Mathematical Aspects of Public Transportation Networks
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Mathematical Aspects of
Public Transportation Networks
Niels Lindner
July 9, 2018
July 9, 2018 1 / 48
Reminder Dates I July 12: no tutorial (excursion) I July 16: evaluation of Problem Set 10 and Test Exam I July 19: no tutorial I July 19/20: block seminar on shortest paths I August 7 (Tuesday): 1st exam I October 8 (Monday): 2nd exam Exams I location: ZIB seminar room I start time: 10.00am I duration: 60 minutes I permitted aids: one A4 sheet with notes July 9, 2018 2 / 48
Chapter 6
Metro Map Drawing
§6.1 Metro Maps
July 9, 2018 3 / 48
§6.1 Metro Maps
Berlin, 1921 (Pharus-Plan)
alt-berlin.info
July 9, 2018 4 / 48
§6.1 Metro Maps
Berlin, 1913 (Hochbahngesellschaft)
Sammlung Mauruszat, u-bahn-archiv.de
July 9, 2018 5 / 48
§6.1 Metro Maps
Berlin, 1933 (BVG)
Sammlung Mauruszat, u-bahn-archiv.de
July 9, 2018 6 / 48
§6.1 Metro Maps
London, 1926 (London Underground)
commons.wikimedia.org
July 9, 2018 7 / 48
§6.1 Metro Maps London, 1933 (Harry Beck) July 9, 2018 8 / 48
§6.1 Metro Maps
Plaque at Finchley Central station
Nick Cooper, commons.wikimedia.org
July 9, 2018 9 / 48§6.1 Metro Maps
Berlin, 1968 (BVG-West)
Sammlung Mauruszat, u-bahn-archiv.de
July 9, 2018 10 / 48§6.1 Metro Maps
Berlin, 2018 (BVG)
Wittenberge RE6 RB55 Kremmen Stralsund/Rostock RE5 RB12 Templin Stadt Groß Schönebeck (Schorfheide) RB27 Stralsund/Schwedt (Oder) RE3 RE66 Szczecin (Stettin) RB24 Eberswalde Legende Barrierefrei durch Berlin
Key to symbols Step-free access
Berlin Vehlefanz
Bärenklau
Sachsenhausen (Nordb)
Oranienburg 1 RB20 :
Lehnitz
Borgsdorf
RB27 Schmachtenhagen
RB27 Wensickendorf
Wandlitzsee
Wandlitz
Rüdnitz
76
RE1 RB22
RE1
S-Bahn-/U-Bahn
urban rail/underground
Bahn-Regionalverkehr
: Aufzug
lift
Rampe
Bernau 2 : RB22 regional rail ramp
Velten (Mark) Birkenwerder 8 : Zühlsdorf Basdorf
Bernau-Friedenstal : Einige Linien halten nicht überall : nur zur S-Bahn
Hohen Neuendorf West RB20 RE5.RB12 Schönwalde (Barnim) Some trains do not stop at all stations only to urban rail
Zepernick :
Hennigsdorf bE : :Hohen Neuendorf 0A Flughafen :
RB
RB55
Bergfelde : Schönfließ Mühlenbeck-Mönchmühle Schönerlinde nur zur U-Bahn
Röntgental :
27
airport only to underground
Heiligensee : :Frohnau Fernbahnhof :
Waidmannslust bF : Buch : nur zum Bahn-Regionalverkehr
Schulzendorf : :Hermsdorf mainline station only to regional rail
Karow RB27 :
Wittenau 8 : ZOB Zentraler Omnibusbahnhof
: Rathaus Reinickendorf bus terminal
Tegel : :Karl-Bonhoeffer- Wilhelmsruh : A B C VBB-Tarifteilbereiche Berlin
:6 Alt-Tegel : Blankenburg VBB fare zones Berlin
Nervenklinik
Borsigwerke
20
Eichborndamm : Alt-Reinickendorf Schönholz : Kein Zugverkehr
B
Stand: 7. Mai 2018
: Pankow-Heinersdorf 27
eE
.R
Holzhauser Str. . RB
RE6
: 6
no service © Berliner Verkehrsbetriebe (BVG)
: Otisstr. :Lindauer Allee . RE6 05.06. - 11.12.2018 600-1-18.1-2
RB
12
: Scharnweberstr. :Paracelsus-Bad . RE5
. RB
RE3
24
:Kurt-Schumacher-Platz 128 Residenzstr. : hE 2 Pankow
Werneuchen RB25
: Afrikanische Str. Franz-Neumann-Platz Osloer Str. 9 : Wollankstr. :
0A Tegel TXL Am Schäfersee Seefeld (Mark)
:Rehberge Blumberg
8
12
TXL 128
Wartenberg gE Blumberg-Rehhahn
Nauen
Seestr. Nauener Platz Vinetastr. : Ahrensfelde Nord
RB10 Pankstr. Ahrensfelde Friedhof
X9 109
RB14 7 Ahrensfelde
Brieselang :Amrumer Str. Leopoldplatz : Bornholmer
TX
RE2 :
:
L
Wismar Str. : Hohenschönhausen
: : Siemens- RE6 RE66 RB27 Mehrower Allee
Finkenkrug Zitadelle Haselhorst Paulsternstr. Rohrdamm damm Halemweg :Beusselstr. : Westhafen :Wedding : Gesundbrunnen Schönhauser Allee : Prenzlauer Allee :
5 Strausberg Nord
<
RE2
Falkensee : Gehrenseestr.
