Metro Maps on Octilinear Grid Graphs - Hannah Bast 1, Patrick Brosi 1 and Sabine Storandt 2 - Uni Freiburg
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Metro Maps on Octilinear Grid Graphs Hannah Bast 1 , Patrick Brosi 1 and Sabine Storandt 2 1 University of Freiburg 2 University of Konstanz EuroVis 2020 - Norrköping, Sweden
Motivation
Tube map - Official London Tube Map
1 2 3 4 5 6 7 8 9
9 8 7 8 7 6 5 8 7
Check before you trav
Chesham
Chalfont &
Latimer
Outside fare zones
Watford Junction
Enfield Town
Cheshunt
Theobalds Grove
Epping
Theydon Bois
9 Outside
fare zones
§ East Ham
No step-free access to the
District line from Tuesday 2
6
Watford High Street Bush Hill Debden Shenfield until early January 2020.
Watford Cockfosters Park Turkey Street
Amersham High Barnet Loughton -----------------------------------------------
A A § Heathrow
5
Chorleywood Bushey Oakwood
Croxley Totteridge & Whetstone Southbury Chingford Buckhurst Hill
TfL Rail customers should c
Rickmansworth Southgate Brentwood
Carpenders Park Terminals 2 & 3 for free rail t
4
Woodside Park Edmonton Green
Moor Park Roding Grange to Terminal 5.
Hatch End Mill Hill East Arnos Grove Valley Hill
West Finchley Silver Street
Highams Park -----------------------------------------------
Northwood Edgware
West Ruislip Headstone Lane Stanmore Bounds Green Chigwell Harold Wood § Hounslow West
White Hart Lane
Northwood Hills
Woodford Hainault Gidea Park Step-free access for manua
Hillingdon Ruislip Harrow & Burnt Oak Finchley Central Wood Green
Canons Park Bruce Grove wheelchairs only.
Ruislip Manor Pinner Wealdstone Harringay Romford
Wood Street South Fairlop -----------------------------------------------
Green South
4
Colindale East Finchley Woodford
3
Uxbridge Ickenham Queensbury Turnpike Lane Lanes Tottenham
North Harrow § Services or access at these
6
Eastcote Barkingside
Kenton Hendon Central Highgate Crouch Blackhorse Snaresbrook Emerson Park
Harrow-
Kingsbury
Hill Road Chadwell subject to variation.
Seven Newbury
on-the-Hill Preston Road Manor House Sisters Heath To check before you travel
Brent Cross Archway Walthamstow Park
Rayners Lane Redbridge
Ruislip Neasden Central tfl.gov.uk/plan-a-journey
Gardens West Northwick Tottenham Goodmayes
3
Harrow Park Golders Green Hale Wanstead Gants
B Outside South Kenton Wembley Park
Dollis Hill
Upper Holloway Stamford Walthamstow Leytonstone
Hill Seven Kings Upminster
B
fare South Harrow
North Wembley
Willesden Green Hampstead
Gospel
Oak Tufnell Park
Arsenal
Finsbury
Hill
Stoke
Queen’s Road Leyton
Midland Road
Ilford Upminster Bridge
zones South Ruislip Wembley Central
Kilburn
West Hampstead Kentish
Holloway Road
Park
Newington
Leytonstone Wanstead
Manor Park Hornchurch
Hampstead Heath Town Kentish Town St James Street
Sudbury Hill Stonebridge Park Rectory High Road Park Woodgrange Elm Park
West Park
Kensal Brondesbury Caledonian Road Clapton Forest
Reading Rise Park Road Leyton
Harlesden Finchley Road Belsize Park Highbury & Gate
2
& Frognal Camden Road Dalston Dagenham
Northolt Islington Kingsland Stratford
Twyford Sudbury Town Willesden Junction Brondesbury Chalk Farm Hackney International East
Maryland
Downs Hackney
Kensal Green Camden Town Caledonian Central Dagenham Heathway
Maidenhead Kilburn Road & Canonbury Stratford
Queen’s Park Finchley Road Barnsbury Becontree
High Road South Hampstead Mornington Homerton Hackney
Alperton
2
Crescent Dalston Junction Wick Upney
Taplow King’s Cross London Fields Stratford
Swiss Cottage St Pancras High Street
