Order in the Underground – Martin N ¨ollenburg and Alexander Wolff
Our Model Our Solution NP-Hardness Order in the Underground – How to Automate the Drawing of Metro Maps ¨ Martin Nollenburg and Alexander Wolff Department of Computer Science Karlsruhe University Dagstuhl Seminar on Spatial Data, 08.03.2006 ¨ Martin Nollenburg and Alexander Wolff 1 32 Drawing Metro Maps Our Model Our Solution NP-Hardness Outline 1 Modeling the Metro Map Problem What is a metro map? Hard and soft constraints 2 Our Solution Mixed-integer programming formulation Experiments Labeling 3 NP-Hardness Rectilinear vs. octilinear drawing Reduction from planar 3-SAT ¨ Martin Nollenburg and Alexander Wolff 2 32 Drawing Metro Maps Our Model Our Solution NP-Hardness What is a metro map? Hard and soft constraints Outline 1 Modeling the Metro Map Problem What is a metro map? Hard and soft constraints 2 Our Solution Mixed-integer programming formulation Experiments Labeling 3 NP-Hardness Rectilinear vs. octilinear drawing Reduction from planar 3-SAT ¨ Martin Nollenburg and Alexander Wolff 3 32 Drawing Metro Maps Our Model Our Solution NP-Hardness What is a metro map? Hard and soft constraints What is a Metro Map? schematic diagram for public transport C B D Chesham Chalfont & Latimer Amersham A 5 Watford Rickmansworth Moor Park Northwood Northwood Hills Ruislip Ruislip Manor Ickenham Burnt Oak Golders Green Wembley Park Hampstead Kensal Rise Brondesbury Kensal Green Queen's Park Kilburn Park Warwick Avenue Royal Oak Westbourne Park Paddington Ealing Broadway 654 3 West Acton Ealing Common South Acton East Acton Holland Park 2 Notting Hill Gate Shepherd's Bush Acton Central Shepherd's Bush Hammersmith 1 Gloucester Road Earl's Court Bank Aldgate East Tower Hill Fenchurch Street 150m Canada Water Surrey Quays Pimlico Southwark Putney Bridge Bus District Hammersmith & City Metropolitan Docklands Light Railway Prince Regent Royal Albert Beckton Park Cyprus Gallions Reach Silvertown Beckton North Woolwich King George V Greenwich Elverson Road Lewisham Oval Clapham High Street 100m Clapham South Airport interchange Connection with Tramlink Custom House London City Airport Deptford Bridge 100m Stockwell Clapham North Connections with National Rail Connections with riverboat services Tooting Bec Under construction National Rail Royal Victoria Pontoon Dock Cutty Sark New Cross Elephant & Castle Kennington Single and return tickets issued for Underground/DLR journeys are not valid between Gunnersbury-Stratford or Custom House-North Woolwich 4 Bus to London City Airport Canning Town West Silvertown North Greenwich for Maritime Greenwich Vauxhall Interchange stations restricted service - see opposite key East London Jubilee 3 East India Blackwall 2 Heron Quays Island Gardens Borough New Cross Gate Southfields Wimbledon Circle Poplar West India Quay Canary Wharf Crossharbour & London Arena Mudchute Lambeth North River Thames East Putney Wimbledon Park Northern Piccadilly Victoria Waterloo & City Central Wapping South Quay Waterloo East Kew Gardens Richmond Bakerloo Upton Park Plaistow West Ham for ExCeL Rotherhithe Bermondsey Waterloo Fulham Broadway Parsons Green for Heathrow Terminal 4 Heathrow Terminal 4 Underground station closed until September 2006. Replacement bus services run from Hatton Cross. Westferry Limehouse Tower Gateway River Thames London Bridge Charing Cross 100m Devons Road All Saints Shadwell Monument Embankment 2 Stepney Green Whitechapel Aldgate Cannon Street Blackfriars Temple Westminster BromleyBow Church by-Bow tion truc cons Heathrow Terminal 4 Key to lines and symbols Bow Road 200m 1 St. Paul's Chancery Lane St. James's Park Victoria South Kensington Shoreditch East Ham er Und ser vice West Kensington Barking Liverpool Street West Brompton Gunnersbury Hornchurch Elm Park Dagenham Heathway Becontree Upney Pudding Mill Lane tion truc cons Turnham Stamford Ravenscourt Green Brook Park Upminster Upminster Bridge Dagenham East Leyton Stratford Mile End Moorgate Covent Garden Leicester Square Mansion House Charing Cross Piccadilly Circus Sloane Square Northfields Boston Manor Chiswick Park Hounslow East Osterley Hounslow Hounslow West Central Heathrow Terminals 1, 2, 3 Hatton Cross 3 Hackney Wick Gants Hill Homerton 2 Bethnal Green Holborn Green Park Knightsbridge Barons Court Russell Square Tottenham Court Road Marble Arch High Street Hyde Park Kensington Corner Kensington (Olympia) Goldhawk Road Acton Town Oxford Circus Dalston Kingsland Old Street Euston 200m Goodge Street Lancaster Bond Queensway Gate Street Hackney Central Canonbury Highbury & Islington Angel Farringdon Barbican Barkingside Leytonstone Caledonian Road & Barnsbury King's Cross St. Pancras Euston Square Fairlop Wanstead er Und South Ealing White City Mornington Crescent Hainault Newbury Park Redbridge Finsbury Park Holloway Road Camden Road Camden Town Bayswater Latimer Road North Acton Arsenal Caledonian Road Chalk Farm Warren Street Regent's Park 4 South Woodford Walthamstow Central Snaresbrook Great Portland Euston Street Baker Street Edgware Marylebone Road Chigwell Grange Hill Blackhorse Road Tottenham Hale 6 5 Roding Valley Woodford Seven Sisters Manor House Tufnell Park Kentish Town 2 Swiss Cottage St. John's Wood Edgware Road Paddington Gospel Oak Kentish Town West Belsize Park 200m Finchley Road Maida Vale Ladbroke Grove Park Royal Kilburn West Hampstead Brondesbury Park Stonebridge Park Harlesden Willesden Junction North Ealing Finchley Road & Frognal Willesden Green Alperton Hanger Lane Wood Green Turnpike Lane Archway Hampstead Heath Dollis Hill Wembley Central Sudbury Hill Sudbury Town Perivale Theydon Bois Debden Loughton Buckhurst Hill Bounds Green Highgate 3 Brent Cross Neasden Northwick Park South Kenton North Wembley South Harrow Epping Oakwood Arnos Grove Finchley Central East Finchley Hendon Central Kingsbury Preston Road Southgate West Finchley Colindale Queensbury Harrowon-the-Hill West Harrow Greenford Cockfosters Totteridge & Whetstone Woodside Park Mill Hill East 4 Edgware Canons Park Kenton North Harrow South Ruislip Northolt Stanmore Harrow & Wealdstone Pinner Eastcote Rayners Lane Ruislip Gardens High Barnet Croxley Chorleywood West Ruislip Hillingdon Uxbridge Closed Sundays Colliers Wood Served by Piccadilly line trains early morning and late evening Morden Clapham Common Brixton 2 100m Balham Tooting Broadway South Wimbledon 3 D C B A 6 5 4 3 2 1 Station in Zone D Bakerloo Station in Zone C Special fares apply for single and return tickets to and from Harrow & Wealdstone. Station in Zone B Central No service Woodford - Hainault after 2000 daily. Station in Zone A Station in Zone 6 and Zone A Station in Zone 6 Circle District Cannon Street open until 2100 Mondays to Fridays. Closed Saturdays and Sundays. Station in Zone 5 Station in Zone 4 Station in both zones Station in Zone 3 Station in Zone 2 Station in both zones Station in Zone 1 3 Points to remember Explanation of zones District East London Hammersmith & City Metropolitan Northern Piccadilly Earl’s Court - Kensington (Olympia) 0700 to 2345 Mondays to Saturdays, 0800 to 2345 Sundays. Shoreditch station opens 0700 to 1030 and 1530 to 2030 Mondays to Fridays. Closed Saturdays. Open from 0700 to 1500 Sundays. No service Whitechapel - Barking early morning or late evening Mondays to Saturdays or all day Sundays. For Chesham change at Chalfont & Latimer on most trains. 4 On Sundays between 1300 and 1730, Camden Town is open for interchange and exit only. No service Uxbridge - Rayners Lane in the early mornings. Replacement bus service between Hatton Cross and Heathrow Terminal 4. Heathrow Terminal 4 Underground station closed until September 2006. Waterloo & City 0615 to 2130 Mondays to Fridays. 0800 to 1830 Saturdays. Closed Sundays. Certain stations are closed on public holidays. ¨ Martin Nollenburg and Alexander Wolff 4 32 Drawing Metro Maps Our Model Our Solution NP-Hardness What is a metro map? Hard and soft constraints What is a Metro Map? schematic diagram for public transport visualizes lines and stations C B D Chesham Chalfont & Latimer Amersham A 5 Watford Rickmansworth Moor Park Northwood Northwood Hills Ruislip Ruislip Manor Ickenham Burnt Oak Golders Green Wembley Park Hampstead Kensal Rise Brondesbury Kensal Green Queen's Park Kilburn Park Warwick Avenue Royal Oak Westbourne Park Paddington Ealing Broadway 654 3 West Acton Ealing Common South Acton East Acton Holland Park 2 Notting Hill Gate Shepherd's Bush Acton Central Shepherd's Bush Hammersmith 1 Gloucester Road Earl's Court Bank Aldgate East Tower Hill Fenchurch Street 150m Canada Water Surrey Quays Pimlico Southwark Putney Bridge Bus District Hammersmith & City Metropolitan Docklands Light Railway Prince Regent Royal Albert Beckton Park Cyprus Gallions Reach Silvertown Beckton North Woolwich King George V Greenwich Elverson Road Lewisham Oval Clapham High Street 100m Clapham South Airport interchange Connection with Tramlink Custom House London City Airport Deptford Bridge 100m Stockwell Clapham North Connections with National Rail Connections with riverboat services Tooting Bec Under construction National Rail Royal Victoria Pontoon Dock Cutty Sark New Cross Elephant & Castle Kennington Single and return tickets issued for Underground/DLR journeys are not valid between Gunnersbury-Stratford or Custom House-North Woolwich 4 Bus to London City Airport Canning Town West Silvertown North Greenwich for Maritime Greenwich Vauxhall Interchange stations restricted service - see opposite key East London Jubilee 3 East India Blackwall 2 Heron Quays Island Gardens Borough New Cross Gate Southfields Wimbledon Circle Poplar West India Quay Canary Wharf Crossharbour & London Arena Mudchute Lambeth North River Thames East Putney Wimbledon Park Northern Piccadilly Victoria Waterloo & City Central Wapping South Quay Waterloo East Kew Gardens Richmond Bakerloo Upton Park Plaistow West Ham for ExCeL Rotherhithe Bermondsey Waterloo Fulham Broadway Parsons Green for Heathrow Terminal 4 Heathrow Terminal 4 Underground station closed