Vorlesung Graphenzeichnen: Order in the Underground or · Vorlesung Graphenzeichnen: Order in the Underground or How to Automate the Drawing of Metro Maps ... mit Perl Regul re Ausdr

Our ModelNP-HardnessOur Solution

Vorlesung Graphenzeichnen:

Order in the Underground orHow to Automate the Drawing of Metro Maps

Martin Nöllenburg

Lehrstuhl für Algorithmik I

26.06.2008

Martin Nöllenburg 1 42 Drawing Metro Maps


Outline

Modeling the Metro Map ProblemWhat is a Metro Map?Hard and Soft Constraints

NP-Hardness: Bad News—Nice ProofRectilinear vs. Octilinear DrawingReduction from PLANAR 3-SAT

MIP Formulation & ExperimentsMixed-Integer Programming FormulationExperimentsLabeling



What is a Metro Map?Hard and Soft Constraints

Outline







What is a Metro Map?

schematic diagram for public transport

visualizes lines and stationsgoal: ease navigation for passengers

“How do I get from A to B?”“Where to get off and change trains?”

distorts geometry and scaleimproves readabilitycompromise betweenschematic road map↔ abstract graph





schematic diagram for public transportvisualizes lines and stations

goal: ease navigation for passengers“How do I get from A to B?”“Where to get off and change trains?”






schematic diagram for public transportvisualizes lines and stationsgoal: ease navigation for passengers









distorts geometry and scale

improves readabilitycompromise betweenschematic road map↔ abstract graph







distorts geometry and scaleimproves readability

compromise betweenschematic road map↔ abstract graph











Why Automate Drawing Metro Maps?

current maps designedmanually

assist graphic designers toimprove/extend mapsmetro map metaphor

metabolic pathwaysweb page mapsproduct lines

VLSI: X-architecturedrawing sketches

[Brandes et al. ’03]





current maps designedmanuallyassist graphic designers toimprove/extend maps

metro map metaphor




P

P

P

P

P

P

P

P

PP

P

P

P

PP

P

P

P

P

P

P

P

P

P

P

P

P

P

P

Under construction

FutureE

xtension

Subject toFundin

g

To Sacramento

To Tracy/Stockton

To Stockton

To Gilroy

SanFranciscoInternationalAirport

OaklandInternationalAirport

AirBART

SanJoseInternationalAirport

19th St /OaklandOakland City Center/12th StLake Merritt

West Oakland

EmbarcaderoMontgomery St

Powell StCivic Center

16th St Mission24th St Mission

Glen Park

Balboa Park

SouthSan Francisco

SanBruno

Millbrae

Daly City

Fruitvale

San Leandro

Bay Fair

HaywardCastro Valley

South Hayward

Union City

Coliseum /Oakland Airport

MacArthurRockridge

OrindaLafayette

Walnut Creek

Pleasant Hill

Concord

Ashby

Downtown Berkeley

El Cerrito PlazaNorth Berkeley

El Cerrito del Norte

North Concord /Martinez

Colma

Dublin/Pleasanton

Richmond

Fremont

Pittsburg /Bay Point

FosterCity

SantaClara

Pacifica

Newark WarmSprings

Fremont

Larkspur

CorteMadera

MillValley

Sausalito

Pinole

SanPablo

Rodeo

Hercules

Danville

SanRamon

Dublin

Livermore

Irvington

Pleasanton

MartinezAntioch

SanFrancisco

SanJose

NT

TP

ACE (Altamont CommuterExpress)

Regional Rail Transfer Point

Capitol Corridor (Amtrak)

San Joaquin (Amtrak)

Caltrain

BART Parking

BART Dublin/Pleasanton–Colma

BART Pittsburg /Bay Point –Colma

BART Fremont –Richmond

BART Fremont –Daly City

BART Richmond –Daly City

T

T

BART System Map

Hwy4

Hwy580

Hwy680

Hwy680

Hwy24

Hwy4

Hwy80

Hwy80

Hwy680

Hwy680

Hwy237

Hwy880

Hwy92

Hwy84

Hwy101

Hwy101

Hwy280

Hwy1

Hwy1





current maps designedmanuallyassist graphic designers toimprove/extend mapsmetro map metaphor

metabolic pathways

web page mapsproduct lines








metabolic pathwaysweb page maps

product lines











2003 OPEN SOURCE ROUTE MAP

Perl/TkPocket Reference

Perl für System-

Administration

Perl in a Nutshell

Web

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VLSI: X-architecture

drawing sketches[Brandes et al. ’03]












More Formally

The Metro Map Problem

Given: planar embedded graph G = (V ,E), V ⊂ R2,line cover L of paths or cycles in G (the metro lines),

Goal: draw G and L nicely.

What is a nice drawing?Look at real-world metro maps drawn by graphic designersand model their design principles as

hard constraints – must be fulfilled,soft constraints – should hold as tightly as possible.




More Formally




What is a nice drawing?

Look at real-world metro maps drawn by graphic designersand model their design principles as





More Formally









More Formally





hard constraints – must be fulfilled,

soft constraints – should hold as tightly as possible.




