling ((cid:96))isanoptimallabelingofT .Inparticular (ρ)istheoptimal
L u(cid:48) L
labelingofT.
DuetothesizeofthesubdivisiontreeT weconsiderO(n2)vertices.Implement-
(cid:48)
ingOptCandidate(u),whichcomputesanoptimalcandidate(cid:96)foru,naively,cre-
atesC(u)explicitly.Weobservethatifuisajunctionvertex,C(u)maycontainO(n2)
labels; O(n2) pairs of road sections of different subtrees of u can be connected by
Algorithm1:ComputinganoptimallabelingofT.
Input:RoadgraphT,whereT isatreewithrootρ.
Output:Optimallabeling (ρ)ofT.
L
1 T(cid:48) computesubdivisiontreeofT
←
2 foreachleafvofT(cid:48)do (v) foreachvertexuofT(cid:48)consideredinabottom-up
L ←∅
traversalofT(cid:48)do
3 (u) (OptCandidate(u))
L ←L
4 return (ρ)
L
10 AndreasGemsa,BenjaminNiedermann,andMartinNöllenburg
horizontallabels.EachlabelcanbeconstructedinO(n)timeusingabreadth-firstsearch.
Thus,foreachvertexutheprocedureOptCandidateneedsinanaiveimplementation
O(n3)time,whichyieldsO(n5)runningtimeintotal.Further,weneedO(n2)storage
to store T . Note that we do not need to store (u) for each vertex u of T , but by
(cid:48) (cid:48)
L
Lemma2wecanreconstructitusing ((cid:96)),where(cid:96)istheoptimalcandidateofu.Tothat
L
endwestoreforeachvertexofT itsoptimalcandidate(cid:96)andw( ((cid:96))).
(cid:48)
L
Theorem2. Foraroadmapwithatreeasunderlyingroadgraph,MAXIDENTIFIED-
ROADScanbesolvedinO(n5)timeusingO(n2)space.
Incasethatallroadsarepaths,Algorithm1runsinO(n4)time,becauseforeach
u V the set C(u) contains O(n) labels. Further, besides the primary objective
(cid:48)
∈
to identify a maximum number of road sections, Chirié [1] also suggested several
additionalsecondaryobjectives,e.g.,labelsshouldoverlapasfewjunctionsaspossible.
Our approach allows us to easily incorporate secondary objectives by changing the
weightfunctionωappropriately.
4.2 ImprovementsonRunningTime
InthispartwedescribehowtherunningtimeofAlgorithm1canbeimprovedtoO(n3)
timebyspeedingupOptCandidate(u)toO(n)time.
Foranedgee=(u,v) E E wecallaverticalcurve(cid:96) E(T)ane-rootedcurve,
(cid:48)
∈ ∪ ⊆
ift((cid:96))=E(u),h((cid:96))liesonaroadsection,andlen(E(e) (cid:96))=min len((cid:96)),len(E(e)) ,
∩ { }
i.e., (cid:96) emanates from E(u) passing through e; for example the red label in Fig. 5b)
is an e-rooted curve. An e-rooted curve (cid:96) is maximal if there is no other e-rooted
curve (cid:96) with len((cid:96)) = len((cid:96)) and ω( ((cid:96))) > ω( ((cid:96))). We observe that in any
(cid:48) (cid:48) (cid:48)
L L
canonicallabelingeachverticallabel(cid:96)isa(u,v)-rootedcurvewith(u,v) E ,and
(cid:48)
∈
each horizontal label (cid:96) can be composed of a (u,v )-rooted curve (cid:96) and a (u,v )-
1 1 2
rootedcurve(cid:96) with(u,v ),(u,v ) E andE(u)isthelowestpointof(cid:96);seeFig.7
2 1 2 (cid:48)
∈
and Fig. 8, respectively. Further, for a vertical curve c in E(T) its distance interval
I(c) is [d ,d ]. Since T is a tree, for every point p of c we have d I(c).
t(c) h(c) p
∈
Two vertical curves c and c superpose 9
ethaacthNeoentxhcteo,rdwiefseIf(ioncrt)ro∩edaIcu(hcce(cid:48)e)da(cid:54)=gde∅a(cid:48)t(;ause,sevtr)Fuicogtfu6rT.e 765c5436cc147c83 74655c26 7 I(c1) I(cI2()c5)I(c4)
allmaximal(u,v)-rootedcurvesasO(n) 23 I(c3)
superposition-freecurvesinE(T ).Inpar- 1 0 5 10
u ρ
ticular,eachofthosecurvesliesonasin-
gleroadsectionsuchthatall(u,v)-rooted Fig.6.Superposingcurves,e.g.,c andc super-
1 2
curvesendingonthatcurvearemaximal poseeachother,whilec andc donot.Thetree
1 5
andidentifythesamenumberofroadsec- isannotatedwithdistancemarks.
tions.Wedefinethisdatastructureasfollows.
Definition1 (Linearization).Lete=(u,v)beanedgeofT.Atuple(L,ω)iscalled
alinearizationofe,ifLisasetofsuperposition-freecurvesandω: L Rsuchthat
→
(1) foreachcurvec Lthereisaroadsectione inT withc E(e),
(cid:48) u (cid:48)
∈ ⊆