until September 2006. Replacement bus services run from Hatton Cross. Westferry Limehouse Tower Gateway River Thames London Bridge Charing Cross 100m Devons Road All Saints Shadwell Monument Embankment 2 Stepney Green Whitechapel Aldgate Cannon Street Blackfriars Temple Westminster BromleyBow Church by-Bow tion truc cons Heathrow Terminal 4 Key to lines and symbols Bow Road 200m 1 St. Paul's Chancery Lane St. James's Park Victoria South Kensington Shoreditch East Ham er Und ser vice West Kensington Barking Liverpool Street West Brompton Gunnersbury Hornchurch Elm Park Dagenham Heathway Becontree Upney Pudding Mill Lane tion truc cons Turnham Stamford Ravenscourt Green Brook Park Upminster Upminster Bridge Dagenham East Leyton Stratford Mile End Moorgate Covent Garden Leicester Square Mansion House Charing Cross Piccadilly Circus Sloane Square Northfields Boston Manor Chiswick Park Hounslow East Osterley Hounslow Hounslow West Central Heathrow Terminals 1, 2, 3 Hatton Cross 3 Hackney Wick Gants Hill Homerton 2 Bethnal Green Holborn Green Park Knightsbridge Barons Court Russell Square Tottenham Court Road Marble Arch High Street Hyde Park Kensington Corner Kensington (Olympia) Goldhawk Road Acton Town Oxford Circus Dalston Kingsland Old Street Euston 200m Goodge Street Lancaster Bond Queensway Gate Street Hackney Central Canonbury Highbury & Islington Angel Farringdon Barbican Barkingside Leytonstone Caledonian Road & Barnsbury King's Cross St. Pancras Euston Square Fairlop Wanstead er Und South Ealing White City Mornington Crescent Hainault Newbury Park Redbridge Finsbury Park Holloway Road Camden Road Camden Town Bayswater Latimer Road North Acton Arsenal Caledonian Road Chalk Farm Warren Street Regent's Park 4 South Woodford Walthamstow Central Snaresbrook Great Portland Euston Street Baker Street Edgware Marylebone Road Chigwell Grange Hill Blackhorse Road Tottenham Hale 6 5 Roding Valley Woodford Seven Sisters Manor House Tufnell Park Kentish Town 2 Swiss Cottage St. John's Wood Edgware Road Paddington Gospel Oak Kentish Town West Belsize Park 200m Finchley Road Maida Vale Ladbroke Grove Park Royal Kilburn West Hampstead Brondesbury Park Stonebridge Park Harlesden Willesden Junction North Ealing Finchley Road & Frognal Willesden Green Alperton Hanger Lane Wood Green Turnpike Lane Archway Hampstead Heath Dollis Hill Wembley Central Sudbury Hill Sudbury Town Perivale Theydon Bois Debden Loughton Buckhurst Hill Bounds Green Highgate 3 Brent Cross Neasden Northwick Park South Kenton North Wembley South Harrow Epping Oakwood Arnos Grove Finchley Central East Finchley Hendon Central Kingsbury Preston Road Southgate West Finchley Colindale Queensbury Harrowon-the-Hill West Harrow Greenford Cockfosters Totteridge & Whetstone Woodside Park Mill Hill East 4 Edgware Canons Park Kenton North Harrow South Ruislip Northolt Stanmore Harrow & Wealdstone Pinner Eastcote Rayners Lane Ruislip Gardens High Barnet Croxley Chorleywood West Ruislip Hillingdon Uxbridge Closed Sundays Colliers Wood Served by Piccadilly line trains early morning and late evening Morden Clapham Common Brixton 2 100m Balham Tooting Broadway South Wimbledon 3 D C B A 6 5 4 3 2 1 Station in Zone D Bakerloo Station in Zone C Special fares apply for single and return tickets to and from Harrow & Wealdstone. Station in Zone B Central No service Woodford - Hainault after 2000 daily. Station in Zone A Station in Zone 6 and Zone A Station in Zone 6 Circle District Cannon Street open until 2100 Mondays to Fridays. Closed Saturdays and Sundays. Station in Zone 5 Station in Zone 4 Station in both zones Station in Zone 3 Station in Zone 2 Station in both zones Station in Zone 1 3 Points to remember Explanation of zones District East London Hammersmith & City Metropolitan Northern Piccadilly Earl’s Court - Kensington (Olympia) 0700 to 2345 Mondays to Saturdays, 0800 to 2345 Sundays. Shoreditch station opens 0700 to 1030 and 1530 to 2030 Mondays to Fridays. Closed Saturdays. Open from 0700 to 1500 Sundays. No service Whitechapel - Barking early morning or late evening Mondays to Saturdays or all day Sundays. For Chesham change at Chalfont & Latimer on most trains. 4 On Sundays between 1300 and 1730, Camden Town is open for interchange and exit only. No service Uxbridge - Rayners Lane in the early mornings. Replacement bus service between Hatton Cross and Heathrow Terminal 4. Heathrow Terminal 4 Underground station closed until September 2006. Waterloo & City 0615 to 2130 Mondays to Fridays. 0800 to 1830 Saturdays. Closed Sundays. Certain stations are closed on public holidays. ¨ Martin Nollenburg and Alexander Wolff 4 32 Drawing Metro Maps Our Model Our Solution NP-Hardness What is a metro map? Hard and soft constraints What is a Metro Map? schematic diagram for public transport visualizes lines and stations goal: ease navigation for passengers “How do I get from A to B?” “Where to get off and change trains?” C B D Chesham Chalfont & Latimer Amersham A 5 Watford Rickmansworth Moor Park Northwood Northwood Hills Ruislip Ruislip Manor Ickenham Burnt Oak Golders Green Wembley Park Hampstead Kensal Rise Brondesbury Kensal Green Queen's Park Kilburn Park Warwick Avenue Royal Oak Westbourne Park Paddington Ealing Broadway 654 3 West Acton Ealing Common South Acton East Acton Holland Park 2 Notting Hill Gate Shepherd's Bush Acton Central Shepherd's Bush Hammersmith 1 Gloucester Road Earl's Court Bank Aldgate East Tower Hill Fenchurch Street 150m Canada Water Surrey Quays Pimlico Southwark Putney Bridge Bus District Hammersmith & City Metropolitan Docklands Light Railway Prince Regent Royal Albert Beckton Park Cyprus Gallions Reach Silvertown Beckton North Woolwich King George V Greenwich Elverson Road Lewisham Oval Clapham High Street 100m Clapham South Airport interchange Connection with Tramlink Custom House London City Airport Deptford Bridge 100m Stockwell Clapham North Connections with National Rail Connections with riverboat services Tooting Bec Under construction National Rail Royal Victoria Pontoon Dock Cutty Sark New Cross Elephant & Castle Kennington Single and return tickets issued for Underground/DLR journeys are not valid between Gunnersbury-Stratford or Custom House-North Woolwich 4 Bus to London City Airport Canning Town West Silvertown North Greenwich for Maritime Greenwich Vauxhall Interchange stations restricted service - see opposite key East London Jubilee 3 East India Blackwall 2 Heron Quays Island Gardens Borough New Cross Gate Southfields Wimbledon Circle Poplar West India Quay Canary Wharf Crossharbour & London Arena Mudchute Lambeth North River Thames East Putney Wimbledon Park Northern Piccadilly Victoria Waterloo & City Central Wapping South Quay Waterloo East Kew Gardens Richmond Bakerloo Upton Park Plaistow West Ham for ExCeL Rotherhithe Bermondsey Waterloo Fulham Broadway Parsons Green for Heathrow Terminal 4 Heathrow Terminal 4 Underground station closed until September 2006. Replacement bus services run from Hatton Cross. Westferry Limehouse Tower Gateway River Thames London Bridge Charing Cross 100m Devons Road All Saints Shadwell Monument Embankment 2 Stepney Green Whitechapel Aldgate Cannon Street Blackfriars Temple Westminster BromleyBow Church by-Bow tion truc cons Heathrow Terminal 4 Key to lines and symbols Bow Road 200m 1 St. Paul's Chancery Lane St. James's Park Victoria South Kensington Shoreditch East Ham er Und ser vice West Kensington Barking Liverpool Street West Brompton Gunnersbury Hornchurch Elm Park Dagenham Heathway Becontree Upney Pudding Mill Lane tion truc cons Turnham Stamford Ravenscourt Green Brook Park Upminster Upminster Bridge Dagenham East Leyton Stratford Mile End Moorgate Covent Garden Leicester Square Mansion House Charing Cross Piccadilly Circus Sloane Square Northfields Boston Manor Chiswick Park Hounslow East Osterley Hounslow Hounslow West Central Heathrow Terminals 1, 2, 3 Hatton Cross 3 Hackney Wick Gants Hill Homerton 2 Bethnal Green Holborn Green Park Knightsbridge Barons Court Russell Square Tottenham Court Road Marble Arch High Street Hyde Park Kensington Corner Kensington (Olympia) Goldhawk Road Acton Town Oxford Circus Dalston Kingsland Old Street Euston 200m Goodge Street Lancaster Bond Queensway Gate Street Hackney Central Canonbury Highbury & Islington Angel Farringdon Barbican Barkingside Leytonstone Caledonian Road & Barnsbury King's Cross St. Pancras Euston Square Fairlop Wanstead er Und South Ealing White City Mornington Crescent Hainault Newbury Park Redbridge Finsbury Park Holloway Road Camden Road Camden Town Bayswater Latimer Road North Acton Arsenal Caledonian Road Chalk Farm Warren Street Regent's Park 4 South Woodford Walthamstow Central Snaresbrook Great Portland Euston Street Baker Street Edgware Marylebone Road Chigwell Grange Hill Blackhorse Road Tottenham Hale 6 5 Roding Valley Woodford Seven Sisters Manor House Tufnell Park Kentish Town 2 Swiss Cottage St. John's Wood Edgware Road Paddington Gospel Oak Kentish Town West Belsize Park 200m Finchley Road Maida Vale Ladbroke Grove Park Royal Kilburn West Hampstead Brondesbury Park Stonebridge Park Harlesden Willesden Junction North Ealing Finchley Road & Frognal Willesden Green Alperton Hanger Lane Wood Green Turnpike Lane Archway Hampstead Heath Dollis Hill Wembley Central Sudbury Hill Sudbury Town Perivale Theydon Bois Debden Loughton Buckhurst Hill Bounds Green Highgate 3 Brent Cross Neasden Northwick Park South Kenton North Wembley South Harrow Epping Oakwood Arnos Grove Finchley Central East Finchley Hendon Central Kingsbury Preston Road Southgate West Finchley Colindale Queensbury Harrowon-the-Hill West Harrow Greenford Cockfosters Totteridge & Whetstone Woodside Park Mill Hill East 4 Edgware Canons Park Kenton North Harrow South Ruislip Northolt Stanmore Harrow & Wealdstone Pinner Eastcote Rayners Lane Ruislip Gardens High Barnet Croxley Chorleywood West Ruislip Hillingdon Uxbridge Closed Sundays Colliers Wood Served by Piccadilly line trains early morning and late evening Morden Clapham Common Brixton 2 100m Balham Tooting Broadway South Wimbledon 3 D C B A 6 5 4 3 2 1 Station in Zone D Bakerloo Station in Zone C Special fares apply for single and return tickets to and from Harrow & Wealdstone. Station in Zone B Central No service Woodford - Hainault after 2000 daily. Station in Zone A Station in Zone 6 and Zone A Station in Zone 6 Circle District Cannon Street open until 2100 Mondays to Fridays. Closed Saturdays and Sundays. Station in Zone 5 Station in Zone 4 Station in both zones Station in Zone 3 Station in Zone 2 Station in both zones Station in Zone 1 3 Points to remember Explanation of zones District East London Hammersmith & City Metropolitan Northern Piccadilly Earl’s Court - Kensington (Olympia) 0700 to 2345 Mondays to Saturdays, 0800 to 2345 Sundays. Shoreditch station opens 0700 to 1030 and 1530 to 2030 Mondays to Fridays. Closed Saturdays. Open from 0700 to 1500 Sundays. No service Whitechapel - Barking early morning or late evening Mondays to Saturdays or all day Sundays. For Chesham change at Chalfont & Latimer on most trains. 