More Formally









Hard Constraints

(H1) preserve embedding of G

(H2) draw all edges as octilinear line segments,i.e. horizontal, vertical or diagonal (45 degrees)

(H3) draw each edge e with length ≥ è(H4) keep edges dmin away from non-incident edges




Hard Constraints

(H1) preserve embedding of G(H2) draw all edges as octilinear line segments,

i.e. horizontal, vertical or diagonal (45 degrees)

(H3) draw each edge e with length ≥ è(H4) keep edges dmin away from non-incident edges




Hard Constraints


i.e. horizontal, vertical or diagonal (45 degrees)(H3) draw each edge e with length ≥ è

(H4) keep edges dmin away from non-incident edges




Hard Constraints


i.e. horizontal, vertical or diagonal (45 degrees)(H3) draw each edge e with length ≥ è(H4) keep edges dmin away from non-incident edges




Soft Constraints

(S1) draw metro lines with few bends

(S2) keep total edge length small(S3) draw each octilinear edge similar to its

geographical orientation: keep its relative position




Soft Constraints

(S1) draw metro lines with few bends(S2) keep total edge length small

(S3) draw each octilinear edge similar to itsgeographical orientation: keep its relative position




Soft Constraints

(S1) draw metro lines with few bends(S2) keep total edge length small(S3) draw each octilinear edge similar to its

geographical orientation: keep its relative position



Rectilinear vs. Octilinear DrawingReduction from PLANAR 3-SAT

Outline







Another Problem

RECTILINEARGRAPHDRAWING Decision ProblemGiven a planar embedded graph G with max degree 4.Is there a drawing of G that

preserves the embedding,uses straight-line edges,is rectilinear?

Theorem (Tamassia SIAMJComp’87)

RECTILINEARGRAPHDRAWING can be solved efficiently.

Proof.By reduction to a flow problem.




Another Problem









Another Problem









Another Problem









Our Problem

METROMAPLAYOUT Decision ProblemGiven a planar embedded graph G with max degree 8.Is there a drawing of G that

preserves the embedding,uses straight-line edges,is octilinear?

Theorem (Nöllenburg MSc’05)METROMAPLAYOUT is NP-hard.

Proof.By Reduction from PLANAR 3-SAT to METROMAPLAYOUT.




Outline of the Reduction

x1 ∨ x2 ∨ x3

x1 ∨ x3 ∨ x4

x1 ∨ x2 ∨ x4

x2 ∨ x3 ∨ x4

x1 x2 x3 x4

Input: planar 3-SAT formula ϕ =(x1 ∨ x3 ∨ x4) ∧ (x1 ∨ x2 ∨ x3) ∧ . . .

Goal: planar embedded graph Gϕ with:Gϕ has a metro map drawing ⇔ ϕ satisfiable.





x1 ∨ x2 ∨ x3

x1 ∨ x3 ∨ x4

x1 ∨ x2 ∨ x4

x2 ∨ x3 ∨ x4

x1 x2 x3 x4







x1 ∨ x2 ∨ x3

x1 ∨ x3 ∨ x4

x1 ∨ x2 ∨ x4

x2 ∨ x3 ∨ x4

x1 x2 x3 x4






Variable Gadget

. . .

. . .

. . .

......

...

...

...

......

......

x = true

x x

x

false true

true




Variable Gadget

. . .

. . .

. . .

......

......

......

...

......

x = false

x x

x

true false

false





x1 ∨ x2 ∨ x3

x1 ∨ x3 ∨ x4

x1 ∨ x2 ∨ x4

x2 ∨ x3 ∨ x4

x1 x2 x3 x4






Clause Gadget




Clause Gadget




Clause Gadget




Clause Gadget




Clause Gadget




Clause Gadget




Clause Gadget




Summary of the Reduction

→. . .

. . .

. . .

......

...

...

...

......

......

x = true

x x

x

false true

true

x1 ∨ x2 ∨ x3

x1 ∨ x3 ∨ x4

x1 ∨ x2 ∨ x4

x2 ∨ x3 ∨ x4

x1 x2 x3 x4

Indeed we have:ϕ satisfiable ⇒ corresponding MM drawing of Gϕ

Gϕ has MM drawing ⇒ satisfying truth assignment of ϕ



MIP FormulationExperimentsLabeling

Outline







Mathematical Programming

Linear Programming: efficientoptimization method for

linear constraintslinear objective functionreal-valued variables

Mixed-Integer Programming (MIP)allows also integer variablesNP-hard in generalstill practical method for manyhard optimization problems

Theorem (Nöllenburg & Wolff GD’05)

The problem MetroMapLayout can be formulated as a MIP s.th.hard constraints → linear constraintssoft constraints → objective function






linear constraintslinear objective functionreal-valued variablesexample:maximize x + 2ysubject toy ≤ 0.9x + 1.5y ≥ 1.4x − 1.3




















































Definitions: Sectors and Coordinates

u

12

3

4

56

0

7v

Sectors– for each vtx. u partition plane into sectors 0–7

here: sec(u, v) = 5 (input)

– number octilinear edge directions accordinglye.g. dir(u, v) = 4 (output)

x

y z1

z2

Coordinatesassign z1- and z2-coordinates to each vertex v :

z1(v) = x(v) + y(v)

z2(v) = x(v)− y(v)





u

12

3

4

56

0

7v




x

y z1

z2


z1(v) = x(v) + y(v)

z2(v) = x(v)− y(v)





u

12

3

4

56

0

7v




x

y z1

z2


z1(v) = x(v) + y(v)

z2(v) = x(v)− y(v)




Octilinearity and Relative Position

Goal

u

12

3

4

56

0

7v

pred

origsucc

Draw edge uvoctilinearlywith minimum length ùv

restricted to 3 directions

How to model this using linear constraints?