25
Jakob-Kaiser-Platz
. RE6
Greifswalder Str. : Raoul-Wallenberg-Str. Strausberg Stadt
RB
RE4.RE6.RB10 :Reinicken- RE3.RE5.RE6 Voltastr. :
Altstadt Spandau
. RB
1> Jungfernheide :
RB20
Humboldthain
: Eberswalder Str. :
Seegefeld : dorfer Str.
10
2 S4 : Bernauer Str. Hegermühle
. RB
RB13
RE3.RE4.RE5 Luftbrücke Plänterwald : Hirschgarten
RE2.RE7.RB14
:Hohenzollerndamm
1 S4
Karl-Marx-
RB
Blissestr. Bayerischer Platz Eisenacher
Birkengrund Waßmannsdorf
Ferch- : Dahlewitz
Ludwigsfelde
Lienewitz Seddin
Thyrow Rangsdorf Flughafen Berlin Brandenburg Airport
Ein Verbund.
Ein Tarif.
Dessau RE7 RB33 Jüterbog Jüterbog RE4 RE3 Lutherstadt Wittenberg/Falkenberg (Elster) Elsterwerda RE5 RE7 Wünsdorf-Waldstadt
Berliner Verkehrsbetriebe, bvg.de
July 9, 2018 11 / 48§6.1 Metro Maps
London, 2018 (Transport for London)
Tube map
1 2 3 4 5 6 7 8 9
Check before you travel
Chesham 9 Chalfont &
Latimer
8 7 Special fares apply
Watford Junction
8 7 6 5 Enfield Town
Cheshunt
Theobalds Grove
8 7 Epping
Theydon Bois
9
Special fares
apply § Brixton
No step-free access until September.
---------------------------------------------------------------------------
Shenfield
Watford High Street Bush Hill Debden
§ Heathrow
A
Amersham
Chorleywood
Watford
Croxley
Bushey
High Barnet
Totteridge & Whetstone
Cockfosters
Oakwood
Park Turkey Street
Southbury Chingford
Loughton
Buckhurst Hill
6 A
TfL Rail customers should change at
Terminals 2 & 3 for free rail transfer to
Rickmansworth Carpenders Park Woodside Park Southgate
Edmonton Green 5 Brentwood
Terminal 5.
---------------------------------------------------------------------------
4
Moor Park Roding Grange
Hatch End Mill Hill East Arnos Grove Valley Hill § Hounslow West
West Finchley
Northwood Silver Street Highams Park Step-free access for manual wheelchairs only.
Chigwell Harold Wood
West Ruislip Headstone Lane Edgware Bounds Green ---------------------------------------------------------------------------
Northwood Hills White Hart Lane
Hillingdon Ruislip Harrow &
Stanmore
Finchley Central Woodford Hainault Gidea Park § Kennington
Wealdstone Wood Green
Ruislip Manor Pinner Burnt Oak Harringay Bruce Grove Romford Bank branch trains will not stop between
Canons Park Wood Street Fairlop
East Finchley Green South Woodford
Uxbridge Ickenham North Harrow Colindale Turnpike Lane Lanes South Tottenham Saturday 26 May and mid-September.
Ruislip
Rayners Lane
Eastcote
Harrow-
on-the-Hill
Northwick
Kenton
Preston
Road
Queensbury
Kingsbury
Hendon Central
Brent Cross
3 Highgate
Archway
Crouch
Hill
Manor House
Seven
Sisters
Tottenham
Hale
Blackhorse
Road
Walthamstow
Central
4 Snaresbrook
Redbridge
Barkingside
Newbury
Park
Goodmayes
Chadwell
Heath
Emerson Park ---------------------------------------------------------------------------
§ Victoria
Step-free access is via the Cardinal Place
Gardens West Gospel Oak entrance.