4 On Sundays between 1300 and 1730, Camden Town is open for interchange and exit only. No service Uxbridge - Rayners Lane in the early mornings. Replacement bus service between Hatton Cross and Heathrow Terminal 4. Heathrow Terminal 4 Underground station closed until September 2006. Waterloo & City 0615 to 2130 Mondays to Fridays. 0800 to 1830 Saturdays. Closed Sundays. Certain stations are closed on public holidays. ¨ Martin Nollenburg and Alexander Wolff 4 32 Drawing Metro Maps Our Model Our Solution NP-Hardness What is a metro map? Hard and soft constraints What is a Metro Map? schematic diagram for public transport visualizes lines and stations goal: ease navigation for passengers “How do I get from A to B?” “Where to get off and change trains?” distorts geometry and scale C B D Chesham Chalfont & Latimer Amersham A 5 Watford Rickmansworth Moor Park Northwood Northwood Hills Ruislip Ruislip Manor Ickenham Burnt Oak Golders Green Wembley Park Hampstead Kensal Rise Brondesbury Kensal Green Queen's Park Kilburn Park Warwick Avenue Royal Oak Westbourne Park Paddington Ealing Broadway 654 3 West Acton Ealing Common South Acton East Acton Holland Park 2 Notting Hill Gate Shepherd's Bush Acton Central Shepherd's Bush Hammersmith 1 Gloucester Road Earl's Court Bank Aldgate East Tower Hill Fenchurch Street 150m Canada Water Surrey Quays Pimlico Southwark Putney Bridge Bus District Hammersmith & City Metropolitan Docklands Light Railway Prince Regent Royal Albert Beckton Park Cyprus Gallions Reach Silvertown Beckton North Woolwich King George V Greenwich Elverson Road Lewisham Oval Clapham High Street 100m Clapham South Airport interchange Connection with Tramlink Custom House London City Airport Deptford Bridge 100m Stockwell Clapham North Connections with National Rail Connections with riverboat services Tooting Bec Under construction National Rail Royal Victoria Pontoon Dock Cutty Sark New Cross Elephant & Castle Kennington Single and return tickets issued for Underground/DLR journeys are not valid between Gunnersbury-Stratford or Custom House-North Woolwich 4 Bus to London City Airport Canning Town West Silvertown North Greenwich for Maritime Greenwich Vauxhall Interchange stations restricted service - see opposite key East London Jubilee 3 East India Blackwall 2 Heron Quays Island Gardens Borough New Cross Gate Southfields Wimbledon Circle Poplar West India Quay Canary Wharf Crossharbour & London Arena Mudchute Lambeth North River Thames East Putney Wimbledon Park Northern Piccadilly Victoria Waterloo & City Central Wapping South Quay Waterloo East Kew Gardens Richmond Bakerloo Upton Park Plaistow West Ham for ExCeL Rotherhithe Bermondsey Waterloo Fulham Broadway Parsons Green for Heathrow Terminal 4 Heathrow Terminal 4 Underground station closed until September 2006. Replacement bus services run from Hatton Cross. Westferry Limehouse Tower Gateway River Thames London Bridge Charing Cross 100m Devons Road All Saints Shadwell Monument Embankment 2 Stepney Green Whitechapel Aldgate Cannon Street Blackfriars Temple Westminster BromleyBow Church by-Bow tion truc cons Heathrow Terminal 4 Key to lines and symbols Bow Road 200m 1 St. Paul's Chancery Lane St. James's Park Victoria South Kensington Shoreditch East Ham er Und ser vice West Kensington Barking Liverpool Street West Brompton Gunnersbury Hornchurch Elm Park Dagenham Heathway Becontree Upney Pudding Mill Lane tion truc cons Turnham Stamford Ravenscourt Green Brook Park Upminster Upminster Bridge Dagenham East Leyton Stratford Mile End Moorgate Covent Garden Leicester Square Mansion House Charing Cross Piccadilly Circus Sloane Square Northfields Boston Manor Chiswick Park Hounslow East Osterley Hounslow Hounslow West Central Heathrow Terminals 1, 2, 3 Hatton Cross 3 Hackney Wick Gants Hill Homerton 2 Bethnal Green Holborn Green Park Knightsbridge Barons Court Russell Square Tottenham Court Road Marble Arch High Street Hyde Park Kensington Corner Kensington (Olympia) Goldhawk Road Acton Town Oxford Circus Dalston Kingsland Old Street Euston 200m Goodge Street Lancaster Bond Queensway Gate Street Hackney Central Canonbury Highbury & Islington Angel Farringdon Barbican Barkingside Leytonstone Caledonian Road & Barnsbury King's Cross St. Pancras Euston Square Fairlop Wanstead er Und South Ealing White City Mornington Crescent Hainault Newbury Park Redbridge Finsbury Park Holloway Road Camden Road Camden Town Bayswater Latimer Road North Acton Arsenal Caledonian Road Chalk Farm Warren Street Regent's Park 4 South Woodford Walthamstow Central Snaresbrook Great Portland Euston Street Baker Street Edgware Marylebone Road Chigwell Grange Hill Blackhorse Road Tottenham Hale 6 5 Roding Valley Woodford Seven Sisters Manor House Tufnell Park Kentish Town 2 Swiss Cottage St. John's Wood Edgware Road Paddington Gospel Oak Kentish Town West Belsize Park 200m Finchley Road Maida Vale Ladbroke Grove Park Royal Kilburn West Hampstead Brondesbury Park Stonebridge Park Harlesden Willesden Junction North Ealing Finchley Road & Frognal Willesden Green Alperton Hanger Lane Wood Green Turnpike Lane Archway Hampstead Heath Dollis Hill Wembley Central Sudbury Hill Sudbury Town Perivale Theydon Bois Debden Loughton Buckhurst Hill Bounds Green Highgate 3 Brent Cross Neasden Northwick Park South Kenton North Wembley South Harrow Epping Oakwood Arnos Grove Finchley Central East Finchley Hendon Central Kingsbury Preston Road Southgate West Finchley Colindale Queensbury Harrowon-the-Hill West Harrow Greenford Cockfosters Totteridge & Whetstone Woodside Park Mill Hill East 4 Edgware Canons Park Kenton North Harrow South Ruislip Northolt Stanmore Harrow & Wealdstone Pinner Eastcote Rayners Lane Ruislip Gardens High Barnet Croxley Chorleywood West Ruislip Hillingdon Uxbridge Closed Sundays Colliers Wood Served by Piccadilly line trains early morning and late evening Morden Clapham Common Brixton 2 100m Balham Tooting Broadway South Wimbledon 3 D C B A 6 5 4 3 2 1 Station in Zone D Bakerloo Station in Zone C Special fares apply for single and return tickets to and from Harrow & Wealdstone. Station in Zone B Central No service Woodford - Hainault after 2000 daily. Station in Zone A Station in Zone 6 and Zone A Station in Zone 6 Circle District Cannon Street open until 2100 Mondays to Fridays. Closed Saturdays and Sundays. Station in Zone 5 Station in Zone 4 Station in both zones Station in Zone 3 Station in Zone 2 Station in both zones Station in Zone 1 3 Points to remember Explanation of zones District East London Hammersmith & City Metropolitan Northern Piccadilly Earl’s Court - Kensington (Olympia) 0700 to 2345 Mondays to Saturdays, 0800 to 2345 Sundays. Shoreditch station opens 0700 to 1030 and 1530 to 2030 Mondays to Fridays. Closed Saturdays. Open from 0700 to 1500 Sundays. No service Whitechapel - Barking early morning or late evening Mondays to Saturdays or all day Sundays. For Chesham change at Chalfont & Latimer on most trains. 4 On Sundays between 1300 and 1730, Camden Town is open for interchange and exit only. No service Uxbridge - Rayners Lane in the early mornings. Replacement bus service between Hatton Cross and Heathrow Terminal 4. Heathrow Terminal 4 Underground station closed until September 2006. Waterloo & City 0615 to 2130 Mondays to Fridays. 0800 to 1830 Saturdays. Closed Sundays. Certain stations are closed on public holidays. ¨ Martin Nollenburg and Alexander Wolff 4 32 Drawing Metro Maps Our Model Our Solution NP-Hardness What is a metro map? Hard and soft constraints What is a Metro Map? schematic diagram for public transport visualizes lines and stations goal: ease navigation for passengers “How do I get from A to B?” “Where to get off and change trains?” distorts geometry and scale C B D Chesham Chalfont & Latimer Amersham A 5 Watford Rickmansworth Moor Park Northwood Northwood Hills Ruislip Ruislip Manor Ickenham Burnt Oak West Harrow Neasden Northwick Park South Kenton North Wembley South Harrow Golders Green Wembley Park Hampstead Paddington Ealing Broadway 654 3 West Acton Ealing Common South Acton East Acton Holland Park 2 Notting Hill Gate Shepherd's Bush Acton Central Shepherd's Bush Hammersmith 1 Gloucester Road Aldgate East Canada Water Surrey Quays Pimlico Southwark Putney Bridge Bus District Hammersmith & City Metropolitan Docklands Light Railway Prince Regent Royal Albert Beckton Park Cyprus Gallions Reach London City Airport Silvertown Beckton North Woolwich King George V Greenwich Deptford Bridge 100m Elverson Road Lewisham Oval Clapham High Street 100m Clapham South Airport interchange Connection with Tramlink Custom House Pontoon Dock Cutty Sark New Cross Elephant & Castle Stockwell Clapham North Connections with National Rail Connections with riverboat services Tooting Bec Under construction National Rail Royal Victoria West Silvertown North Greenwich for Maritime Greenwich Vauxhall Kennington restricted service - see opposite key 4 Bus to London City Airport Canning Town East India Blackwall 2 Heron Quays Island Gardens