Binary Variables

αpred(u, v) + αorig(u, v) + αsucc(u, v) = 1





Goal

u

12

3

4

56

0

7v

pred

origsucc




Binary Variables






Goal

u

12

3

4

56

0

7v

pred

origsucc




Binary Variables






u

12

3

4

56

0

7v

pred

origsucc

Predecessor Sector

y(u)− y(v) ≤ M(1− αpred(u, v))−y(u) + y(v) ≤ M(1− αpred(u, v))

x(u)− x(v) ≥ −M(1− αpred(u, v)) + ùv





u

12

3

4

56

0

7v

pred

origsucc

Predecessor Sector


x(u)− x(v) ≥ −M(1− αpred(u, v)) + ùv

How does this work?





u

12

3

4

56

0

7v

pred

origsucc

Predecessor Sector


x(u)− x(v) ≥ −M(1− αpred(u, v)) + ùv

How does this work?Case 1: αpred(u, v) = 0

y(u)− y(v) ≤ M−y(u) + y(v) ≤ M

x(u)− x(v) ≥ ùv −M





u

12

3

4

56

0

7v

pred

origsucc

Predecessor Sector


x(u)− x(v) ≥ −M(1− αpred(u, v)) + ùv

How does this work?Case 2: αprev(u, v) = 1

y(u)− y(v) ≤ 0−y(u) + y(v) ≤ 0

x(u)− x(v) ≥ ùv





u

12

3

4

56

0

7v

pred

origsucc

Original Sector

z2(u)− z2(v) ≤ M(1− αorig(u, v))−z2(u) + z2(v) ≤ M(1− αorig(u, v))

z1(u)− z1(v) ≥ −M(1− αorig(u, v)) + 2ùv

Successor Sector

x(u)− x(v) ≤ M(1− αsucc(u, v))−x(u) + x(v) ≤ M(1− αsucc(u, v))

y(u)− y(v) ≥ −M(1− αsucc(u, v)) + ùv





u

12

3

4

56

0

7v

pred

origsucc

Original Sector

z2(u)− z2(v) ≤ M(1− αorig(u, v))−z2(u) + z2(v) ≤ M(1− αorig(u, v))

z1(u)− z1(v) ≥ −M(1− αorig(u, v)) + 2ùv

u

12

3

4

56

0

7v

pred

origsucc

Successor Sector

x(u)− x(v) ≤ M(1− αsucc(u, v))−x(u) + x(v) ≤ M(1− αsucc(u, v))

y(u)− y(v) ≥ −M(1− αsucc(u, v)) + ùv




Preserving the Embedding

DefinitionTwo planar drawings of G have the same embedding if theinduced orderings on the neighbors of each vertex are equal.

Same Embedding

1

2

3

4

5

2

14

3

5





DefinitionTwo planar drawings of G have the same embedding if theinduced orderings on the neighbors of each vertex are equal.

Different Embeddings

1

2

3

4

5 1

2

3

4

5





Constraints (Example)

N(v) = {u1,u2,u3,u4}circular input order: u1 < u2 < u3 < u4 < u1

All but one of the following inequalities must hold

dir(v ,u1) < dir(v ,u2) < dir(v ,u3) < dir(v ,u4) < dir(v ,u1)

Input

vu1

u2

u3u4

Output

v

u1

u2

u3u4

0

13

7





Constraints (Example)

N(v) = {u1,u2,u3,u4}circular input order: u1 < u2 < u3 < u4 < u1

All but one of the following inequalities must hold

dir(v ,u1) ≮ dir(v ,u2) < dir(v ,u3) < dir(v ,u4) < dir(v ,u1)

Input

vu1

u2

u3u4

Output

v

u1

u2

u3u4

0

13

7Martin Nöllenburg 24 42 Drawing Metro Maps



Planarity

Observation

For octilinear, straight edge e1non-intersecting edge e2 must be placed

east, northeast, north, northwest,west, southwest, south, or southeast

E

Constraintsmodel as MIP with binary variablesneed planarity constraints for each pair of non-incidentedges




Planarity

Observation



NE





Planarity

Observation



N





Planarity

Observation



NW





Planarity

Observation



W





Planarity

Observation



SW





Planarity

Observation



S





Planarity

Observation



SE





Planarity

Observation



SE





Objective Function

Objective Functioncorresponds to soft constraints (S1)–(S3)weighted sum of individual cost functions

minimize λbends costbends + λlength costlength + λrelpos costrelpos

Line Bends (S1)