Haggerston Barking
Burnham Greenford Kilburn Park Cambridge Heath
Paddington Edgware Road Marylebone St John’s Wood Euston
East Ham
Maida Vale Hoxton Bethnal Green Pudding Abbey
Slough Warwick Angel Mill Lane Road
Perivale Avenue Upton
Bethnal Green Mile End
C Edgware
Great Portland Euston Old Street Bow Park C
Baker Street Street Square Warren Road
Langley Park Royal Road Street Plaistow
Royal Oak Farringdon Shoreditch
Hanger High Street Bromley- West
65 4 3 2 1
Lane Regent’s Park Stepney Green
Iver Westbourne Park Goodge by-Bow Ham
Acton Russell Bow
Main Line Street Square Barbican Church
Ladbroke Grove Bayswater Aldgate
Oxford East Whitechapel
Bond Street
2 2/3 3 4
Circus Moorgate Devons Road
West Latimer Road Liverpool Star Lane
1
Drayton Hayes & North Ealing Street
Harlington Marble Arch Tottenham Holborn Chancery Lane
Hanwell Langdon Park
East White Shepherd’s Notting Court Road
Acton City Bush Hill Gate Bank Aldgate
Southall West Covent Garden All Saints Canning
Lancaster Town
Ealing North Holland Queensway Gate St Paul’s Royal
Ealing West
Acton Acton Park Green Park Leicester Square Limehouse Poplar Victoria
Broadway Wood Lane Monument
Hyde Park Corner
Shepherd’s Piccadilly Cannon Street Custom House for ExCeL
Acton Central Circus Tower Shadwell Westferry Blackwall East
Bush Market High Street Kensington India
Kensington Hill Tower Emirates
Ealing Common Mansion House Fenchurch Street
Goldhawk Road (Olympia) Knightsbridge Charing Gateway Wapping Royal Prince Regent
South Acton West India Docks
Cross
D South Barons Gloucester Blackfriars River Thames
Quay
Royal Albert
D
Ealing Acton Town Hammersmith Court Road Canary
Westminster West Silvertown
Victoria Temple Rotherhithe Wharf
Beckton Park
Northfields
Chiswick Turnham Stamford Ravenscourt West Earl’s South Sloane St James’s Embankment London Canada North Emirates
Bermondsey Greenwich
Park Green Brook Park Kensington Court Kensington Square Park Bridge Water Heron Quays Greenwich Cyprus
Boston Manor Pontoon Dock
Peninsula
Osterley South Quay Gallions Reach
London
Waterloo City Airport
2 2
Hounslow East Gunnersbury West Brompton
Crossharbour Beckton Key to lines
Hounslow Central Pimlico King George V
1
Hounslow Southwark Surrey Quays Mudchute Bakerloo
West Kew Gardens Borough
Heathrow Fulham Broadway Island Gardens Central
Hatton Cross Lambeth North 00/0000/P
Terminals 2 & 3
Parsons Green Imperial Wharf
Putney Bridge Woolwich Arsenal Circle
Cutty Sark for
Richmond Maritime Greenwich
New Cross District
6 5 4 3
Queens Road
E E
3 4
East Putney Peckham Greenwich
Vauxhall New Cross Gate Hammer
Elephant & Castle
Heathrow Southfields Deptford Bridge
2
Terminal 5
Peckham Rye
Brockley Jubilee
Heathrow Wimbledon Park Kennington Elverson Road
Terminal 4 Clapham Wandsworth Honor Oak Park Metropo
Wimbledon Oval Lewisham
Junction Road
Stockwell Forest Hill
Clapham High Street Northern
3
Denmark Hill
Clapham North Sydenham
Piccadilly
Key to symbols Explanation of zones Clapham Common
Avenue
4
Dundonald Penge West Harrington Road Road
Interchange stations 9 Station in Zone 9 Road Clapham South
Brixton Beckenham
Junction
Victoria
Anerley Birkbeck Beckenham
Internal interchange 8 Station in Zone 8 Balham Road Waterloo
3
Station in Zone 7 Crystal Palace
Under a 10 minute walk between stations 7 Norwood Junction Elmers End
5
Step-free access from street to train
Station in both zones Tooting Bec DLR
6 Station in Zone 6
Arena
Merton Park Tooting Broadway
Step-free access from street to platform