Borough New Cross Gate Southfields Interchange stations Single and return tickets issued for Underground/DLR journeys are not valid between Gunnersbury-Stratford or Custom House-North Woolwich 3 Devons Road Poplar West India Quay Canary Wharf Crossharbour & London Arena Mudchute Lambeth North River Thames East London Wapping South Quay Wimbledon Jubilee Upton Park Plaistow West Ham for ExCeL Rotherhithe Bermondsey East Putney Circle Westferry Limehouse Tower Gateway River Thames Wimbledon Park Northern Piccadilly Victoria Waterloo & City Central 2 All Saints Shadwell Tower Hill Fenchurch Street 150m London Bridge Charing Cross 100m Waterloo East Kew Gardens Richmond Bakerloo BromleyBow Church by-Bow Stepney Green Whitechapel Aldgate Monument Embankment Waterloo Parsons Green for Heathrow Terminal 4 Heathrow Terminal 4 Underground station closed until September 2006. Replacement bus services run from Hatton Cross. Key to lines and symbols Bow Road tion truc cons Heathrow Terminal 4 1 Bank Cannon Street Blackfriars Temple Westminster Dagenham Heathway Becontree East Ham 200m St. James's Park Victoria Liverpool Street St. Paul's Chancery Lane Leicester Square Mansion House Charing Cross Piccadilly Circus Sloane Square South Kensington Pudding Mill Lane er Und ser vice Earl's Court Fulham Broadway Hornchurch Elm Park Dagenham East Upney Barking Shoreditch Covent Garden Green Park Knightsbridge Barons Court West Kensington Upminster Upminster Bridge Gants Hill Leyton Stratford Mile End Moorgate West Brompton Gunnersbury 3 Hackney Wick Homerton 2 tion truc cons Turnham Stamford Ravenscourt Green Brook Park Dalston Kingsland Bethnal Green Holborn Tottenham Court Road Marble Arch High Street Hyde Park Kensington Corner Kensington (Olympia) Goldhawk Road Acton Town Northfields Boston Manor Chiswick Park Hounslow East Osterley Hounslow Hounslow West Central Heathrow Terminals 1, 2, 3 Hatton Cross Newbury Park Wanstead Hackney Central Canonbury Highbury & Islington Russell Square Barkingside Leytonstone Caledonian Road & Barnsbury Old Street Euston 200m Oxford Circus Fairlop Redbridge Finsbury Park Holloway Road Angel Farringdon Barbican Goodge Street Lancaster Bond Queensway Gate Street 4 Snaresbrook Arsenal Kentish Town King's Cross St. Pancras Euston Square Hainault er Und South Ealing White City Tufnell Park South Woodford Walthamstow Central Caledonian Road Mornington Crescent Warren Street Regent's Park Chigwell Grange Hill Blackhorse Road Tottenham Hale 6 5 Roding Valley Woodford Seven Sisters Manor House Camden Road Camden Town Great Portland Euston Street Baker Street Edgware Marylebone Road Bayswater Latimer Road North Acton Gospel Oak Chalk Farm 2 Swiss Cottage St. John's Wood Edgware Road Paddington Kentish Town West Belsize Park 200m Finchley Road Maida Vale Ladbroke Grove Park Royal Kilburn West Hampstead Kensal Rise Brondesbury Kensal Green Queen's Park Kilburn Park Warwick Avenue Royal Oak Westbourne Park North Ealing Finchley Road & Frognal Willesden Green Brondesbury Park Stonebridge Park Harlesden Willesden Junction Alperton Hanger Lane Wood Green Turnpike Lane Archway Hampstead Heath Dollis Hill Wembley Central Sudbury Town Perivale Theydon Bois Debden Loughton Buckhurst Hill Bounds Green Highgate 3 Brent Cross improves readability Epping Oakwood Arnos Grove Finchley Central East Finchley Hendon Central Kingsbury Preston Road Southgate West Finchley Colindale Queensbury Harrowon-the-Hill Sudbury Hill Greenford Cockfosters Totteridge & Whetstone Woodside Park Mill Hill East 4 Edgware Canons Park Kenton North Harrow South Ruislip Northolt Stanmore Harrow & Wealdstone Pinner Eastcote Rayners Lane Ruislip Gardens High Barnet Croxley Chorleywood West Ruislip Hillingdon Uxbridge Closed Sundays Colliers Wood Served by Piccadilly line trains early morning and late evening Morden Clapham Common Brixton 2 100m Balham Tooting Broadway South Wimbledon 3 D C B A 6 5 4 3 2 1 Station in Zone D Bakerloo Station in Zone C Special fares apply for single and return tickets to and from Harrow & Wealdstone. Station in Zone B Central No service Woodford - Hainault after 2000 daily. Station in Zone A Station in Zone 6 and Zone A Station in Zone 6 Circle District Cannon Street open until 2100 Mondays to Fridays. Closed Saturdays and Sundays. Station in Zone 5 Station in Zone 4 Station in both zones Station in Zone 3 Station in Zone 2 Station in both zones Station in Zone 1 3 Points to remember Explanation of zones District East London Hammersmith & City Metropolitan Northern Piccadilly Earl’s Court - Kensington (Olympia) 0700 to 2345 Mondays to Saturdays, 0800 to 2345 Sundays. Shoreditch station opens 0700 to 1030 and 1530 to 2030 Mondays to Fridays. Closed Saturdays. Open from 0700 to 1500 Sundays. No service Whitechapel - Barking early morning or late evening Mondays to Saturdays or all day Sundays. For Chesham change at Chalfont & Latimer on most trains. 4 On Sundays between 1300 and 1730, Camden Town is open for interchange and exit only. No service Uxbridge - Rayners Lane in the early mornings. Replacement bus service between Hatton Cross and Heathrow Terminal 4. Heathrow Terminal 4 Underground station closed until September 2006. Waterloo & City 0615 to 2130 Mondays to Fridays. 0800 to 1830 Saturdays. Closed Sundays. Certain stations are closed on public holidays. ¨ Martin Nollenburg and Alexander Wolff 4 32 Drawing Metro Maps Our Model Our Solution NP-Hardness What is a metro map? Hard and soft constraints What is a Metro Map? schematic diagram for public transport visualizes lines and stations goal: ease navigation for passengers “How do I get from A to B?” “Where to get off and change trains?” distorts geometry and scale C B D Chesham Chalfont & Latimer Amersham A 5 Watford Rickmansworth Moor Park Northwood Northwood Hills Ruislip Ruislip Manor Ickenham Burnt Oak West Harrow Neasden Northwick Park South Kenton North Wembley South Harrow Golders Green Wembley Park Hampstead Paddington White City West Acton South Acton East Acton Holland Park 2 1 Gloucester Road Russell Square Shoreditch 1 Aldgate East Charing Cross 100m Bermondsey Canada Water Surrey Quays Pimlico Southwark Putney Bridge Bus District Hammersmith & City Metropolitan Docklands Light Railway Custom House Prince Regent Royal Albert Beckton Park Cyprus Gallions Reach Silvertown Beckton North Woolwich King George V Cutty Sark Greenwich Elverson Road Lewisham Oval Clapham High Street 100m Clapham South Airport interchange Connection with Tramlink Royal Victoria Pontoon Dock London City Airport Deptford Bridge 100m Stockwell Clapham North Connections with National Rail Connections with riverboat services Tooting Bec Under construction National Rail 2 Heron Quays Crossharbour & London Arena Mudchute New Cross Elephant & Castle Kennington Single and return tickets issued for Underground/DLR journeys are not valid between Gunnersbury-Stratford or Custom House-North Woolwich 4 Bus to London City Airport Canning Town West Silvertown North Greenwich for Maritime Greenwich Vauxhall Interchange stations restricted service - see opposite key East London Upton Park Plaistow West Ham 3 East India Blackwall West India Quay Canary Wharf Island Gardens Borough New Cross Gate Southfields Wimbledon Jubilee Wapping Lambeth North River Thames East Putney Circle Poplar for ExCeL Rotherhithe South Quay Wimbledon Park Northern Piccadilly Victoria Waterloo & City Central Westferry Limehouse Tower Gateway London Bridge Waterloo Waterloo East Kew Gardens Richmond Bakerloo BromleyBow Church by-Bow Devons Road All Saints Shadwell Tower Hill Fenchurch Street 150m 2 Stepney Green Whitechapel Aldgate Monument River Thames Fulham Broadway Parsons Green for Heathrow Terminal 4 Heathrow Terminal 4 Underground station closed until September 2006. Replacement bus services run from Hatton Cross. Bow Road compromise between schematic road map ↔ abstract graph tion truc cons Heathrow Terminal 4 Key to lines and symbols Dagenham Heathway Becontree East Ham er Und ser vice Gunnersbury Pudding Mill Lane Liverpool Street Bank Embankment West Brompton Hounslow Hounslow West Central Heathrow Terminals 1, 2, 3 Hatton Cross Hornchurch Elm Park Dagenham East Upney Barking Moorgate Cannon Street Blackfriars Temple Westminster Upminster Upminster Bridge Gants Hill Leyton Stratford Mile End 200m St. James's Park Victoria St. Paul's Chancery Lane Covent Garden Leicester Square Mansion House Charing Cross Piccadilly Circus Sloane Square South Kensington 3 Hackney Wick Homerton 2 tion truc cons Earl's Court Dalston Kingsland Bethnal Green Holborn Tottenham Court Road Green Park Knightsbridge West Kensington Newbury Park Wanstead Hackney Central Canonbury Highbury & Islington Euston 200m Oxford Circus Marble Arch High Street Hyde Park Kensington Corner Kensington (Olympia) Barons Court Barkingside er Und Hammersmith Turnham Stamford Ravenscourt Green Brook Park Fairlop Leytonstone Caledonian Road & Barnsbury Old Street Hainault Redbridge Finsbury Park Holloway Road Angel Farringdon Barbican Goodge Street Lancaster Bond Queensway Gate Street Notting Hill Gate Shepherd's Bush Acton Central Shepherd's Bush Goldhawk Road Acton Town 4 Snaresbrook Arsenal Kentish Town King's Cross St. Pancras Euston Square Northfields Boston Manor Chiswick Park Hounslow East Osterley Tufnell Park South Woodford Walthamstow Central Caledonian Road Mornington Crescent Warren Street Regent's Park Chigwell Grange Hill Blackhorse Road Tottenham Hale 6 5 Roding Valley Woodford Seven Sisters Manor House Camden Road Camden Town Great Portland Euston Street Baker Street Edgware Marylebone Road Bayswater Latimer Road North Acton 3 South Ealing Gospel Oak Chalk Farm 2 Swiss Cottage St. John's Wood Edgware Road Paddington Kentish Town West Belsize Park 200m Finchley Road Maida Vale Ladbroke Grove Park Royal Ealing Common Kilburn West Hampstead Kensal Rise Brondesbury Kensal Green Queen's Park Kilburn Park Warwick Avenue Royal Oak Westbourne Park North Ealing Ealing Broadway 654 Finchley Road & Frognal Willesden Green Brondesbury Park