Edges uv and vw on a line L ∈ Ldraw as straight as possibleincreasing cost bend(u, v ,w) forincreasing acuteness of ∠(uv , vw)




Objective Function



Line Bends (S1)

vu3

w Edges uv and vw on a line L ∈ Ldraw as straight as possibleincreasing cost bend(u, v ,w) forincreasing acuteness of ∠(uv , vw)




Objective Function



Line Bends (S1)

vu

2w





Objective Function



Line Bends (S1)

vu

1w Edges uv and vw on a line L ∈ L

draw as straight as possibleincreasing cost bend(u, v ,w) forincreasing acuteness of ∠(uv , vw)




Objective Function



Line Bends (S1)

vu 0 w





Objective Function



Line Bends (S1)

vu

1w





Objective Function



Line Bends (S1)

vu

2w





Objective Function



Line Bends (S1)

vu

3w





Objective Function



Line Bends (S1)

vu

3w


costbends =∑L∈L

∑uv ,vw∈L

bend(u, v ,w)




Objective Function

Total Edge Length (S2)

costlength =∑

uv∈E

length(uv)

Relative Position (S3)

only three directions possible

charge 1 if edge deviates fromoriginal sector




Objective Function


costlength =∑

uv∈E

length(uv)


u

12

3

4

56

0

7v

only three directions possible

charge 1 if edge deviates fromoriginal sector




Objective Function


costlength =∑

uv∈E

length(uv)


u

12

3

4

56

0

7v

1

only three directions possiblecharge 1 if edge deviates fromoriginal sector




Objective Function


costlength =∑

uv∈E

length(uv)


u

12

3

4

56

0

7v

1

0





Objective Function


costlength =∑

uv∈E

length(uv)


u

12

3

4

56

0

7v

1

01





Objective Function


costlength =∑

uv∈E

length(uv)


u

12

3

4

56

0

7v

1

01


costrelpos =∑

uv∈E

relpos(uv)




Summary of the MIP

hard constraints:octilinearityminimum edge length(partially) relative positionpreservation of embeddingplanarity

soft constraints:

minimize λbends costbends+λlength costlength+λrelpos costrelpos

models METROMAPLAYOUT as MIPin total O(|V |2) constraints and variables




Summary of the MIP


soft constraints:






Summary of the MIP


soft constraints:


models METROMAPLAYOUT as MIP

in total O(|V |2) constraints and variables




Summary of the MIP


soft constraints:






Summary of the MIP


soft constraints:






Speed-Up Techniques: Reduce Graph Size

metro graphs have many degree-2 verticeswant to optimize line straightness

Idea 1 collapse all degree-2 vertices

low flexibility

Idea 2 keep two joints

higher flexibilitymore similar to input







low flexibilityIdea 2 keep two joints































Idea 1 collapse all degree-2 verticeslow flexibility









































Idea 2 keep two jointshigher flexibilitymore similar to input




Speed-Up Techniques: Reduce MIP Size

O(|V |2) planarity constraints (for each pair of edges...)in practice 95–99% of constraints

Observation 1consider only pairs of edges incident to the same facestill O(|V |2) constraints

Observation 2in practice no or only few crossings due to soft constraints
































Speed-Up Techniques: Callback Functions

MIP optimizer CPLEX offers advanced callback functionsadd required planarity constraints on the fly

Algorithm1 start solving MIP without planarity constraints2 for each new solution s

1 interrupt CPLEX2 if s is not planar

add planarity constraints for edges that intersect in sreject s

elseaccept s

3 continue solving the MIP (until optimal)




Results – Vienna

Input Input |V | |E | fcs. |L|normal 90 96reduced 44 50 8 5




Results – Vienna


↓MIP constr. var.

normal 39,363 9,960faces 23,226 6,048callback? 1,875 872




Results – Vienna


↓MIP constr. var.

normal 39,363 9,960faces 23,226 6,048callback? 1,875 872

?) 21 seconds w/o proof of opt.




Results – Vienna

Input Output (21 sec.)




Results – Vienna

Official maps Output (21 sec.)




Results – SydneyInput Input |V | |E | fcs. |L|

normal 174 183reduced 67 76

11 10






11 10

↓MIP constr. var.

normal 93,620 23,389faces 52,568 13,437callback? 4,147 6,741






11 10

↓MIP constr. var.


?) 33 seconds w/o proof of opt.?) constr. of 4 edge pairs added




Results – SydneyInput Output (33 sec.)




Results – SydneyOfficial map Output (33 sec.)




Sydney: Related Work[Hong et al. GD’04] (7.6 sec.) Our output (33 sec.)




Sydney: Related Work[Stott, Rodgers IV’04] (4 min.) Our output (33 sec.)




Sydney: Related Work[Stott, Rodgers IV’04] (28 min.) Our output (33 sec.)




Large Maps

London




Large Maps

London (16 min.)




Large Maps

Tokyo




Large Maps

Tokyo (4:50 hrs.)