Emirates
National Rail
5 Station in Zone 5
Station in Zone 4
Colliers Wood West Croydon Woodside
London Trams cable car
F 4 F
Airport
3
Station in both zones
South Wimbledon
Reeves Corner Centrale Wellesley
Road Lebanon
Blackhorse Lane
fare zone (special f
River services interchange Station in Zone 3 Addiscombe
Road London O
Victoria Coach Station 2 Station in Zone 2
Morden Phipps Belgrave Mitcham Mitcham Beddington Therapia Ampere Waddon Wandle Church George East Sandilands TfL Rail
Emirates Air Line cable car Station in both zones
Road Bridge Walk Junction Lane Lane Way Marsh Park Street Street Croydon
Outside fare zones, Oyster not valid 1 Station in Zone 1 Morden Lloyd Park
4 1
London T
Zonal tint values: Coombe Lane Addington King Henry’s New
Village Fieldway Drive Addington
Transport for London January 2020 Reg. user No. 00/0000/P Zones 10% Gravel Hill District o
some public
1 2 3 4 Zone 2/3
5 15% 6 7 8 9
Zone numbers 10%Goals
Given an input line graph G = (V, E, L) with edge lines L(e),
render an octilinear drawing of G automatically and fast
2Goals
Given an input line graph G = (V, E, L) with edge lines L(e),
render an octilinear drawing of G automatically and fast
Input line graph G 2Goals
Given an input line graph G = (V, E, L) with edge lines L(e),
render an octilinear drawing of G automatically and fast
→
Input line graph G Octilinear drawing of G 2Goals (ctd.)
Allow arbitrary (but optimal) number of edge bends to
circumvent obstacles and approximate geographical courses
3Goals (ctd.)
Allow arbitrary (but optimal) number of edge bends to
circumvent obstacles and approximate geographical courses
Octilinear embedding
3Goals (ctd.)
Allow arbitrary (but optimal) number of edge bends to
circumvent obstacles and approximate geographical courses
Lake
vs.
Octilinear embedding Octilinear drawing
3Octilinear
Tube map Embedding vs. Octilinear Drawing
1 2 3 4 5 6 7 8 9
9 8 7 8 7 6 5 8 7
Check before you trav
Chesham
Chalfont &
Latimer
Outside fare zones
Watford Junction
Enfield Town
Cheshunt
Theobalds Grove
Epping
Theydon Bois
9 Outside
fare zones
§ East Ham
No step-free access to the
District line from Tuesday 2
6
Watford High Street Bush Hill Debden Shenfield until early January 2020.
Watford Cockfosters Park Turkey Street
Amersham High Barnet Loughton -----------------------------------------------
A A § Heathrow
5
Chorleywood Bushey Oakwood
Croxley Totteridge & Whetstone Southbury Chingford Buckhurst Hill
TfL Rail customers should c
Rickmansworth Southgate Brentwood
Carpenders Park Terminals 2 & 3 for free rail t
4
Woodside Park Edmonton Green
Moor Park Roding Grange to Terminal 5.
Hatch End Mill Hill East Arnos Grove Valley Hill
West Finchley Silver Street
Highams Park -----------------------------------------------
Northwood Edgware
West Ruislip Headstone Lane Stanmore Bounds Green Chigwell Harold Wood § Hounslow West
White Hart Lane
Northwood Hills
Woodford Hainault Gidea Park Step-free access for manua
Hillingdon Ruislip Harrow & Burnt Oak Finchley Central Wood Green
Canons Park Bruce Grove wheelchairs only.
Ruislip Manor Pinner Wealdstone Harringay Romford
Wood Street South Fairlop -----------------------------------------------
Green South
4
Colindale East Finchley Woodford
3
Uxbridge Ickenham Queensbury Turnpike Lane Lanes Tottenham
North Harrow § Services or access at these
6
Eastcote Barkingside
Kenton Hendon Central Highgate Crouch Blackhorse Snaresbrook Emerson Park
Harrow-
Kingsbury
Hill Road Chadwell subject to variation.