Stonebridge Park Harlesden Willesden Junction Alperton Hanger Lane Wood Green Turnpike Lane Archway Hampstead Heath Dollis Hill Wembley Central Sudbury Town Perivale Theydon Bois Debden Loughton Buckhurst Hill Bounds Green Highgate 3 Brent Cross improves readability Epping Oakwood Arnos Grove Finchley Central East Finchley Hendon Central Kingsbury Preston Road Southgate West Finchley Colindale Queensbury Harrowon-the-Hill Sudbury Hill Greenford Cockfosters Totteridge & Whetstone Woodside Park Mill Hill East 4 Edgware Canons Park Kenton North Harrow South Ruislip Northolt Stanmore Harrow & Wealdstone Pinner Eastcote Rayners Lane Ruislip Gardens High Barnet Croxley Chorleywood West Ruislip Hillingdon Uxbridge Closed Sundays Colliers Wood Served by Piccadilly line trains early morning and late evening Morden Clapham Common Brixton 2 100m Balham Tooting Broadway South Wimbledon 3 D C B A 6 5 4 3 2 1 Station in Zone D Bakerloo Station in Zone C Special fares apply for single and return tickets to and from Harrow & Wealdstone. Station in Zone B Central No service Woodford - Hainault after 2000 daily. Station in Zone A Station in Zone 6 and Zone A Station in Zone 6 Circle District Cannon Street open until 2100 Mondays to Fridays. Closed Saturdays and Sundays. Station in Zone 5 Station in Zone 4 Station in both zones Station in Zone 3 Station in Zone 2 Station in both zones Station in Zone 1 3 Points to remember Explanation of zones District East London Hammersmith & City Metropolitan Northern Piccadilly Earl’s Court - Kensington (Olympia) 0700 to 2345 Mondays to Saturdays, 0800 to 2345 Sundays. Shoreditch station opens 0700 to 1030 and 1530 to 2030 Mondays to Fridays. Closed Saturdays. Open from 0700 to 1500 Sundays. No service Whitechapel - Barking early morning or late evening Mondays to Saturdays or all day Sundays. For Chesham change at Chalfont & Latimer on most trains. 4 On Sundays between 1300 and 1730, Camden Town is open for interchange and exit only. No service Uxbridge - Rayners Lane in the early mornings. Replacement bus service between Hatton Cross and Heathrow Terminal 4. Heathrow Terminal 4 Underground station closed until September 2006. Waterloo & City 0615 to 2130 Mondays to Fridays. 0800 to 1830 Saturdays. Closed Sundays. Certain stations are closed on public holidays. ¨ Martin Nollenburg and Alexander Wolff 4 32 Drawing Metro Maps Our Model Our Solution NP-Hardness What is a metro map? Hard and soft constraints More Formally The Metro Map Problem Given: Goal: planar embedded graph G = (V , E), V ⊂ R2 , line cover L of paths or cycles in G (the metro lines), draw G and L nicely. ¨ Martin Nollenburg and Alexander Wolff 5 32 Drawing Metro Maps Our Model Our Solution NP-Hardness What is a metro map? Hard and soft constraints More Formally The Metro Map Problem Given: Goal: planar embedded graph G = (V , E), V ⊂ R2 , line cover L of paths or cycles in G (the metro lines), draw G and L nicely. What is a nice drawing? ¨ Martin Nollenburg and Alexander Wolff 5 32 Drawing Metro Maps Our Model Our Solution NP-Hardness What is a metro map? Hard and soft constraints More Formally The Metro Map Problem Given: Goal: planar embedded graph G = (V , E), V ⊂ R2 , line cover L of paths or cycles in G (the metro lines), draw G and L nicely. What is a nice drawing? Look at real-world metro maps drawn by graphic designers and model their design principles as ¨ Martin Nollenburg and Alexander Wolff 5 32 Drawing Metro Maps Our Model Our Solution NP-Hardness What is a metro map? Hard and soft constraints More Formally The Metro Map Problem Given: Goal: planar embedded graph G = (V , E), V ⊂ R2 , line cover L of paths or cycles in G (the metro lines), draw G and L nicely. What is a nice drawing? Look at real-world metro maps drawn by graphic designers and model their design principles as hard constraints – must be fulfilled, ¨ Martin Nollenburg and Alexander Wolff 5 32 Drawing Metro Maps Our Model Our Solution NP-Hardness What is a metro map? Hard and soft constraints More Formally The Metro Map Problem Given: Goal: planar embedded graph G = (V , E), V ⊂ R2 , line cover L of paths or cycles in G (the metro lines), draw G and L nicely. What is a nice drawing? Look at real-world metro maps drawn by graphic designers and model their design principles as hard constraints – must be fulfilled, soft constraints – should hold as tightly as possible. ¨ Martin Nollenburg and Alexander Wolff 5 32 Drawing Metro Maps Our Model Our Solution NP-Hardness What is a metro map? Hard and soft constraints Hard Constraints (H1) preserve embedding of G ¨ Martin Nollenburg and Alexander Wolff 6 32 Drawing Metro Maps Our Model Our Solution NP-Hardness What is a metro map? Hard and soft constraints Hard Constraints (H1) preserve embedding of G (H2) draw all edges as octilinear line segments, i.e. parallel to a coordinate axes or at 45◦ degrees ¨ Martin Nollenburg and Alexander Wolff 6 32 Drawing Metro Maps Our Model Our Solution NP-Hardness What is a metro map? Hard and soft constraints Hard Constraints (H1) preserve embedding of G (H2) draw all edges as octilinear line segments, i.e. parallel to a coordinate axes or at 45◦ degrees (H3) draw each edge e with length ≥ `e ¨ Martin Nollenburg and Alexander Wolff 6 32 Drawing Metro Maps Our Model Our Solution NP-Hardness What is a metro map? Hard and soft constraints Hard Constraints (H1) preserve embedding of G (H2) draw all edges as octilinear line segments, i.e. parallel to a coordinate axes or at 45◦ degrees (H3) draw each edge e with length ≥ `e (H4) keep vertices dmin away from non-incident edges ¨ Martin Nollenburg and Alexander Wolff 6 32 Drawing Metro Maps Our Model Our Solution NP-Hardness What is a metro map? Hard and soft constraints Soft Constraints (S1) draw metro lines with few bends ¨ Martin Nollenburg and Alexander Wolff 7 32 Drawing Metro Maps Our Model Our Solution NP-Hardness What is a metro map? Hard and soft constraints Soft Constraints (S1) draw metro lines with few bends (S2) keep total edge length small C B D Chesham Chalfont & Latimer Amersham A 5 Watford Rickmansworth Moor Park Northwood Northwood Hills Ruislip Ruislip Manor Ickenham Burnt Oak Harrowon-the-Hill South Ruislip South Harrow Wembley Park Hampstead Hanger Lane South Acton East Acton Holland Park 2 Acton Town Hammersmith 1 Barons Court Russell Square West Kensington Gloucester Road Earl's Court Sloane Square Gunnersbury Blackfriars Canada Water Surrey Quays Pimlico Southwark Putney Bridge Bu s Circle Hammersmith & City Metropolitan Docklands Light Railway ¨ Martin Nollenburg and Alexander Wolff 7 32 National Rail Prince Regent Royal Albert 2 Beckton Park Cyprus Gallions Reach London City Airport Silvertown Beckton North Woolwich King George V Cutty Sark Greenwich Deptford Bridge 100m Elverson Road Lewisham Stockwell Oval Clapham North Connections with National Rail Clapham High Street 100m Connections with riverboat services Clapham South Airport interchange Connection with Tramlink Custom House Pontoon Dock Crossharbour & London Arena Mudchute New Cross Elephant & Castle Kennington restricted service - see opposite key Royal Victoria North Greenwich Heron Quays for Maritime Greenwich Vauxhall Tooting Bec Under construction Single and return tickets issued for Underground/DLR journeys are not valid between Gunnersbury-Stratford or Custom House-North Woolwich 4 Bus to London City Airport Canning Town West Silvertown West India Quay Canary Wharf Island Gardens Borough New Cross Gate Southfields Interchange stations Northern Piccadilly Victoria Waterloo & City Central District Wapping Lambeth North River Thames East London Upton Park Plaistow West Ham 3 East India Blackwall South Quay Waterloo Fulham Broadway Wimbledon Jubilee Poplar for ExCeL Rotherhithe Bermondsey East Putney Bakerloo BromleyBow Church by-Bow Devons Road All Saints Westferry River Thames Wimbledon Park Key to lines and symbols 2 Stepney Green Whitechapel Limehouse Tower Gateway London Bridge Embankment Waterloo East Kew Gardens Richmond Bow Road East Ham n ctio stru con ser vic e Aldgate East Shadwell Tower Hill Fenchurch Street 150m Charing Cross 100m Parsons Green Hatton Cross for Heathrow Terminal 4 Heathrow Terminal 4 Underground station closed until September 2006. Replacement bus services run from Hatton Cross. Pudding Mill Lane der Un Heathrow Terminal 4 Shoreditch 1 Aldgate Monument West Brompton Hounslow Central Heathrow Terminals 1, 2, 3 Barking Mile End Liverpool Street Bank Cannon Street Temple Westminster Dagenham Heathway Becontree 200m St. James's Park Victoria South Kensington Hornchurch Elm Park Dagenham East Upney n ctio stru con Hounslow West Turnham Stamford Ravenscourt Green Brook Park Upminster Upminster Bridge Gants Hill Leyton Stratford Moorgate St. Paul's Chancery Lane Covent Garden Leicester Square Mansion House Charing Cross Piccadilly Circus Northfields Boston Manor Chiswick Park Hounslow East Osterley Bethnal Green Holborn Tottenham Court Road Green Park Knightsbridge 3 Hackney Wick Barkingside Homerton 2 Old Street Euston 200m Oxford Circus Marble Arch High Street Hyde Park Kensington Corner Kensington (Olympia) Farringdon Barbican Goodge Street Lancaster Bond Queensway Gate Street Notting Hill Gate Shepherd's Bush Acton Central Shepherd's Bush Goldhawk Road Wanstead Dalston Kingsland Fairlop der Un South Ealing White City West Acton Snaresbrook Hackney Central Canonbury Highbury & Islington Hainault Newbury Park Redbridge Leytonstone Caledonian Road & Barnsbury Angel 6 5 4 South Woodford Walthamstow Central Tottenham Hale Finsbury Park Holloway Road King's Cross St. Pancras Euston Square Bayswater Latimer Road North Acton 3 Mornington Crescent Warren Street Regent's Park Edgware Marylebone Road Blackhorse Road Caledonian Road Camden Road Camden Town Great Portland Euston Street Baker Street Arsenal Kentish Town Chalk Farm 2 Swiss Cottage St. John's Wood Manor House Tufnell Park Kentish Town West Belsize Park 200m Finchley Road Edgware Road Paddington Ladbroke Grove Park Royal Ealing Common Kilburn West Hampstead