Labeling

unlabeled metro map oflittle use in practice

labelsoccupy spacemay not overlap

static map labeling isNP-hard

[Tollis, Kakoulis ’01]want to combine layout andlabeling for better results

N8

N7

N6

N5

N4

N3

N2

N1

N4




Labeling

unlabeled metro map oflittle use in practicelabels

occupy spacemay not overlap



N8

N7

N6

N5

N4

N3

N2

N1

Zentral-station

Neuhäuser TorDetmolderTor

Michaelstr.Heiersstr.

Am Bogen

Kasseler StraßeWesterntor

Maspernplatz

Gierstor

NordbahnhofHeinrichskirche

UhlandstraßeSteubenstraße

Gottfried-Keller-Weg

Mörikestraße

Englische Siedlung

Berline

r Ring

Langer Weg

Fröbelstr.

Piepen-turmweg

Schulze-Delitzsch-Straße

Lipps

pring

er Str

.

Kirchb

rede

Buchh

olz

Papenberg

Postweg

Kloster

garte

n

Ronc

allipl

atz

Beksch

er Berg

Druhe

imer

Straß

e

Holtgre

vens

traße

Bf Kasseler Tor

Schulbrede

Hilligenbusch

Uni/Schöne Aussicht

Am Laugrund

Corveyer Weg

Bahneinschnitt

Hochs

tiftss

tr.

Südr

ing

Uni/Sü

dring

Im SpiringsfeldeGertru-denstr.

KilianplatzQuerweg

Stephanusstraße

Elisabethkirche

Kilian

bad

Mälzerstr.Abtsbrede

Sighardstr.Ilseweg

Breslau

er Str

.

Walden

burge

r Str.

Grüner Weg

Frank

furte

r Weg

BetriebshofBarkhausen

StembergKleestraßeIm Tigg

WinkelsgartenFixberg

Weikenberg

Delbrücker WegTriftweg Im

Lich

tenfel

de

Voessingweg

Neuenheerser WegLiethstaudamm

Auf der LiethBürener Weg

Borgentreicher WegPeckelsheimer Weg

Vinsebecker WegWeißdornweg

Mistelweg

Bussardweg

Kaukenberg

Kleing

. Dah

ler W

eg

Iggen

haus

er Weg

Lang

efeld

Bergso

hle

Dahler

Heid

e

Dahl P

ost

Braken

berg

Lülingsberg

Pastorskamp

Win-friedstr.Josefs-krankenhausFrauenklinik

Im BruchhofRummelsberg

Am Zollhaus Zwetschenweg

Ludwigsfelder Ring

Am NiesenteichDr.-Mertens-Weg

Anhalter WegGerold

Füllers HeideStadtheide

Talle-TerrassenWaldweg

SchlesierwegIm Vogtland

Zum KampeMarienloh Mitte

SennewegVon-Dript-Weg

Dören-Park

Schwabenweg

Hauptbahnhof

Klöcknerstraße

PontanusstraßeTechn. Rathaus

WestfriedhofIm Lohfeld

Riemecker Feld

DelpstraßeHeinz-Nixdorf-Ring

Alme Aue

Am Almerfeld

OstalleeCarl-Diem-StraßeWewerstraße

Blumenstraße

Spritzenhaus

Von-Ketteler-StraßeElsen Schule

BohlenwegGasthof Zur Heide

MühlenteichstraßeKarl-Arnold-Straße

Sander Straße

FürstenwegRolandsweg

Freibad/SchützenplatzFerrariweg

MuseumsForumPadersee

An der KapelleFürstenallee

Marienloher Str.

RothewegGustav-Schultze-StraßeBonifatiuskircheWürttemberger WegIngolstädter Weg

An der

Talle

Schloß Neuhaus

AntoniusstraßeUrbanstraße

Verner Straße

Almeri

ng

Mersch

weg

Mentro

pstra

ße

Mittelw

eg

Gesam

tschu

le

Fries

enweg

TÜV

Abzw. F

ischt

eiche

Schil

lerstr

aße

Hohe Kamp

Sande Kirche

Ostenländer Straße

Anemonenweg

Sande Schule

Dirksfeld

Hagebuttenweg

Sunderkampstr.

Sander Friedhof

Dubelohstraße

Schatenweg

HatzfelderPlatz

Memelstraße

Mastbruch Schule

Mastbruch Gaststätte

Dietrichstraße

Husarenstraße

Habichtsweg

Fasanenweg

Bahnkreuzung

Thunebrücke

Hauptwache

Salvatorstraße

Pionierweg

Infanterieweg

Schleswiger WegWilseder Weg

PirolwegAusbesserungswerk

Schützenplatz Nord

Obernheideweg

Lesteweg

*

* Inbetriebnahme nach baulicher Fertigstellung

Marienloh

Neuenbeken

Benhausen

Kaukenberg

Auf derLieth

Dahl

Wewer

Elsen

Gesseln

Sande

Schloß Neuhaus

Mastbruch

Sennelager

Gültig ab 13.02.2005

In den Nächten von Freitag auf Samstag, Samstag auf Sonntag und vor

ausgewählten Feiertagenab Zentralstation Paderborn

www.padersprinter.de

N4 Dahl

Nachtliniennetz [Nachtbus]im Stadtgebiet Paderborn

www.nachtbusse-paderborn.de




Labeling




[Tollis, Kakoulis ’01]

want to combine layout andlabeling for better results

N8

N7

N6

N5

N4

N3

N2

N1

Zentral-station



Am Bogen


Maspernplatz

Gierstor




Mörikestraße

Englische Siedlung

Berline

r Ring

Langer Weg

Fröbelstr.