Seven Newbury
on-the-Hill Preston Road Manor House Sisters Heath To check before you travel
Brent Cross Archway Walthamstow Park
Rayners Lane Redbridge
Ruislip Neasden Central tfl.gov.uk/plan-a-journey
Gardens West Northwick Tottenham Goodmayes
3
Harrow Park Golders Green Hale Wanstead Gants
B Outside South Kenton Wembley Park
Dollis Hill
Upper Holloway Stamford Walthamstow Leytonstone
Hill Seven Kings Upminster
B
fare South Harrow
North Wembley
Willesden Green Hampstead
Gospel
Oak Tufnell Park
Arsenal
Finsbury
Hill
Stoke
Queen’s Road Leyton
Midland Road
Ilford Upminster Bridge
zones South Ruislip Wembley Central
Kilburn
West Hampstead Kentish
Holloway Road
Park
Newington
Leytonstone Wanstead
Manor Park Hornchurch
Hampstead Heath Town Kentish Town St James Street
Sudbury Hill Stonebridge Park Rectory High Road Park Woodgrange Elm Park
West Park
Kensal Brondesbury Caledonian Road Clapton Forest
Reading Rise Park Road Leyton
Harlesden Finchley Road Belsize Park Highbury & Gate
2
& Frognal Camden Road Dalston Dagenham
Northolt Islington Kingsland Stratford
Twyford Sudbury Town Willesden Junction Brondesbury Chalk Farm Hackney International East
Maryland
Downs Hackney
Kensal Green Camden Town Caledonian Central Dagenham Heathway
Maidenhead Kilburn Road & Canonbury Stratford
Queen’s Park Finchley Road Barnsbury Becontree
High Road South Hampstead Mornington Homerton Hackney
Alperton
2
Crescent Dalston Junction Wick Upney
Taplow King’s Cross London Fields Stratford
Swiss Cottage St Pancras High Street
Haggerston Barking
Burnham Greenford Kilburn Park Cambridge Heath
Paddington Edgware Road Marylebone St John’s Wood Euston
East Ham
Maida Vale Hoxton Bethnal Green Pudding Abbey
Slough Warwick Angel Mill Lane Road
Perivale Avenue Upton
Bethnal Green Mile End
C Edgware
Great Portland Euston Old Street Bow Park C
Baker Street Street Square Warren Road
Langley Park Royal Road Street Plaistow
Royal Oak Farringdon Shoreditch
Hanger High Street Bromley- West
65 4 3 2 1
Lane Regent’s Park Stepney Green
Iver Westbourne Park Goodge by-Bow Ham
Acton Russell Bow
Main Line Street Square Barbican Church
Ladbroke Grove Bayswater Aldgate
Oxford East Whitechapel
Bond Street
2 2/3 3 4
Circus Moorgate Devons Road
West Latimer Road Liverpool Star Lane
1
Drayton Hayes & North Ealing Street
Harlington Marble Arch Tottenham Holborn Chancery Lane
Hanwell Langdon Park
East White Shepherd’s Notting Court Road
Acton City Bush Hill Gate Bank Aldgate
Southall West Covent Garden All Saints Canning
Lancaster Town
Ealing North Holland Queensway Gate St Paul’s Royal
Ealing West
Acton Acton Park Green Park Leicester Square Limehouse Poplar Victoria
Broadway Wood Lane Monument
Hyde Park Corner
Shepherd’s Piccadilly Cannon Street Custom House for ExCeL
Acton Central Circus Tower Shadwell Westferry Blackwall East
Bush Market High Street Kensington India
Kensington Hill Tower Emirates
Ealing Common Mansion House Fenchurch Street
Goldhawk Road (Olympia) Knightsbridge Charing Gateway Wapping Royal Prince Regent
South Acton West India Docks
Cross
D South Barons Gloucester Blackfriars River Thames
Quay
Royal Albert
D
Ealing Acton Town Hammersmith Court Road Canary
Westminster West Silvertown
Victoria Temple Rotherhithe Wharf
Beckton Park
Northfields
Chiswick Turnham Stamford Ravenscourt West Earl’s South Sloane St James’s Embankment London Canada North Emirates
Bermondsey Greenwich
Park Green Brook Park Kensington Court Kensington Square Park Bridge Water Heron Quays Greenwich Cyprus
Boston Manor Pontoon Dock
Peninsula
Osterley South Quay Gallions Reach
London
Waterloo City Airport
2 2
Hounslow East Gunnersbury West Brompton
Crossharbour Beckton Key to lines
Hounslow Central Pimlico King George V
1
Hounslow Southwark Surrey Quays Mudchute Bakerloo
West Kew Gardens Borough