Maida Vale Paddington North Ealing Ealing Broadway 654 Finchley Road & Frognal Willesden Green Kensal Rise Brondesbury Kensal Green Queen's Park Kilburn Park Warwick Avenue Royal Oak Westbourne Park Gospel Oak Chigwell Grange Hill Woodford Seven Sisters Turnpike Lane Archway Hampstead Heath Dollis Hill Brondesbury Park Stonebridge Park Harlesden Willesden Junction Alperton Wood Green Highgate 3 Golders Green Neasden Wembley Central Sudbury Hill Sudbury Town Perivale Theydon Bois Roding Valley Buckhurst Hill Bounds Green East Finchley Brent Cross Debden Loughton Arnos Grove Finchley Central Hendon Central Kingsbury Preston Road Northwick Park South Kenton North Wembley Epping Oakwood Southgate West Finchley Colindale Queensbury West Harrow Greenford Cockfosters Woodside Park Mill Hill East 4 Edgware Canons Park Kenton North Harrow Rayners Lane Northolt Stanmore Harrow & Wealdstone Pinner Eastcote Ruislip Gardens High Barnet Totteridge & Whetstone Croxley Chorleywood West Ruislip Hillingdon Uxbridge Closed Sundays Colliers Wood Served by Piccadilly line trains early morning and late evening Morden Clapham Common Brixton 2 100m Balham Tooting Broadway South Wimbledon 3 Points to remember Explanation of zones D C B A 6 5 4 3 2 1 Station in Zone D Bakerloo Station in Zone C Station in Zone B Station in Zone A Station in Zone 6 and Zone A Station in Zone 6 Station in Zone 5 Station in Zone 4 Station in both zones Station in Zone 3 Station in Zone 2 Station in both zones Station in Zone 1 3 Special fares apply for single and return tickets to and from Harrow & Wealdstone. Central No service Woodford - Hainault after 2000 daily. Circle District Cannon Street open until 2100 Mondays to Fridays. Closed Saturdays and Sundays. District Earl’s Court - Kensington (Olympia) 0700 to 2345 Mondays to Saturdays, 0800 to 2345 Sundays. East London Hammersmith & City Metropolitan Northern Piccadilly Shoreditch station opens 0700 to 1030 and 1530 to 2030 Mondays to Fridays. Closed Saturdays. Open from 0700 to 1500 Sundays. No service Whitechapel - Barking early morning or late evening Mondays to Saturdays or all day Sundays. For Chesham change at Chalfont & Latimer on most trains. 4 On Sundays between 1300 and 1730, Camden Town is open for interchange and exit only. No service Uxbridge - Rayners Lane in the early mornings. Replacement bus service between Hatton Cross and Heathrow Terminal 4. Heathrow Terminal 4 Underground station closed until September 2006. Waterloo & City 0615 to 2130 Mondays to Fridays. 0800 to 1830 Saturdays. Closed Sundays. Certain stations are closed on public holidays. Drawing Metro Maps Our Model Our Solution NP-Hardness What is a metro map? Hard and soft constraints 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 ¨ Martin Nollenburg and Alexander Wolff 7 32 Drawing Metro Maps Our Model Our Solution NP-Hardness Mixed-integer programming formulation Experiments Labeling Outline 1 Modeling the Metro Map Problem What is a metro map? Hard and soft constraints 2 Our Solution Mixed-integer programming formulation Experiments Labeling 3 NP-Hardness Rectilinear vs. octilinear drawing Reduction from planar 3-SAT ¨ Martin Nollenburg and Alexander Wolff 8 32 Drawing Metro Maps Mixed-integer programming formulation Experiments Labeling Our Model Our Solution NP-Hardness Mathematical Programming Linear Programming: efficient optimization method for linear constraints and objective function, real-valued variables (domain R). ¨ Martin Nollenburg and Alexander Wolff 9 32 Drawing Metro Maps Mixed-integer programming formulation Experiments Labeling Our Model Our Solution NP-Hardness Mathematical Programming Linear Programming: efficient optimization method for linear constraints and objective function, real-valued variables (domain R). Mixed-Integer Programming (MIP): allows also integer variables (domain Z), solution NP-hard in general. ¨ Martin Nollenburg and Alexander Wolff 9 32 Drawing Metro Maps Mixed-integer programming formulation Experiments Labeling Our Model Our Solution NP-Hardness Mathematical Programming Linear Programming: efficient optimization method for linear constraints and objective function, real-valued variables (domain R). Mixed-Integer Programming (MIP): allows also integer variables (domain Z), solution NP-hard in general. Still a practical method for many hard optimizat. problems. ¨ Martin Nollenburg and Alexander Wolff 9 32 Drawing Metro Maps Mixed-integer programming formulation Experiments Labeling Our Model Our Solution NP-Hardness Mathematical Programming Linear Programming: efficient optimization method for linear constraints and objective function, real-valued variables (domain R). Mixed-Integer Programming (MIP): allows also integer variables (domain Z), solution NP-hard in general. Still a practical method for many hard optimizat. problems. ¨ Theorem (Nollenburg & Wolff GD’05) The problem MetroMapLayout can be formulated as a MIP s.th. linear constraints → hard constraints, objective function → soft constraints. ¨ Martin Nollenburg and Alexander Wolff 9 32 Drawing Metro Maps Our Model Our Solution NP-Hardness Mixed-integer programming formulation Experiments Labeling Example: Octilinearity and Relative Position 2 Sectors – for each vtx. u partition plane into sectors 0–7 1 3 u 0 4 v 5 here: sec(u, v ) = 5 (input) 7 6 ¨ Martin Nollenburg and Alexander Wolff 10 32 Drawing Metro Maps Our Model Our Solution NP-Hardness Mixed-integer programming formulation Experiments Labeling Example: Octilinearity and Relative Position 2 Sectors – for each vtx. u partition plane into sectors 0–7 1 3 u 0 here: sec(u, v ) = 5 (input) 4 v 5 7 6 – number octilinear edge directions accordingly e.g. dir(u, v ) = 4 (output) ¨ Martin Nollenburg and Alexander Wolff 10 32 Drawing Metro Maps Our Model Our Solution NP-Hardness Mixed-integer programming formulation Experiments Labeling Example: Octilinearity and Relative Position 2 Sectors – for each vtx. u partition plane into sectors 0–7 1 3 u 0 here: sec(u, v ) = 5 (input) 4 v 7 5 6 – number octilinear edge directions accordingly e.g. dir(u, v ) = 4 (output) y Coordinates assign z1 - and z2 -coordinates to each vertex v : z1 x z1 (v ) = x(v ) + y(v ) z2 (v ) = x(v ) − y(v ) z2 ¨ Martin Nollenburg and Alexander Wolff 10 32 Drawing Metro Maps Our Model Our Solution NP-Hardness Mixed-integer programming formulation Experiments Labeling Example: Octilinearity and Relative Position Goal 2 prev Draw edge uv 1 3 with minimum length `uv u 0 4 v orig 5 restricted to 3 directions 7 6 next ¨ Martin Nollenburg and Alexander Wolff 11 32 Drawing Metro Maps Our Model Our Solution NP-Hardness Mixed-integer programming formulation Experiments Labeling Example: Octilinearity and Relative Position Goal 2 prev Draw edge uv 1 3 with minimum length `uv u 0 4 v orig 5 restricted to 3 directions 7 6 next How to model this using linear constraints? ¨ Martin Nollenburg and Alexander Wolff 11 32 Drawing Metro Maps Our Model Our Solution NP-Hardness Mixed-integer programming formulation Experiments Labeling Example: Octilinearity and Relative Position Goal 2 prev Draw edge uv 1 3 with minimum length `uv u 0 4 v orig 5 restricted to 3 directions 7 6 next How to model this using linear constraints? Binary Variables αprev (u, v ) + αorig (u, v ) + αnext (u, v ) = 1 ¨ Martin Nollenburg and Alexander Wolff 11 32 Drawing Metro Maps Our Model Our Solution NP-Hardness Mixed-integer programming formulation Experiments Labeling Example: Octilinearity and Relative Position Previous Sector 2 1 3 prev u 0 4 v orig 5 7 6 next y (u) − y(v ) ≤ M(1 − αprev (u, v )) −y(u) + y (v ) ≤ M(1 − αprev (u, v )) x(u) − x(v ) ≥ −M(1 − αprev (u, v )) + `uv ¨ Martin Nollenburg and Alexander Wolff 12 32 Drawing Metro Maps Our Model Our Solution NP-Hardness Mixed-integer programming formulation Experiments Labeling Example: Octilinearity and Relative Position Previous Sector 2 1 3 prev u 0 4 v orig 5 7 6 next y (u) − y(v ) ≤ M(1 − αprev (u, v )) −y(u) + y (v ) ≤ M(1 − αprev (u, v )) x(u) − x(v ) ≥ −M(1 − αprev (u, v )) + `uv How does this work? ¨ Martin Nollenburg and Alexander Wolff 12 32 Drawing Metro Maps Our Model Our Solution NP-Hardness Mixed-integer programming formulation Experiments Labeling Example: Octilinearity and Relative Position Previous Sector 2 1 3 prev u 0 4 v orig 5 7 6 next y (u) − y(v ) ≤ M(1 − αprev (u, v )) −y(u) + y (v ) ≤ M(1 − αprev (u, v )) x(u) − x(v ) ≥ −M(1 − αprev (u, v )) + `uv How does this work? Case 1: αprev (u, v ) = 0 y(u) − y(v ) ≤ M −y(u) + y(v ) ≤ M x(u) − x(v ) ≥ `uv − M ¨ Martin Nollenburg and Alexander Wolff 12 32 Drawing Metro Maps Our Model Our Solution NP-Hardness Mixed-integer programming formulation Experiments Labeling Example: Octilinearity and Relative Position Previous Sector 2 1 3 prev u 0 4 v orig 5 7 6 next y (u) − y(v ) ≤ M(1 − αprev (u, v )) −y(u) + y (v ) ≤ M(1 − αprev (u, v )) x(u) − x(v ) ≥ −M(1 − αprev (u, v )) + `uv How does this work? Case 2: αprev (u, v ) = 1 y(u) − y(v ) ≤ 0 −y(u) + y(v ) ≤ 0 x(u) − x(v ) ≥ `uv ¨ Martin Nollenburg and Alexander Wolff 12 32 Drawing Metro Maps Our Model Our Solution NP-Hardness Mixed-integer programming formulation Experiments Labeling Example: Octilinearity and Relative Position Original Sector 2 1 3 prev u 0 4 v orig 5 7 6 next 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 )) + 2`uv ¨ Martin Nollenburg and Alexander Wolff 13 32 Drawing Metro Maps Our Model Our Solution NP-Hardness Mixed-integer programming formulation Experiments Labeling Example: Octilinearity and Relative Position Original Sector 2 1 3 prev u 0 4 v orig 5 7 6 next Next Sector 2 1 3 prev u 0 4 v orig 5 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 )) + 2`uv 7 6 next x(u) − x(v ) ≤ M(1 − αnext (u, v )) −x(u) + x(v ) ≤ M(1 − αnext (u, v )) y (u) − y(v ) ≥ −M(1 − αnext (u, v )) + `uv ¨ Martin Nollenburg and Alexander Wolff 13 32 Drawing Metro Maps Our Model Our Solution NP-Hardness Mixed-integer programming formulation Experiments Labeling Summary of the Model The above constraints enforce octilinearity, minimum edge length, (partially) relative