Piepen-turmweg


Lipps

pring

er Str

.

Kirchb

rede

Buchh

olz

Papenberg

Postweg

Kloster

garte

n

Ronc

allipl

atz

Beksch

er Berg

Druhe

imer

Straß

e

Holtgre

vens

traße

Bf Kasseler Tor

Schulbrede

Hilligenbusch


Am Laugrund

Corveyer Weg

Bahneinschnitt

Hochs

tiftss

tr.

Südr

ing

Uni/Sü

dring


KilianplatzQuerweg

Stephanusstraße

Elisabethkirche

Kilian

bad


Sighardstr.Ilseweg

Breslau

er Str

.

Walden

burge

r Str.

Grüner Weg

Frank

furte

r Weg




Weikenberg


Lich

tenfel

de

Voessingweg





Mistelweg

Bussardweg

Kaukenberg

Kleing

. Dah

ler W

eg

Iggen

haus

er Weg

Lang

efeld

Bergso

hle

Dahler

Heid

e

Dahl P

ost

Braken

berg

Lülingsberg

Pastorskamp




Ludwigsfelder Ring


Anhalter WegGerold






Dören-Park

Schwabenweg

Hauptbahnhof

Klöcknerstraße



Riemecker Feld


Alme Aue

Am Almerfeld


Blumenstraße

Spritzenhaus




Sander Straße





Marienloher Str.


An der

Talle

Schloß Neuhaus


Verner Straße

Almeri

ng

Mersch

weg

Mentro

pstra

ße

Mittelw

eg

Gesam

tschu

le

Fries

enweg

TÜV

Abzw. F

ischt

eiche

Schil

lerstr

aße

Hohe Kamp

Sande Kirche


Anemonenweg

Sande Schule

Dirksfeld

Hagebuttenweg

Sunderkampstr.

Sander Friedhof

Dubelohstraße

Schatenweg

HatzfelderPlatz

Memelstraße

Mastbruch Schule


Dietrichstraße

Husarenstraße

Habichtsweg

Fasanenweg

Bahnkreuzung

Thunebrücke

Hauptwache

Salvatorstraße

Pionierweg

Infanterieweg



Schützenplatz Nord

Obernheideweg

Lesteweg

*


Marienloh

Neuenbeken

Benhausen

Kaukenberg

Auf derLieth

Dahl

Wewer

Elsen

Gesseln

Sande

Schloß Neuhaus

Mastbruch

Sennelager





N4 Dahl






Labeling





N8

N7

N6

N5

N4

N3

N2

N1

Zentral-station



Am Bogen


Maspernplatz

Gierstor




Mörikestraße

Englische Siedlung

Berline

r Ring

Langer Weg

Fröbelstr.

Piepen-turmweg


Lipps

pring

er Str

.

Kirchb

rede

Buchh

olz

Papenberg

Postweg

Kloster

garte

n

Ronc

allipl

atz

Beksch

er Berg

Druhe

imer

Straß

e

Holtgre

vens

traße

Bf Kasseler Tor

Schulbrede

Hilligenbusch


Am Laugrund

Corveyer Weg

Bahneinschnitt

Hochs

tiftss

tr.

Südr

ing

Uni/Sü

dring


KilianplatzQuerweg

Stephanusstraße

Elisabethkirche

Kilian

bad


Sighardstr.Ilseweg

Breslau

er Str

.

Walden

burge

r Str.

Grüner Weg

Frank

furte

r Weg




Weikenberg


Lich

tenfel

de

Voessingweg





Mistelweg

Bussardweg

Kaukenberg

Kleing

. Dah

ler W

eg

Iggen

haus

er Weg

Lang

efeld

Bergso

hle

Dahler

Heid

e

Dahl P

ost

Braken

berg

Lülingsberg

Pastorskamp




Ludwigsfelder Ring


Anhalter WegGerold






Dören-Park

Schwabenweg

Hauptbahnhof

Klöcknerstraße



Riemecker Feld


Alme Aue

Am Almerfeld


Blumenstraße

Spritzenhaus




Sander Straße





Marienloher Str.


An der

Talle

Schloß Neuhaus


Verner Straße

Almeri

ng

Mersch

weg

Mentro

pstra

ße

Mittelw

eg

Gesam

tschu

le

Fries

enweg

TÜV

Abzw. F

ischt

eiche

Schil

lerstr

aße

Hohe Kamp

Sande Kirche


Anemonenweg

Sande Schule

Dirksfeld

Hagebuttenweg

Sunderkampstr.