Heathrow Fulham Broadway Island Gardens Central
Hatton Cross Lambeth North 00/0000/P
Terminals 2 & 3
Parsons Green Imperial Wharf
Putney Bridge Woolwich Arsenal Circle
Cutty Sark for
Richmond Maritime Greenwich
New Cross District
6 5 4 3
Queens Road
E E
3 4
East Putney Peckham Greenwich
Vauxhall New Cross Gate Hammer
Elephant & Castle
Heathrow Southfields Deptford Bridge
2
Terminal 5
Peckham Rye
Brockley Jubilee
Heathrow Wimbledon Park Kennington Elverson Road
Terminal 4 Clapham Wandsworth Honor Oak Park Metropo
Wimbledon Oval Lewisham
Junction Road
Stockwell Forest Hill
Clapham High Street Northern
3
Denmark Hill
Clapham North Sydenham
Piccadilly
Key to symbols Explanation of zones Clapham Common
Avenue
4
Dundonald Penge West Harrington Road Road
Interchange stations 9 Station in Zone 9 Road Clapham South
Brixton Beckenham
Junction
Victoria
Anerley Birkbeck Beckenham
Internal interchange 8 Station in Zone 8 Balham Road Waterloo
3
Station in Zone 7 Crystal Palace
Under a 10 minute walk between stations 7 Norwood Junction Elmers End
5
Step-free access from street to train
Station in both zones Tooting Bec DLR
6 Station in Zone 6
Arena
Merton Park Tooting Broadway
Step-free access from street to platform
Emirates
National Rail
5 Station in Zone 5
Station in Zone 4
Colliers Wood West Croydon Woodside
London Trams cable car
F 4 F
Airport
3
Station in both zones
South Wimbledon
Reeves Corner Centrale Wellesley
Road Lebanon
Blackhorse Lane
fare zone (special f
River services interchange Station in Zone 3 Addiscombe
Road London O
Victoria Coach Station 2 Station in Zone 2
Morden Phipps Belgrave Mitcham Mitcham Beddington Therapia Ampere Waddon Wandle Church George East Sandilands TfL Rail
Emirates Air Line cable car Station in both zones
Road Bridge Walk Junction Lane Lane Way Marsh Park Street Street Croydon
Outside fare zones, Oyster not valid 1 Station in Zone 1 Morden Lloyd Park
4 4
London T
Zonal tint values: Coombe Lane Addington King Henry’s New
Village Fieldway Drive Addington
Transport for London January 2020 Reg. user No. 00/0000/P Zones 10% Gravel Hill District o
some public
1 2 3 4 Zone 2/3
5 15% 6 7 8 9
Zone numbers 10%Octilinear Embedding vs. Octilinear Drawing
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Excerpt of regional train map of southwestern Germany
5Existing Approaches
1. Local search / Hill climbing (Stott 2004)
6Existing Approaches
1. Local search / Hill climbing (Stott 2004)
2. Linear Programming (Nöllenburg, Wolff, 2005)
6Existing Approaches
1. Local search / Hill climbing (Stott 2004)
2. Linear Programming (Nöllenburg, Wolff, 2005)
3. Least Squares Optimization (Wang, Chi, 2011)
6Existing Approaches
1. Local search / Hill climbing (Stott 2004)
2. Linear Programming (Nöllenburg, Wolff, 2005)
3. Least Squares Optimization (Wang, Chi, 2011)
6Existing Approaches
1. Local search / Hill climbing (Stott 2004)
2. Linear Programming (Nöllenburg, Wolff, 2005)
3. Least Squares Optimization (Wang, Chi, 2011)
Problems:
• Octilinearity not guaranteed [1]
6Existing Approaches
1. Local search / Hill climbing (Stott 2004)
2. Linear Programming (Nöllenburg, Wolff, 2005)
3. Least Squares Optimization (Wang, Chi, 2011)
Problems:
• Octilinearity not guaranteed [1]
• Produce octilinear embeddings, so no bends along edges
(can be mitigated by adding explicit bend nodes) [1, 2, 3]
6Existing Approaches
1. Local search / Hill climbing (Stott 2004)
2. Linear Programming (Nöllenburg, Wolff, 2005)
3. Least Squares Optimization (Wang, Chi, 2011)
Problems:
• Octilinearity not guaranteed [1]
• Produce octilinear embeddings, so no bends along edges
(can be mitigated by adding explicit bend nodes) [1, 2, 3]
• Are often too slow for practical purposes (on-demand
maps, editors) [1, 2]
6Our Approach
Basic idea: Formulate as a set of shortest path problems.