Harrow Park Golders Green Hampstead Wanstead Gants
Stamford ---------------------------------------------------------------------------
3
Heath Hill Seven Kings Upminster
South Kenton Upper Holloway Hill Walthamstow
B South Harrow Wembley Park
Neasden Hampstead
Arsenal Queen’s Road Leyton
Leytonstone
Ilford Upminster Bridge
B § Services or access at these stations are
North Wembley Tufnell Park Finsbury Midland Road
Dollis Hill Park Stoke
Manor Park
subject to variation.
South Ruislip Wembley Central Holloway Road Newington Hornchurch
Kentish St James Street Leytonstone Wanstead Please search ‘TfL stations’ for full details.
Willesden Green Finchley Road Town West Kentish Town High Road Woodgrange
Sudbury Hill Stonebridge Park & Frognal Belsize Park Rectory Park Elm Park
Road Clapton Park
Kilburn Caledonian Road Highbury & Leyton Forest
Harlesden Dalston Gate
Kensal Brondesbury West Chalk Farm Islington Kingsland
Camden Stratford Dagenham
Northolt Willesden Junction Rise Park Hampstead Road East
Sudbury Town
2
Hackney International Maryland
Caledonian Downs Hackney
Camden Town Central Dagenham Heathway
Brondesbury Finchley Road Road & Canonbury
Kensal Green Barnsbury Stratford Becontree
Swiss Cottage Mornington Dalston Junction Homerton Hackney
Alperton Queen’s Park Kilburn South Crescent Wick
High Road Hampstead St John’s Wood Upney
King’s Cross
2
London Fields Stratford
Greenford St Pancras Haggerston High Street Barking
Kilburn Park Paddington Marylebone Cambridge Heath
Edgware Road Baker Great Portland Euston
Street East Ham
Maida Vale Street Hoxton Bethnal Green Pudding Abbey
Warwick Angel Mill Lane Road
Perivale Avenue Upton
Bethnal Green Mile End
C Edgware Warren Street
Euston Old Street Bow Park C
Square Road
Royal Oak Road Plaistow
Farringdon Shoreditch
Hanger Lane Regent’s Park High Street Bromley- West
Park Royal
Acton
Main Line
Westbourne Park
Ladbroke Grove
Latimer Road
Bayswater 1 Bond
Street
Oxford
Circus
Goodge
Street
Russell
Square Barbican
Moorgate Liverpool
Aldgate
East Whitechapel
Stepney Green
Devons Road
Bow
Church
by-Bow Ham
Star Lane
2 2/3 3 4
North Ealing Marble Arch Street
Hanwell Tottenham Holborn Chancery Lane Langdon Park
Hayes & Southall
West
Ealing
Ealing West
Acton
North
Acton
East
Acton
White
City
Shepherd’s
Bush
Holland
Park
Notting
Hill Gate
Queensway
Lancaster
Gate
Green Park
Court Road
Covent Garden
Leicester Square
St Paul’s
Bank
1 Aldgate
Limehouse
All Saints
Poplar
Canning
Town
Royal
Victoria
5 Harlington
4 3 Broadway
Ealing Common
Acton Central
South Acton
Wood Lane
Shepherd’s
Bush Market Kensington
(Olympia)
Hyde Park Corner
High Street Kensington
Knightsbridge
Piccadilly
Circus
Charing
Cannon Street
Mansion House
Monument Tower
Hill
Fenchurch Street Tower
Gateway
Shadwell
Wapping
Westferry
West India
Blackwall East
India Emirates
Royal
Docks
Custom House for ExCeL
Prince Regent
Goldhawk Road Cross Quay
D South Barons Gloucester Blackfriars River Thames West Silvertown Royal Albert
D
Ealing Acton Town Hammersmith Court Road St James’s
Victoria Park Temple Rotherhithe
Canary Wharf
Beckton Park
18
Embankment North Emirates
Northfields Chiswick Westminster London Canada
20
Turnham Stamford Ravenscourt West Earl’s South Sloane Bermondsey Greenwich Pontoon
Bridge Water Greenwich
ay
Park Green Brook Park Kensington Court Kensington Square Heron Quays Peninsula Dock Cyprus
M
Boston Manor
on
Tr
Osterley
nd
Waterloo South Quay London Gallions Reach
an
r Lo
Gunnersbury City Airport
sp
Hounslow East West Brompton
ort
2
fo
Crossharbour Key to lines
2
Beckton
ort
fo
Hounslow Central King George V