Sander Friedhof

Dubelohstraße

Schatenweg

HatzfelderPlatz

Memelstraße

Mastbruch Schule


Dietrichstraße

Husarenstraße

Habichtsweg

Fasanenweg

Bahnkreuzung

Thunebrücke

Hauptwache

Salvatorstraße

Pionierweg

Infanterieweg



Schützenplatz Nord

Obernheideweg

Lesteweg

*


Marienloh

Neuenbeken

Benhausen

Kaukenberg

Auf derLieth

Dahl

Wewer

Elsen

Gesseln

Sande

Schloß Neuhaus

Mastbruch

Sennelager





N4 Dahl






Modeling Labels

Model labels as special metro lines:

put all labels between apair of interchange stationsinto one parallelogram,

allow parallelograms tochange sides,bad news: a lot moreplanarity constraintsgood news: callbackmethod helps

v

w

r

s

t

u

Frankfurt (Main)

Mainz

Heidelberg

Karlsruhe

Mannheim




Modeling Labels


put all labels between apair of interchange stationsinto one parallelogram,allow parallelograms tochange sides,

bad news: a lot moreplanarity constraintsgood news: callbackmethod helps

v

w

r

s

t

u

Frankfurt (Main)

Mainz

Heidelberg

Karlsruhe

Mannheim




Modeling Labels


put all labels between apair of interchange stationsinto one parallelogram,allow parallelograms tochange sides,

bad news: a lot moreplanarity constraintsgood news: callbackmethod helps

Frankfurt (Main)

Mainz

Heidelberg

Karlsruhe

Mannheim

vr

s

t

u w




Modeling Labels


put all labels between apair of interchange stationsinto one parallelogram,allow parallelograms tochange sides,bad news: a lot moreplanarity constraints

good news: callbackmethod helps

Frankfurt (Main)

Mainz

Heidelberg

Karlsruhe

Mannheim

vr

s

t

u w




Modeling Labels


put all labels between apair of interchange stationsinto one parallelogram,allow parallelograms tochange sides,bad news: a lot moreplanarity constraintsgood news: callbackmethod helps

Frankfurt (Main)

Mainz

Heidelberg

Karlsruhe

Mannheim

vr

s

t

u w




Labeling – MontréalInput Input |V | |E | fcs. |L|

normal 65 66reduced 20 21 3 4labeled 62 74 14






↓MIP constr. var.







↓MIP constr. var.


?) 17 minutes w/o proof of opt.constr. of 60 edge pairs added




Labeling – MontréalInput Output (17 min.)

Georges-Vanier

Lucien L’Allier

Bonaventure

Square-Victoria

Place d’Armes

Champ-de-MarsSnowdon

Lionel Groulx

Place-Saint-Henri

Vendome

Villa-Maria

Beaudry

Papineau

Frontenac

Prefontaine

Joliette

Pie-IX

Viau

Assomption

Cadillac

Langelier

Radisson

Honore-Beaugrand

Jarry

Cremazie

Sauve

Henri-Bourassa

Cote-Sainte-Catherine

Plamondon

Namur

De La Savanne

Du College

Cote-Vertu

Beaubien

Rosemont

Laurier

Mont-Royal

Sherbrooke

Jean-Drapeau

Longueil

Charlevoix

LaSalle

De L’Eglise

Verdun

Jolicoeur

Monk

Angrignon

De Castelnau

Parc

Acadie

Outremont

Edouard-Montpetit

Universite-de-Montreal

Cote-des-NeigesBerri-UQAM

Fabre

D’Iberville

Saint-Michel

Jean-Talon

Saint-Laurent

Place-des-Arts

McGill

Peel

Guy-Concordia

Atwater




Labeling – Montréal

Georges-Vanier

Lucien L’Allier

Bonaventure

Square-Victoria

Place d’Armes

Champ-de-MarsSnowdon

Lionel Groulx

Place-Saint-Henri

Vendome

Villa-Maria

Beaudry

Papineau

Frontenac

Prefontaine

Joliette

Pie-IX

Viau

Assomption

Cadillac

Langelier

Radisson

Honore-Beaugrand

Jarry

Cremazie

Sauve

Henri-Bourassa

Cote-Sainte-Catherine

Plamondon

Namur

De La Savanne

Du College

Cote-Vertu

Beaubien

Rosemont

Laurier

Mont-Royal

Sherbrooke

Jean-Drapeau

Longueil

Charlevoix

LaSalle

De L’Eglise

Verdun

Jolicoeur

Monk

Angrignon

De Castelnau

Parc

Acadie

Outremont

Edouard-Montpetit

Universite-de-Montreal

Cote-des-NeigesBerri-UQAM

Fabre

D’Iberville

Saint-Michel

Jean-Talon

Saint-Laurent

Place-des-Arts

McGill

Peel

Guy-Concordia

Atwater




Labeling – ViennaInput Input |V | |E | fcs. |L|







↓MIP constr. var.







↓MIP constr. var.


?) 1 day w/o proof of opt.constr. of 160 edge pairs added




Labeling – ViennaInput Output (1 day)

Museumsquartier

Vorgartenstrasse

Donauinsel

VIC Kaisermuehlen

Alte Donau

Kagran

Kagraner Platz

Rennbahnweg

Aderklaaer Strasse

Grossfeldsiedlung

Leopoldau

Rochu

sgas

se

Kardin

al-Nag

el-Plat

z

Schlac

htha

usga

sse

Erdbe

rg

Gasom

eter

Zipper

erstr

asse

Enkpla

tz

Simm

ering

Taubstummengasse

Suedtiroler Platz

Keplerplatz

Reumannplatz

Schwedenplatz

Spittelau

Land

stras

se

Rossauer Laende

Friedensbruecke

Neuba

ugas

se

Ziegler

gass

e

Meid

ling

Haupt

stras

se

Schoe

nbru

nn

Hietzin

g

Braun

schw

eigga

sse

Unter

St.