7Our Approach
Basic idea: Formulate as a set of shortest path problems.
t s0 t0
1. Build octilinear grid graph on
which edge bends in paths t0
are penalized
s
s
7Our Approach
Basic idea: Formulate as a set of shortest path problems.
t s0 t0
1. Build octilinear grid graph on
which edge bends in paths t0
are penalized
s
s
2. For each e ∈ E, e = (s, t):
define start and target grid ? ? ψ(s)
s S
nodes S, T and find the
shortest path from S to T. T T
Make path an obstacle. t t
dˆ
7Our Approach
Basic idea: Formulate as a set of shortest path problems.
t s0 t0
1. Build octilinear grid graph on
which edge bends in paths t0
are penalized
s
s
2. For each e ∈ E, e = (s, t):
define start and target grid ? ? ψ(s)
s S
nodes S, T and find the
shortest path from S to T. T T
Make path an obstacle. t t
dˆ
7Our Approach
Basic idea: Formulate as a set of shortest path problems.
t s0 t0
1. Build octilinear grid graph on
which edge bends in paths t0
are penalized
s
s
2. For each e ∈ E, e = (s, t):
define start and target grid ? ? ψ(s)
s S
nodes S, T and find the
shortest path from S to T. T T
Make path an obstacle. t t
dˆ
Key question: Should the shortest paths be determined
simultaneously or iteratively (then: in which order)?
7Graph Modeling - Detail
ψ0x,y 90◦ 45◦
ψ1x,y
ω0,2
x,y
ψ7x,y ω1,2
135◦ x,y
ω2,7
x,y
2,6
ψ2x,y 180◦ ωx,y
ψ6x,y ψx,y ω2x,y ψ2x,y
ω2,5
x,y
135◦ ω2,3
x,y
ψ5x,y
ω2,4
ψ3x,y x,y
ψ4x,y 90◦ 45◦
8Graph Modeling - Detail
ψ0x,y 90◦ 45◦
ψ1x,y
ω0,2
x,y
ψ7x,y ω1,2
135◦ x,y
ω2,7
x,y
2,6
ψ2x,y 180◦ ωx,y
ψ6x,y ψx,y ω2x,y ψ2x,y
ω2,5
x,y
135◦ ω2,3
x,y
ψ5x,y
ω2,4
ψ3x,y x,y
ψ4x,y 90◦ 45◦
• Each grid node is extended by explicit bend edges
between port nodes. Edge costs reflect the angle
8Graph Modeling - Detail
ψ0x,y 90◦ 45◦
ψ1x,y
ω0,2
x,y
ψ7x,y ω1,2
135◦ x,y
ω2,7
x,y
2,6
ψ2x,y 180◦ ωx,y
ψ6x,y ψx,y ω2x,y ψ2x,y
ω2,5
x,y
135◦ ω2,3
x,y
ψ5x,y
ω2,4
ψ3x,y x,y
ψ4x,y 90◦ 45◦
• Each grid node is extended by explicit bend edges
between port nodes. Edge costs reflect the angle
• To arrive at a grid node, we add sink edges
8Graph Modeling - Detail
ψ0x,y 90◦ 45◦
ψ1x,y
ω0,2
x,y
ψ7x,y ω1,2
135◦ x,y
ω2,7
x,y
2,6
ψ2x,y 180◦ ωx,y
ψ6x,y ψx,y ω2x,y ψ2x,y
ω2,5
x,y
135◦ ω2,3
x,y
ψ5x,y
ω2,4
ψ3x,y x,y
ψ4x,y 90◦ 45◦
• Each grid node is extended by explicit bend edges
between port nodes. Edge costs reflect the angle
• To arrive at a grid node, we add sink edges
• Sink edge costs reflect (1) the angle between the sink and
continued lines on already settled sinks and (2) a node
move penalty
8Iterative Calculation
9Iterative Calculation
9Iterative Calculation
9Iterative Calculation
9Iterative Calculation
9Iterative Calculation
9Iterative Calculation
9Iterative Calculation
9Iterative Calculation
9Iterative Calculation
9Iterative Calculation
9Iterative Calculation
9Iterative Calculation
9Iterative Calculation
9Iterative Calculation
9Iterative Calculation
9Iterative Calculation
9Iterative Calculation
9Iterative Calculation
9Iterative Calculation
9Iterative Calculation
9Iterative Calculation
9Iterative Calculation
9Iterative Calculation
9Iterative Calculation
9Iterative Calculation
9Iterative Calculation
9Iterative Calculation
9Iterative Calculation
9Linear Program (LP) or Approximate Approach (A)?