r Lo
sp
Southwark Surrey Quays Bakerloo
an
Mudchute
nd
Hounslow
Tr
1
on
West Kew Gardens Pimlico Borough
M
Heathrow Fulham Broadway Lambeth North Island Gardens Central
ay
Terminals 2 & 3 Hatton Cross
20
Parsons Green
18
Imperial Wharf
Richmond Circle
Putney Bridge Cutty Sark for Woolwich
Maritime Greenwich Arsenal
Queens Road District
E E
6 5 4 3 East Putney Peckham New Greenwich
Heathrow
Terminal 5
Southfields
Vauxhall
Elephant & Castle Cross Gate
Brockley
New Cross Deptford Bridge
3 4 Hammersmith & City
Jubilee
2
Peckham Rye Elverson Road
Wimbledon Park Kennington
Heathrow
Terminal 4 Clapham Wandsworth Oval Honor Oak Park Metropolitan
Wimbledon Junction Road Lewisham
Stockwell Forest Hill Beckenham
Clapham High Street Denmark Hill Birkbeck Road Northern
Key to symbols
Dundonald
Road
Clapham North
Clapham Common
Brixton
3 Sydenham
Penge West
Avenue
Road
Beckenham
Junction
Piccadilly
Victoria
4
Harrington Road
Explanation of zones Clapham South
Balham Anerley
Interchange stations 9 Station in Zone 9 Elmers End Waterloo & City
Crystal Palace Norwood Junction
8
3
Station in Zone 8
Step-free access from street to train Arena
Station in Zone 7 Tooting Bec DLR
5
7
Step-free access from street to platform
6
Station in both zones
Merton Park
Tooting Broadway Woodside London Trams Emirates Air Line
National Rail
5
Station in Zone 6
Station in Zone 5
Colliers Wood West Croydon
Blackhorse Lane fare zone cable car
F South Wimbledon F (special fares apply)
Airport 4 Station in Zone 4 Reeves Corner Centrale Wellesley
Station in both zones Road Lebanon Addiscombe
Riverboat services 3
Road London Overground
Station in Zone 3
Victoria Coach Station 2 Station in Zone 2 Morden Phipps Belgrave Mitcham Mitcham Beddington Therapia Ampere Waddon Wandle Church George East Sandilands
Morden Road Bridge Walk Junction Lane Lane Way Marsh Park Street Street Croydon
TfL Rail
Station in both zones
Lloyd Park Gravel Addington King Henry’s New
Emirates Air Line cable car 1
4
Station in Zone 1
Hill Village Fieldway Drive Addington London Trams
Coombe Lane
Transport for London May 2018 District open weekends and on
some public holidays
1 2 3 4 5 6 7 8 9
MAYOR OF LONDON
Online maps are strictly for personal use only. To license the Tube map for commercial use please visit tfl.gov.uk/maplicensing
Transport for London, tfl.gov.uk
July 9, 2018 12 / 48§6.1 Metro Maps
Saint Petersburg, 2018
Петербургский Метрополитен, metro.spb.ru
July 9, 2018 13 / 48Chapter 6
Metro Map Drawing
§6.2 Planar Graphs
July 9, 2018 14 / 48§6.2 Planar Graphs
Planar Embeddings
Let G = (V , E ) be a (simple) graph.
Definition
A planar embedding of G consists of an injective map ψ : V → R2 , and a
family of continuous and injective functions fe : [0, 1] → R2 for each edge
e ∈ E such that
I fvw (0) = ψ(v ) and fvw (1) = ψ(w ) for all vw ∈ E ,
S
I fe ((0, 1)) ∩ e 0 6=e fe 0 ([0, 1]) = ∅ for all e ∈ E .
G is called planar if it admits some planar embedding.
Remark
This embeds a graph into the plane using simple Jordan curves.
July 9, 2018 15 / 48§6.2 Planar Graphs
Faces
Definition
A connected component of R2 \
S
e∈E fe ([0, 1]) is called a face.
Observations
I There is exactly one unbounded face (Jordan curve theorem).
I The boundary of a bounded face F gives rise to a face circuit CF in G .
Lemma
{CF | F is a bounded face} is an undirected cycle basis of G .
Proof.
Let C be a circuit in G . Then R2 \ e∈E (C ) fe ([0, 1]) is the union of a
S
bounded and an unbounded component. Let F1 , . . . , Fk be the faces
containedPin the bounded component. Then C = CF1 + · · · + CFk .