Veit

Ober S

t. Veit

Huette

ldorf

Schot

tenr

ing

Schweg

lerstr

asse

John

stras

se

Huette

ldorfe

r Stra

sse

Kendle

rstra

sse

Ottakr

ing

Heiligenstadt

Gumpendorfer StrasseKarlsplatz

Praterstern

Nussdorfer Strasse

Waehringer Strasse Volksoper

Michelbeuern AKH

Alser Strasse

Josefstaedter Strasse

Thaliastrasse

Burggasse Stadthalle

Stube

ntor

Mar

gare

teng

uerte

l

Pilgra

mga

sse

Kette

nbru

ecke

ngas

se

Wes

tbah

nhof

Herre

ngas

se

Nestroyplatz

Schottentor Universitaet

Rathaus

Stadtpark

Messe

Trabrennstrasse

Stadion

Donaustadtbr?cke

Seestern

Stadlau

Hardeggasse

Donauspital

Aspernstrasse

Jaegerstrasse

Dresdner Strasse

Handelskai

Neue Donau

Floridsdorf

Steph

ansp

latz C

ity

Laen

genf

eldga

sse

Tabor

stras

se

Volksth

eate

r

Niederhofstrasse

Philadelphiabruecke

Tscherttegasse

Am Schoepfwerk

Alterlaa

Erlaaer Strasse

Perfektastrasse

Siebenhirten




Labeling – Vienna

Museumsquartier

Vorgartenstrasse

Donauinsel

VIC Kaisermuehlen

Alte Donau

Kagran

Kagraner Platz

Rennbahnweg

Aderklaaer Strasse

Grossfeldsiedlung

Leopoldau

Rochu

sgas

se

Kardin

al-Nag

el-Plat

z

Schlac

htha

usga

sse

Erdbe

rg

Gasom

eter

Zipper

erstr

asse

Enkpla

tz

Simm

ering

Taubstummengasse

Suedtiroler Platz

Keplerplatz

Reumannplatz

Schwedenplatz

Spittelau

Land

stras

se

Rossauer Laende

Friedensbruecke

Neuba

ugas

se

Ziegler

gass

e

Meid

ling

Haupt

stras

se

Schoe

nbru

nn

Hietzin

g

Braun

schw

eigga

sse

Unter

St.

Veit

Ober S

t. Veit

Huette

ldorf

Schot

tenr

ing

Schweg

lerstr

asse

John

stras

se

Huette

ldorfe

r Stra

sse

Kendle

rstra

sse

Ottakr

ing

Heiligenstadt

Gumpendorfer StrasseKarlsplatz

Praterstern

Nussdorfer Strasse

Waehringer Strasse Volksoper

Michelbeuern AKH

Alser Strasse

Josefstaedter Strasse

Thaliastrasse

Burggasse Stadthalle

Stube

ntor

Mar

gare

teng

uerte

l

Pilgra

mga

sse

Kette

nbru

ecke

ngas

se

Wes

tbah

nhof

Herre

ngas

se

Nestroyplatz

Schottentor Universitaet

Rathaus

Stadtpark

Messe

Trabrennstrasse

Stadion

Donaustadtbr?cke

Seestern

Stadlau

Hardeggasse

Donauspital

Aspernstrasse

Jaegerstrasse

Dresdner Strasse

Handelskai

Neue Donau

Floridsdorf

Steph

ansp

latz C

ity

Laen

genf

eldga

sse

Tabor

stras

se

Volksth

eate

r

Niederhofstrasse

Philadelphiabruecke

Tscherttegasse

Am Schoepfwerk

Alterlaa

Erlaaer Strasse

Perfektastrasse

Siebenhirten




To Do: Rectangular Stations & Multi-Edges




Summary (Metro Maps)

METROMAPLAYOUT is NP-hard.Formulated and implemented MIP.Results comparable to manually designed maps.Reduced MIP size & runtime drastically.Combined layout and labeling.Our MIP can draw any kind of sketch “nicely”.

To Do

rectangular stationsmulti-edgesuser interaction (e.g. fixing certain edge directions)




Summary (Metro Maps)

METROMAPLAYOUT is NP-hard.Formulated and implemented MIP.Results comparable to manually designed maps.Reduced MIP size & runtime drastically.Combined layout and labeling.Our MIP can draw any kind of sketch “nicely”.

To Do

rectangular stationsmulti-edgesuser interaction (e.g. fixing certain edge directions)




Thank you for the attention.

Any questions?


Documents

Vorlesung Graphenzeichnen: Order in the Underground or · Vorlesung Graphenzeichnen: Order in the Underground or How to Automate the Drawing of Metro Maps ... mit Perl Regul re Ausdr