Final target values and approximation error δ of linear
program (LP-2) and approximate approach (A-2) when degree 2
nodes where contracted first
LP-2 t A-2 t δ
Freiburg 144.6 11m 146.5 73ms 1.3%
Vienna 170.5 13h 175.1 171ms 2.7%
Stuttgart 383.2 12h 399.2 510ms 4.1%
Berlin 315.4 20h 326.0 513ms 3.4%
Sydney 360.6 7h 361.4 250ms 0.2%
London ≥669.2 — 758.3 2.1 s ≤14%
10Linear Program (LP) or Approximate Approach (A)?
Linear Program (LP) 11Linear Program (LP) or Approximate Approach (A)?
Linear Program (LP) Approximate Approach (A) 11Results
Vienna, drawn with approx. approach in 202 ms. 12Results
Berlin, drawn with approx. approach in 764 ms. 13Results
Stuttgart, drawn with approx. approach in 843 ms. 14Results
Chesham
Epping
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ood
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nt
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lfo
Cha
Cho
Amersham
Watford
Rickmansworth Debden
High Barnet Cockfosters
Croxley
Oakwood
Loughton
Totteridge & Whetstone
Southgate
Moor Park Buckhurst Hill
Woodside Park
Arnos Grove
Stanmore
Northwood
k
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Bu
Edgware
West Finchley Roding Valley
Bounds Green
Mill Hill East Chigwell
Canons Park
Northwood Hills
Colindale
Woodford Grange Hill
Hainault
Wood Green
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South Woodford
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d
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North Acton Ladbroke Grove
Ald
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Canning Town
gat
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wa
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cha
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nor
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k
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loo
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thw
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ter
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stm
don
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High Street Kensington Knightsbridge
Bri
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Hounslow East Borough
dge
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rth
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h No
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bet
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s Par
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en
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Victoria Elephant & Castle
Bro
rt
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k
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ord
m
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nha
ons
mm
mf
th
ane
Bar
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Ken
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Squ
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sin
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are
gto
n
West Kensington
Pimlico
Rav
ens
cou
rt
Par
West Brompton Oval
k
Hounslow West Vauxhall
Gunnersbury
3
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als
min
Ter
Hatton Cross Fulham Broadway
row
Kew Gardens
ath
He
Heathrow Terminal 5 Stockwell
Richmond
Parsons Green
Brixton
Clapham North
Heathrow Terminal 4 Putney Bridge
Clapham Common
East Putney
Clapham South
Balham
Southfields
Tooting Bec
Wimbledon Park
Tooting Broadway
Wimbledon
Colliers Wood
South Wimbledon
Morden
London (labeled), drawn with approx. approach in 2.7 s.
15Comparison to other work
Sydney light rail network
Nöllenburg et al. Wang et al. (2011) Wang et al. (2016) Our approach
t = 23m t = 816ms t ≈ 150ms * t = 370ms
* No time was reported, given time is for a network of similar size (Berlin)
16Results
17Results
18Future Work
Our approach allows for several refinements and
improvements:
• Labeling only rudimentary so far and optimized
separately
19Future Work
Our approach allows for several refinements and
improvements:
• Labeling only rudimentary so far and optimized
separately
• Use grid graphs with different node densities
19Future Work
Our approach allows for several refinements and
improvements:
• Labeling only rudimentary so far and optimized
separately
• Use grid graphs with different node densities
• Include previously developed clustering techniques for
local search
19Future Work
Our approach allows for several refinements and
improvements:
• Labeling only rudimentary so far and optimized
separately
• Use grid graphs with different node densities
• Include previously developed clustering techniques for
local search
• Use different base grids
19Outlook: Orthoradial Base Grids
Neckargröningen Remseck, Remseck am Neckar
4
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Aldingen Mühle, Remseck am Neckar
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Münster Rathaus
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Sieglestraße Kraftwerk Münster
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Sonnenberg U8
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Fauststraße
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Europaplatz
Ci
Leinfelden Unteraichen, Leinfelden-Echterdingen
BW
En
U6 U6
Leinfelden Bahnhof, Leinfelden-Echterdingen
Fasanenhof SchelmenwasenThank you!
Thank you!
http://octi.informatik.uni-freiburg.de
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