Suppose F bounded face λF CF = 0 for some λF ∈ F2 . If e is an edge
between a bounded face F and the outer face, then λF = 0. Proceed by
induction on the number of bounded faces.
July 9, 2018 16 / 48§6.2 Planar Graphs
Euler’s formula
Let G be a planar graph with n vertices, m edges, f faces and c connected
components.
Corollary (Euler, 1758)
f − 1 = m − n + c.
Lemma
Suppose that G is connected and n ≥ 3.
(1) 2m ≥ 3f (2) m ≤ 3n − 6 (3) f ≤ 2n − 4
Proof.
P P
(1) Handshake: 2m = v ∈V deg(v ) = F face #{vertices along F } ≥ 3f .
If n ≥ 3, then the outer face contains at least 3 vertices.
(2) 2m ≥ 3f = 3(m − n + 2) = 3m − 3n + 6 ⇒ m ≤ 3n − 6.
(3) 3n − 6 ≥ m = n + f − 2 ⇒ f ≤ 2n − 4.
July 9, 2018 17 / 48§6.2 Planar Graphs
K5 and K3,3
Lemma
The complete graph K5 is not planar.
Proof.
5
We have n = 5 and m = 2 = 10, so m = 10 > 9 = 3 · 5 − 6 = 3n − 6.
Lemma
The complete bipartite graph K3,3 is not planar.
Proof.
Every circuit in K3,3 has length at least 4. In particular, if K3,3 was planar,
then 2m ≥ 4(f − 1) + 3 = 4(m − n + 1) + 3 = 4m − 4n + 7 and therefore
2m ≤ 4n − 7. But 2m = 2 · 32 = 18 and 4n − 7 = 4 · 6 − 7 = 17.
July 9, 2018 18 / 48§6.2 Planar Graphs Minors Definition Let G and M be undirected graphs. M is a minor of G if a series of the following operations transforms G into M: I delete a vertex I delete an edge I contract an edge Theorem (Wagner, 1937) An undirected graph is planar if and only if it contains neither K5 nor K3,3 as a minor. Proof. (⇒) Any minor of a planar graph must be planar. (⇐) Omitted. July 9, 2018 19 / 48
§6.2 Planar Graphs More on Minors Theorem (Robertson/Seymour, 2004) A family of graphs is closed under taking minors if and only if it has a finite number of minimal forbidden minors. Example Planar graphs: K5 , K3,3 Forests: C3 (circuit on 3 vertices) Theorem (Robertson/Seymour, 1995) For any fixed minor M, there is a O(n3 ) algorithm deciding whether a graph on n vertices has M as a minor. This is nice, but the proof is not constructive. For planarity testing, there are better (and explicit) algorithms: Theorem (Hopcroft/Tarjan, 1974) Planarity testing can be done in O(n) time. July 9, 2018 20 / 48
§6.2 Planar Graphs
Subdivisions
Definition
Let G and M be undirected graphs. M is a subdivision of G if it is
obtained from G by consecutively replacing an edge uw with two edges
uv , vw , inserting a new vertex v .
u w u v w
→
The reverse process is called smoothing.
Theorem (Kuratowski, 1930)
An undirected graph is planar if and only if it contains no subgraph that is
a subdivision of K5 or K3,3 .
Proof.
(⇒) Any subgraph and any smoothing of a planar graph must be planar.
(⇐) Omitted.
July 9, 2018 21 / 48§6.2 Planar Graphs 2-bases Observation Let G be a planar embedded graph. Then every edge is contained in at most two face circuits. Definition A cycle basis B of a graph is called a 2-basis if every edge is contained in at most two cycles of B. Theorem (MacLane, 1937) A graph is planar if and only if it admits a 2-basis. Proof (O’Neill, 1973). Planar graphs have 2-bases. The following would prove the converse: (1) Admitting a 2-basis is closed under taking subgraphs and smoothings. (2) K5 and K3,3 do not admit a 2-basis. Then any graph with a 2-basis does not contain a subdivision of K5 or K3,3 , and must hence be planar by Kuratowski’s theorem. July 9, 2018 22 / 48
§6.2 Planar Graphs
MacLane’s planarity criterion
Proof of (1).
Let {C1 , . . . , Cµ } be a 2-basis of a graph G = (V , E ).
Deleting an edge e ∈ E : If e is not contained in any cycle, nothing
happens. Otherwise this reduces the cyclomatic number by 1. If e is
contained in a single cycle (w.l.o.g. C1 ), then {C2 , . . . , Cµ } is a 2-basis of
(V , E \ {e}). If e is contained in two cycles (w.l.o.g. C1 , C2 ), then
{C1 + C2 , . . . , Cµ } is a 2-basis of (V , E \ {e}).