position ¨ Martin Nollenburg and Alexander Wolff 14 32 Drawing Metro Maps Our Model Our Solution NP-Hardness Mixed-integer programming formulation Experiments Labeling Summary of the Model The above constraints enforce octilinearity, minimum edge length, (partially) relative position Similarly: preservation of embedding planarity ¨ Martin Nollenburg and Alexander Wolff 14 32 Drawing Metro Maps Our Model Our Solution NP-Hardness Mixed-integer programming formulation Experiments Labeling Summary of the Model The above constraints enforce octilinearity, minimum edge length, (partially) relative position Similarly: preservation of embedding planarity Soft constraints: weighted sum in objective function minimize λbends costbends + λlength costlength + λdir costdir ¨ Martin Nollenburg and Alexander Wolff 14 32 Drawing Metro Maps Our Model Our Solution NP-Hardness Mixed-integer programming formulation Experiments Labeling Summary of the Model The above constraints enforce octilinearity, minimum edge length, (partially) relative position Similarly: preservation of embedding planarity Soft constraints: weighted sum in objective function minimize λbends costbends + λlength costlength + λdir costdir In total O(|V |2 ) constraints and variables ¨ Martin Nollenburg and Alexander Wolff 14 32 Drawing Metro Maps Our Model Our Solution NP-Hardness Mixed-integer programming formulation Experiments Labeling Summary of the Model The above constraints enforce octilinearity, minimum edge length, (partially) relative position Similarly: preservation of embedding planarity Soft constraints: weighted sum in objective function minimize λbends costbends + λlength costlength + λdir costdir In total O(|V |2 ) constraints and variables ¨ Martin Nollenburg and Alexander Wolff 14 32 Drawing Metro Maps Our Model Our Solution NP-Hardness Mixed-integer programming formulation Experiments Labeling Results – Vienna Input ¨ Martin Nollenburg and Alexander Wolff 15 32 Input |V | |E| lines normal reduced 90 44 96 50 5 Drawing Metro Maps Our Model Our Solution NP-Hardness Mixed-integer programming formulation Experiments Labeling Results – Vienna Input Input |V | |E| lines normal reduced 90 44 96 50 5 ↓ ¨ Martin Nollenburg and Alexander Wolff 15 32 MIP constr. var. normal improved heuristic 1? heuristic 2 39363 23226 5703 1875 9960 6048 1800 872 Drawing Metro Maps Our Model Our Solution NP-Hardness Mixed-integer programming formulation Experiments Labeling Results – Vienna Input Input |V | |E| lines normal reduced 90 44 96 50 5 ↓ MIP constr. var. normal improved heuristic 1? heuristic 2? 39363 23226 5703 1875 9960 6048 1800 872 ?) ¨ Martin Nollenburg and Alexander Wolff 15 32 29 seconds w/o proof of opt. Drawing Metro Maps Our Model Our Solution NP-Hardness Mixed-integer programming formulation Experiments Labeling Results – Vienna Input ¨ Martin Nollenburg and Alexander Wolff Output 15 32 Drawing Metro Maps Our Model Our Solution NP-Hardness Mixed-integer programming formulation Experiments Labeling Results – Vienna Official map ¨ Martin Nollenburg and Alexander Wolff Output 15 32 Drawing Metro Maps Our Model Our Solution NP-Hardness Mixed-integer programming formulation Experiments Labeling Results – Sydney Input Input normal reduced |V | |E| lines 174 62 183 71 10 Our Model Our Solution NP-Hardness Mixed-integer programming formulation Experiments Labeling Results – Sydney Input |V | |E| lines 174 62 183 71 10 Input normal reduced ↓ ¨ Martin Nollenburg and Alexander Wolff 16 32 MIP constr. var. normal improved heuristic 1 heuristic 2? 81416 45182 6242 3041 20329 11545 2105 1329 Drawing Metro Maps Our Model Our Solution NP-Hardness Mixed-integer programming formulation Experiments Labeling Results – Sydney Input |V | |E| lines 174 62 183 71 10 Input normal reduced ↓ MIP constr. var. normal improved heuristic 1? heuristic 2? 81416 45182 6242 3041 20329 11545 2105 1329 ?) ¨ Martin Nollenburg and Alexander Wolff 16 32 22 minutes w/o proof of opt. Drawing Metro Maps Our Model Our Solution NP-Hardness Mixed-integer programming formulation Experiments Labeling Results – Sydney Input ¨ Martin Nollenburg and Alexander Wolff Output 16 32 Drawing Metro Maps Our Model Our Solution NP-Hardness Mixed-integer programming formulation Experiments Labeling Results – Sydney Official map ¨ Martin Nollenburg and Alexander Wolff Output 16 32 Drawing Metro Maps Our Model Our Solution NP-Hardness Mixed-integer programming formulation Experiments Labeling Sydney: Related Work [HMdN-GD04] 7.6 sec. ¨ Martin Nollenburg and Alexander Wolff Output (22 min.) 17 32 Drawing Metro Maps Our Model Our Solution NP-Hardness Mixed-integer programming formulation Experiments Labeling Sydney: Related Work [SR-IV04] reduced: 4 min. ¨ Martin Nollenburg and Alexander Wolff Output (22 min.) 17 32 Drawing Metro Maps Our Model Our Solution NP-Hardness Mixed-integer programming formulation Experiments Labeling Sydney: Related Work [SR-IV04] normal: 28 min. ¨ Martin Nollenburg and Alexander Wolff Output (22 min.) 17 32 Drawing Metro Maps Eb e or d N rb Lin ac da h Zw c h in g N en ec b k er Bin arg g ( B au era ad ch e M n) os ba ch -N ec ka re lz Labeling Mixed-integer programming formulation Experiments Labeling Se ck Zim ach m Ad ern els he im Our Model Our Solution NP-Hardness Neckarhausen Neckarsteinach M o M sb os ac ba h ch W e N (B st ec ad ka e rb n ) ur ke D n a O Au llau be er rs b a ch c h Eic eff ho lenz lz he im Osterburken Hirschhorn (Neckar) Neckargemuend Altstadt N L an U-M nh it eim te H bf ck en he im er sla is A- M Schifferstadt Sued Se W Fr a pe ye r Ka HD-Karlstor HD-Weststadt/Suedstadt Heidelberg Hbf MA-Neckarau hs oc H HD-Schlierbach/Ziegelhausen nk ute rn H bf e eid nste en in eid tha (Pfa en l lz La fe ) m ls br N ec eu h s t N tad (Pfa eu t s (W lz) H tad ein as ts B str Bo loc oeb .) H bf eh h (P ig Sc l-Ig falz hif ge ) lh Lim fers eim b tad LU urg t -R erh LU hein of -M go Lu un en dw de he ig nhe im sh im afe n (R he M in A)H R bf an gie M M rba A- A- hn F F h H rie rie of D dr dr -P ic ic fa hs hs ffe fe fe ng ld ld ru Su nd ed /W ie bli ng en Neckargemuend HD-Orthopaedie Model labels as special metro lines: MA-Rheinau M Speyer Nord-West Schwetzingen Speyer Hbf Oftersheim Speyer Sued Hockenheim Berghausen (Pfalz) Neulussheim HD-Kirchheim/Rohrbach Heiligenstein (Pfalz) Waghaeusel St. Ilgen-Sandhausen Wiesental Wiesloch-Walldorf Lingenfeld R Bad Schoenborn-Kronau Bad Schoenborn Sued G G er m er s he heim in Ph sh ilip eim ra be H psb n- utt ur N eu enh g do eim rf N G or ra d be nN eu do rf Rot-Malsch Karlsdorf Ubstadt-Weiher Bruchsal Bruchsal Bildungszentrum Friedrichstal (Baden) Blankenloch Untergrombach KA-Hagsfeld Weingarten (Baden) Ka rls ru he -D ur la ch Karlsruhe Hbf ¨ Martin Nollenburg and Alexander Wolff 18 32 Drawing Metro Maps Eb e or d N rb Lin ac da h Zw c h in g N en ec b k er Bin arg g ( B au era ad ch e M n) os ba ch -N ec ka re lz Labeling Mixed-integer programming formulation Experiments Labeling Se ck Zim ach m Ad ern els he im Our Model Our Solution NP-Hardness M o M sb os ac ba h ch W e N (B st ec ad ka e rb n ) ur ke D n a O Au llau be er rs b a ch c h Eic eff ho lenz lz he im Osterburken Hirschhorn (Neckar) Neckarhausen Neckarsteinach Neckargemuend Altstadt N L an U-M nh it eim te H bf ck en he im er sla is A- M Schifferstadt Sued Se W Fr a pe ye r Ka HD-Karlstor HD-Weststadt/Suedstadt Heidelberg Hbf MA-Neckarau hs oc H HD-Schlierbach/Ziegelhausen nk ute rn H bf e eid nste en in eid tha (Pfa en l lz La fe ) m ls br N ec eu h s t N tad (Pfa eu t s (W lz) H tad ein as ts B str Bo loc oeb .) H bf eh h (P ig Sc l-Ig falz hif ge ) lh Lim fers eim b tad LU urg t -R erh LU hein of -M go Lu un en dw de he ig nhe im sh im afe n (R he M in A)H R bf an gie M M rba A- A- hn F F h H rie rie of D dr dr -P ic ic fa hs hs ffe fe fe ng ld ld ru Su nd ed /W ie bli ng en Neckargemuend HD-Orthopaedie Model labels as special metro lines: MA-Rheinau M Speyer Nord-West Schwetzingen Speyer Hbf Oftersheim Speyer Sued Hockenheim Berghausen (Pfalz) Neulussheim HD-Kirchheim/Rohrbach Heiligenstein (Pfalz) Waghaeusel St. Ilgen-Sandhausen Wiesental Wiesloch-Walldorf Lingenfeld R Bad Schoenborn-Kronau Bad Schoenborn Sued G G er m er s he heim in Ph sh ilip eim ra be H psb n- utt ur N eu enh g do eim rf N G or ra d be nN eu do rf Rot-Malsch Karlsdorf put all labels between a pair of interchange stations into one parallelogram, Ubstadt-Weiher Bruchsal Bruchsal Bildungszentrum Friedrichstal (Baden) Blankenloch Untergrombach KA-Hagsfeld Weingarten (Baden) Ka rls ru he -D ur la ch Karlsruhe Hbf ¨ Martin Nollenburg and Alexander Wolff 18 32 Drawing Metro Maps Eb e or d N rb Lin ac da h Zw c h in g N en ec b k er Bin arg g ( B au era ad ch e M n) os ba ch -N ec ka re lz Labeling Mixed-integer programming formulation Experiments Labeling Se ck Zim ach m Ad ern els he im Our Model Our Solution NP-Hardness M o M sb os ac ba h ch W e N (B st ec ad ka e rb n ) ur ke D n a O Au llau be er rs b a ch c h Eic eff ho lenz lz he im Osterburken Hirschhorn (Neckar) Neckarhausen Neckarsteinach Neckargemuend Altstadt N L an U-M nh it eim te H bf ck en he im er sla is A- M Schifferstadt Sued Se W Fr a pe ye r Ka HD-Karlstor HD-Weststadt/Suedstadt Heidelberg Hbf MA-Neckarau hs oc H HD-Schlierbach/Ziegelhausen nk ute rn H bf e eid nste en in eid tha (Pfa en l lz La fe ) m ls br N ec eu h s t N tad (Pfa eu t s (W lz) H tad ein as ts B str Bo loc oeb .) H bf eh h (P ig Sc l-Ig falz hif ge ) lh Lim fers eim b tad LU urg t -R erh LU hein of -M go Lu un en dw de he ig nhe im sh im afe n (R he M in A)H R bf an gie M M rba A- A- hn F F h H rie rie of D dr dr -P ic ic fa hs hs ffe fe fe ng ld ld ru Su nd ed /W ie bli ng en Neckargemuend HD-Orthopaedie Model labels as special metro lines: MA-Rheinau M Speyer Nord-West Schwetzingen Speyer Hbf Oftersheim Speyer Sued Hockenheim Berghausen (Pfalz) Neulussheim HD-Kirchheim/Rohrbach Heiligenstein (Pfalz) Waghaeusel St. Ilgen-Sandhausen Wiesental Wiesloch-Walldorf Lingenfeld R Bad Schoenborn-Kronau Bad Schoenborn Sued G G er m er s he heim in Ph sh