Deleting a vertex: Delete first all adjacent edges. Removing an isolated
vertex does not affect the cyclomatic number.
Smoothing uv and vw to uw : The columns of uv and vw in the cycle
matrix are the same, so smoothing does not change anything about the
2-basis.
July 9, 2018 23 / 48§6.2 Planar Graphs
MacLane’s planarity criterion
Proof of (2).
C6 } be a 2-basis for K5 . Set C7 := 6i=1 Ci . Observe that
P
Let {C1 , . . . ,P
C7 6= 0 and 7i=1 Ci,e = 0 ∈ F2 for all e ∈ E . In particular, every edge is
contained in exactly two of the cycles C1 , . . . , C7 . Hence
P 7
i=1 |E (Ci )| = 2|E (K5 )| = 20.
On the other hand, |E (Ci )| ≥ 3 for all i, so 7i=1 |E (Ci )| ≥ 21.
P
P4
P5 let {C1 , . . . , C4 } be a 2-basis
Now P5for K3,3 . Set C5 := i=1 Ci . Again
i=1 Ci,e = 0 for all P e ∈ E , so i=1 |E (Ci )| = 2|E (K3,3 )| = 18. Since
|E (Ci )| ≥ 4 for all i, 5i=1 |E (Ci )| ≥ 20.
July 9, 2018 24 / 48§6.2 Planar Graphs
More on 2-bases
Lemma
The cycle matrix of a 2-basis of a directed graph is totally unimodular. In
particular, any 2-basis is integral.
Proof.
Orient all cycles counter-clockwise w.r.t. some planar embedding. Proceed
by induction on the size q of a quadratic submatrix A of the cycle matrix.
The case q = 1 is clear. If q ≥ 2, there are two cases:
(1) A contains a column with a single non-zero entry. Use Laplace
expansion and induction.
(2) All columns of A have at least two non-zero entries. Since we have a
2-basis, there are exactly two non-zeros, a +1 and a −1. So all rows of
A add up to 0, showing det A = 0.
July 9, 2018 25 / 48§6.2 Planar Graphs
Graph drawings
Let G be a planar graph.
Types of planar embeddings
I polygonal: All edges are embedded as polygonal arcs.
I straight line: All edges are drawn as straight line segments.
I rectilinear : All edges are drawn as straight line segments with slopes
k · 90◦ , k ∈ Z.
I octilinear : All edges are drawn as straight line segments with slopes
k · 45◦ , k ∈ Z.
July 9, 2018 26 / 48§6.2 Planar Graphs
Combinatorial type
Definition
Let G = (V , E ) be a planar graph with some planar embedding. For each
vertex v ∈ V , the embedding defines an ordering of the neighborhood of v
by counter-clockwise sorting of the edges incident to v . This is the
combinatorial type of the embedding.
B D
A A
D C B C
A: (B, D, C) A: (D, B, C)
This defines an equivalence relation on planar embeddings of G .
July 9, 2018 27 / 48§6.2 Planar Graphs
Straight line drawings
Theorem (Fáry, 1948)
Any simple planar graph has a straight line drawing.
Proof.
Let G be a simple connected planar graph with n ≥ 3 vertices, embedded
into the plane. W.l.o.g. G is maximally planar, i.e., m = 3n − 6, and every
bounded face is a triangle.
Claim: If uvw is a triangle, then there is a straight line embedding of the
same combinatorial type where u, v , w are the vertices along the outer face.
This is easy for n = 3. Thus let n ≥ 4, and let uvw be a triangle. Not all
of u, v , w have degree 2 (connectedness). Assume that the other n − 3
vertices have degree at least 6. Then
X
deg(v ) ≥ 6(n − 3) + 7 = 6n − 11 > 6n − 12 = 2m,
v ∈V
contradicting the Handshaking Lemma.
July 9, 2018 28 / 48§6.2 Planar Graphs Straight line drawings Proof (cont.) In particular, we find a vertex z different from u, v , w with deg(z) ≤ 5. Now remove z from the graph and retriangulate the new face F . The new graph has n − 1 vertices and – by induction – admits a straight line embedding of the same combinatorial type where u, v , w are the vertices along the outer face. In particular, the face F has become a simple polygon with at most 5 sides. By the art gallery theorem (with b5/3c = 1 guards), there is a point inside this polygon that can be connected to all vertices of F by non-crossing straight lines (Short proof: Triangulations of simple polygons admit a 3-coloring.) Theorem (De Fraysseix/Pach/Pollack, 1990) Straight line drawings on a grid can be found in linear time. July 9, 2018 29 / 48
§6.2 Planar Graphs
Straight line drawings: Example
z
delete z
−−−−→
−−−−−→
induction
insert z
←−−−−
July 9, 2018 30 / 48§6.2 Planar Graphs Rectilinear and octilinear drawings Let G be a graph. Theorem (Garg/Tamassia, 2001) It is NP-hard to decide whether G admits a rectilinear drawing. Theorem (Tamassia, 1987) Fix some planar embedding of G . There is a polynomial-time algorithm that decides whether G admits a rectilinear drawing preserving the combinatorial type. Theorem (Nöllenburg, 2005) Fix some planar embedding of G . It is NP-hard to decide whether G admits an octilinear drawing preserving the combinatorial type. Both NP-hardness proofs reduce (variants of) the 3-SAT problem. Remark Clearly, if G admits a rectilinear (octilinear) drawing, then deg(v ) ≤ 4 (≤ 8) for all v ∈ V (G ). July 9, 2018 31 / 48
Chapter 6
Metro Map Drawing
§6.3 Octilinear Layout Computation
July 9, 2018 32 / 48§6.3 Octilinear Layout Computation
Requirements
Design Principles (taken from Nöllenburg, 2011)
I Preserve the combinatorial type.
I Octilinear drawing: All edges are drawn as line segments with slopes
k · 45◦ for k ∈ {0, 1, . . . , 7}.
I Lines should avoid sharp bends, and pass straight through
interchanges.
I Ensure a minimum distance between stations, and stations and
non-incident edges.
I Minimize geometric distortion.
I Use uniform edge lengths.
I Use large angular resolution.
I Place station labels in a readable way.
I ...
July 9, 2018 33 / 48§6.3 Octilinear Layout Computation
Metro Map Layout Problem
Input
I a line network consisting of a graph G = (V , E ) of maximum degree 8
and a line cover L
I a planar embedding of G , e.g., by geographical coordinates
Output
I a map ψ : V → R2 inducing an octilinear drawing of G
I satisfying/optimizing design principles
Solution Methods
I metaheuristics (hill climbing, simulated annealing, ant colonies, . . . )
I local optimization: least squares
I global optimization: mixed integer programming
July 9, 2018 34 / 48§6.3 Octilinear Layout Computation
U-Bahn Berlin: geographical layout
173 vertices, 184 edges, 10 lines
July 9, 2018 35 / 48§6.3 Octilinear Layout Computation
U-Bahn Berlin: curvilinear layout
least squares method (Wang/Chi 2011, Wang/Peng 2016)
July 9, 2018 36 / 48§6.3 Octilinear Layout Computation
U-Bahn Berlin: octilinear layout?
least squares method (Wang/Chi 2011, Wang/Peng 2016), naive python/cvxopt implementation: 36 s
July 9, 2018 37 / 48§6.3 Octilinear Layout Computation Octilinear Layout MIP We will use the formulation due to Nöllenburg (2005): Hard constraints (feasibility) I octilinearity I combinatorial type preservation I minimum edge length I minimum distance for non-adjacent edges In particular, a feasible solution guarantees octilinearity. Soft constraints (objective) I bend minimization I preservation of relative positions for adjacent stations I minimum total edge length July 9, 2018 38 / 48
§6.3 Octilinear Layout Computation
Octilinear Layout MIP: Basic variables
I vertex coordinates xv , yv ∈ [0, |V |] for v ∈ V
I additional vertex coordinates zv := xv + yv , wv := xv − yv
I edge directions dirvw , dirwv ∈ {0, 1, . . . , 7} for {v , w } ∈ E
hj ki
,w )
I original directions secvw := ^(v 45 ◦ + 1
2 ,
8
where ^(v , w ) ∈ (−180◦ , 180◦ ] is the slope of the edge {v , w }
Nöllenburg (2005)
July 9, 2018 39 / 48§6.3 Octilinear Layout Computation
Octilinear Layout MIP: Directions
I binary variables αi,vw for i ∈ {−1, 0, 1} and {v , w } ∈ E such that
αi,vw = 1 ⇔ edge {v , w } is in original direction +i · 45◦
α−1,vw + α0,vw + α1,vw = 1, {v , w } ∈ E
dirvw + Mαi,vw ≤ M + [secvw + i]8 , i ∈ {−1, 0, 1}, {v , w } ∈ E
dirvw − Mαi,vw ≥ −M + [secvw + i]8 , i ∈ {−1, 0, 1}, {v , w } ∈ E
dirwv + Mαi,vw ≤ M + [secvw + i + 4]8 , i ∈ {−1, 0, 1}, {v , w } ∈ E
dirwv − Mαi,vw ≥ −M + [secvw + i + 4]8 , i ∈ {−1, 0, 1}, {v , w } ∈ E
M 0 is a large constant
I relating directions with coordinates, e.g., for secvw = 2 and α0,vw :
xv − xw + Mα0,vw ≤ M
xv − xw − Mα0,vw ≥ −M
yv − yw + Mα0,vw ≤ M − minimum edge length
If α0,vw = 1, then xv = xw and yw ≥ yv + minimum edge length.
July 9, 2018 40 / 48§6.3 Octilinear Layout Computation
Octilinear Layout MIP: Planar embedding
I preservation of combinatorial type: binary variables βi,v for
i ∈ {1, . . . , deg(v )} and v ∈ V ,
deg(v )
X
βi,v = 1, v ∈ V,
i=1
dirv ,wj − dirv ,w[j−1] + Mβj (v ) ≥ 1, v ∈ V , j = 1, . . . , deg(v ),
deg(v )
where w1 , . . . , wdeg(v ) are the neighbors of v in counter-clockwise
order
I planarity constraints: modeled by binary variables γi,e,e 0 for
i ∈ {0, . . . , 7} and e, e 0 ∈ E non-incident (large number!)
July 9, 2018 41 / 48§6.3 Octilinear Layout Computation
Octilinear Layout MIP: Objective
I total edge length objective: by new variables lenvw , vw ∈ E , e.g., for
secvw = 2 and α0,vw :
yv − yw + Mα0,vw = M − lenvw
If α0,vw = 1, then xv = xw , and yw ≥ yv = lenvw . Edge lengths are
measured w.r.t. | · |∞ : a diagonal line segment (x, y ) → (x + 1, y + 1)
has length 1.
I bend objective: for each three consecutive stations (u, v , w ) on a line
of the network, add the constraints
(
|diruv − dirvw | if |diruv − dirvw | ≤ 4
benduvw =
8 − |diruv − dirvw | if |diruv − dirvw | ≥ 5
I relative position objective:
−M · relvw ≤ dirvw − secvw ≤ M · relvw , vw ∈ E
P P P
I objective function: λ1 vw lenvw + λ2 uvw benduvw + λ3 vw relvw
July 9, 2018 42 / 48§6.3 Octilinear Layout Computation
Octilinear Layout MIP: Practical Aspects
Size
Suppose that G has n vertices and m edges. Let m0 := `∈L |E (`)|. Then
P
the MIP formulation uses uses O(n + m0 + m2 ) variables and constraints.
Example
The Berlin U-Bahn example needs
I 3 046 variables (1 623 binary, 546 integer, 877 continuous) and 7 149
constraints without planarity constraints.
I 137 158 variables (135 735 binary, 546 integer, 877 continuous) and
560 361 constraints with planarity constraints
Pre-processing
I Do not use all planarity constraints (heuristics, lazy constraints).
I Contract vertices of degree 2.
July 9, 2018 43 / 48§6.3 Octilinear Layout Computation
Crossing minimization
Consider a line network (G , L). The metro line crossing minimization
problem is to find an ordering of the lines at each station such that the
total number of crossings of lines is minimized.
Results
I The problem is in general NP-hard (Fink/Pupyrev, 2013).
I There is an integer programming formulation (Asquith et. al., 2008).
I The key step is to determine the ordering of the lines at their
terminals.
I For a fixed ordering at the terminals, there is a polynomial-time
algorithm.
July 9, 2018 44 / 48§6.3 Octilinear Layout Computation U-Bahn Berlin: octilinear layout! mixed integer programming method (Nöllenburg 2005), solution found by CPLEX 12.7.1 after 66 s, optimality: 15 min 55 s July 9, 2018 45 / 48
§6.3 Octilinear Layout Computation S-Bahn Berlin: geographical layout July 9, 2018 46 / 48
§6.3 Octilinear Layout Computation
S-Bahn Berlin: curvilinear layout
least squares method, naive python/cvxopt implementation: 31 s
July 9, 2018 47 / 48§6.3 Octilinear Layout Computation
S-Bahn Berlin: octilinear layout
mixed integer programming method, solution found by CPLEX 12.7.1 after 25 s, optimality gap: ≤ 3%
July 9, 2018 48 / 48Sie können auch lesen