ilip eim ra be H psb n- utt ur N eu enh g do eim rf N G or ra d be nN eu do rf Rot-Malsch Karlsdorf put all labels between a pair of interchange stations into one parallelogram, Ubstadt-Weiher Bruchsal Bruchsal Bildungszentrum Friedrichstal (Baden) Blankenloch Untergrombach KA-Hagsfeld Weingarten (Baden) Ka rls ru he -D ur la ch Karlsruhe Hbf allow parallelograms to change sides, ¨ Martin Nollenburg and Alexander Wolff 18 32 Drawing Metro Maps Eb e or d N rb Lin ac da h Zw c h in g N en ec b k er Bin arg g ( B au era ad ch e M n) os ba ch -N ec ka re lz Labeling Mixed-integer programming formulation Experiments Labeling Se ck Zim ach m Ad ern els he im Our Model Our Solution NP-Hardness M o M sb os ac ba h ch W e N (B st ec ad ka e rb n ) ur ke D n a O Au llau be er rs b a ch c h Eic eff ho lenz lz he im Osterburken Hirschhorn (Neckar) Neckarhausen Neckarsteinach Neckargemuend Altstadt N L an U-M nh it eim te H bf ck en he im er sla is A- M Schifferstadt Sued Se W Fr a pe ye r Ka HD-Karlstor HD-Weststadt/Suedstadt Heidelberg Hbf MA-Neckarau hs oc H HD-Schlierbach/Ziegelhausen nk ute rn H bf e eid nste en in eid tha (Pfa en l lz La fe ) m ls br N ec eu h s t N tad (Pfa eu t s (W lz) H tad ein as ts B str Bo loc oeb .) H bf eh h (P ig Sc l-Ig falz hif ge ) lh Lim fers eim b tad LU urg t -R erh LU hein of -M go Lu un en dw de he ig nhe im sh im afe n (R he M in A)H R bf an gie M M rba A- A- hn F F h H rie rie of D dr dr -P ic ic fa hs hs ffe fe fe ng ld ld ru Su nd ed /W ie bli ng en Neckargemuend HD-Orthopaedie Model labels as special metro lines: MA-Rheinau M Speyer Nord-West Schwetzingen Speyer Hbf Oftersheim Speyer Sued Hockenheim Berghausen (Pfalz) Neulussheim HD-Kirchheim/Rohrbach Heiligenstein (Pfalz) Waghaeusel St. Ilgen-Sandhausen Wiesental Wiesloch-Walldorf Lingenfeld R Bad Schoenborn-Kronau Bad Schoenborn Sued G G er m er s he heim in Ph sh ilip eim ra be H psb n- utt ur N eu enh g do eim rf N G or ra d be nN eu do rf Rot-Malsch Karlsdorf put all labels between a pair of interchange stations into one parallelogram, Ubstadt-Weiher Bruchsal Bruchsal Bildungszentrum Friedrichstal (Baden) Blankenloch Untergrombach KA-Hagsfeld Weingarten (Baden) Ka rls ru he -D ur la ch Karlsruhe Hbf allow parallelograms to change sides, ¨ Martin Nollenburg and Alexander Wolff problem: a lot more planarity constraints :-( 18 32 Drawing Metro Maps Mixed-integer programming formulation Experiments Labeling Our Model Our Solution NP-Hardness Labeling Henri-Bourassa Honore-Beaugrand Sauve Saint-Michel Cremazie Radisson D’Iberville Jarry Langelier Fabre Cadillac Jean-Talon Assomption Viau De Castelnau Pie-IX Beaubien Parc Joliette Rosemont Acadie Prefontaine Laurier Outremont Cote-Vertu Frontenac Mont-Royal Edouard-Montpetit Du College Papineau Sherbrooke Universite-de-Montreal De La Savanne Beaudry Cote-des-Neiges Berri-UQAM Namur Plamondon Jean-Drapeau Saint-Laurent Cote-Sainte-Catherine Snowdon Place-des-Arts Longueil Champ-de-Mars McGill Solution: Place d’Armes Villa-Maria Peel Square-Victoria Guy-Concordia Vendome Bonaventure Atwater Lucien L’Allier Place-Saint-Henri Georges-Vanier solve MIP without planarity constraints Lionel Groulx Charlevoix LaSalle De L’Eglise Verdun Jolicoeur Monk Angrignon ¨ Martin Nollenburg and Alexander Wolff 19 32 Drawing Metro Maps Mixed-integer programming formulation Experiments Labeling Our Model Our Solution NP-Hardness Labeling Henri-Bourassa Honore-Beaugrand Sauve Saint-Michel Cremazie Radisson D’Iberville Jarry Langelier Fabre Cadillac Jean-Talon Assomption Viau De Castelnau Pie-IX Beaubien Parc Joliette Rosemont Acadie Prefontaine Laurier Outremont Cote-Vertu Frontenac Mont-Royal Edouard-Montpetit Du College Papineau Sherbrooke Universite-de-Montreal De La Savanne Beaudry Cote-des-Neiges Berri-UQAM Namur Plamondon Jean-Drapeau Saint-Laurent Cote-Sainte-Catherine Snowdon Place-des-Arts Longueil Champ-de-Mars McGill Solution: Place d’Armes Villa-Maria Peel Square-Victoria Guy-Concordia Vendome Bonaventure Atwater Lucien L’Allier Place-Saint-Henri Georges-Vanier Lionel Groulx solve MIP without planarity constraints use CPLEX callback fct. Charlevoix LaSalle De L’Eglise Verdun Jolicoeur Monk Angrignon ¨ Martin Nollenburg and Alexander Wolff 19 32 Drawing Metro Maps Mixed-integer programming formulation Experiments Labeling Our Model Our Solution NP-Hardness Labeling Henri-Bourassa Honore-Beaugrand Sauve Saint-Michel Cremazie Radisson D’Iberville Jarry Langelier Fabre Cadillac Jean-Talon Assomption Viau De Castelnau Pie-IX Beaubien Parc Joliette Rosemont Acadie Prefontaine Laurier Outremont Cote-Vertu Frontenac Mont-Royal Edouard-Montpetit Du College Papineau Sherbrooke Universite-de-Montreal De La Savanne Beaudry Cote-des-Neiges Berri-UQAM Namur Plamondon Jean-Drapeau Saint-Laurent Cote-Sainte-Catherine Snowdon Place-des-Arts Longueil Champ-de-Mars McGill Solution: Place d’Armes Villa-Maria Peel Square-Victoria Guy-Concordia Vendome Bonaventure Atwater Lucien L’Allier Place-Saint-Henri Georges-Vanier Lionel Groulx solve MIP without planarity constraints use CPLEX callback fct. Charlevoix LaSalle De L’Eglise test which edge pairs cross Verdun Jolicoeur Monk Angrignon ¨ Martin Nollenburg and Alexander Wolff 19 32 Drawing Metro Maps Mixed-integer programming formulation Experiments Labeling Our Model Our Solution NP-Hardness Labeling Henri-Bourassa Honore-Beaugrand Sauve Saint-Michel Cremazie Radisson D’Iberville Jarry Langelier Fabre Cadillac Jean-Talon Assomption Viau De Castelnau Pie-IX Beaubien Parc Joliette Rosemont Acadie Prefontaine Laurier Outremont Cote-Vertu Frontenac Mont-Royal Edouard-Montpetit Du College Papineau Sherbrooke Universite-de-Montreal De La Savanne Beaudry Cote-des-Neiges Berri-UQAM Namur Plamondon Jean-Drapeau Saint-Laurent Cote-Sainte-Catherine Snowdon Place-des-Arts Longueil Champ-de-Mars McGill Solution: Place d’Armes Villa-Maria Peel Square-Victoria Guy-Concordia Vendome Bonaventure Atwater Lucien L’Allier Place-Saint-Henri Georges-Vanier Lionel Groulx use CPLEX callback fct. Charlevoix LaSalle De L’Eglise test which edge pairs cross Verdun Jolicoeur Monk add corresponding constr. Angrignon ¨ Martin Nollenburg and Alexander Wolff solve MIP without planarity constraints 19 32 Drawing Metro Maps Mixed-integer programming formulation Experiments Labeling Our Model Our Solution NP-Hardness Labeling Henri-Bourassa Honore-Beaugrand Sauve Saint-Michel Cremazie Radisson D’Iberville Jarry Langelier Fabre Cadillac Jean-Talon Assomption Viau De Castelnau Pie-IX Beaubien Parc Joliette Rosemont Acadie Prefontaine Laurier Outremont Cote-Vertu Frontenac Mont-Royal Edouard-Montpetit Du College Papineau Sherbrooke Universite-de-Montreal De La Savanne Beaudry Cote-des-Neiges Berri-UQAM Namur Plamondon Jean-Drapeau Saint-Laurent Cote-Sainte-Catherine Snowdon Place-des-Arts Longueil Champ-de-Mars McGill Solution: Place d’Armes Villa-Maria Peel Square-Victoria Guy-Concordia Vendome Bonaventure Atwater Lucien L’Allier Place-Saint-Henri Georges-Vanier Lionel Groulx solve MIP without planarity constraints use CPLEX callback fct. Charlevoix LaSalle De L’Eglise test which edge pairs cross Verdun Jolicoeur Monk add corresponding constr. Angrignon continue to solve MIP ¨ Martin Nollenburg and Alexander Wolff 19 32 Drawing Metro Maps Mixed-integer programming formulation Experiments Labeling Our Model Our Solution NP-Hardness Labeling Henri-Bourassa Honore-Beaugrand Sauve Saint-Michel Cremazie Radisson D’Iberville Jarry Langelier Fabre Cadillac Jean-Talon Assomption Viau De Castelnau Pie-IX Beaubien Parc Joliette Rosemont Acadie Prefontaine Laurier Outremont Cote-Vertu Frontenac Mont-Royal Edouard-Montpetit Du College Papineau Sherbrooke Universite-de-Montreal De La Savanne Beaudry Cote-des-Neiges Berri-UQAM Namur Plamondon Jean-Drapeau Saint-Laurent Cote-Sainte-Catherine Snowdon Place-des-Arts Longueil Champ-de-Mars McGill Solution: Place d’Armes Villa-Maria Peel Square-Victoria Guy-Concordia Vendome Bonaventure Atwater Lucien L’Allier Place-Saint-Henri Georges-Vanier Lionel Groulx use CPLEX callback fct. Charlevoix LaSalle De L’Eglise test which edge pairs cross Verdun Jolicoeur Monk add corresponding constr. Angrignon continue to solve MIP ´ Montreal: 17 min. ¨ Martin Nollenburg and Alexander Wolff solve MIP without planarity constraints 19 32 Drawing Metro Maps Mixed-integer programming formulation Experiments Labeling Our Model Our Solution NP-Hardness Floridsdorf Labeling Neue Donau Handelskai Leopoldau Dresdner Strasse Grossfeldsiedlung Jaegerstrasse Heiligenstadt Aderklaaer Strasse Rennbahnweg Spittelau Kagraner Platz Kagran Friedensbruecke Nussdorfer Strasse Alte Donau Rossauer Laende VIC Kaisermuehlen ho tte nr in g Waehringer Strasse Volksoper Donauinsel Vorgartenstrasse Sc Michelbeuern AKH Praterstern ra ss e Alser Strasse Ta bo r Schottentor Universitaet Thaliastrasse Messe st Josefstaedter Strasse Rathaus Trabrennstrasse Stadion Nestroyplatz Donaustadtbr?cke Schwedenplatz Burggasse Stadthalle Seestern ds La n Donauspital r rd Ro in ch a u Sc l-N sg hla ag as ch el-P se th au latz sg as Er se G db Zip as er pe om g re et rs er tra En sse Sim kp la m tz er in g C St ub tz pla ns ha Aspernstrasse Ka St ep ue H Museumsquartier Hardeggasse en to ity eu as ba se ug as se Vo lk st he at er Zie N gle rg Ke Ott a tte ndle krin g ld or rstr fe as r S se Sc Jo tras h hw ns se eg tra le ss rs tra e ss e H W er es re tb a ng hn as tra ss e se ho f Stadlau Stadtpark as se ng Karlsplatz ke Taubstummengasse ec ss e ru nb tte Suedtiroler Platz Ke gr Pil ar M am re ga ge en La H ue O tteld be r or U St f un nte . Ve sc r S it hw t. eig Veit ga s M eid S Hie se lin cho tz in g H enb g au ru pt nn st ra ss e Br a ga te n nf eld gu ga er ss te l e Gumpendorfer Strasse Keplerplatz Reumannplatz Niederhofstrasse Philadelphiabruecke Tscherttegasse Am Schoepfwerk Alterlaa Erlaaer Strasse Perfektastrasse Wien: 1 day. Siebenhirten ¨ Martin Nollenburg and Alexander Wolff 20 32 Drawing Metro Maps Our Model Our Solution NP-Hardness Mixed-integer programming formulation Experiments Labeling Labeling Tokyo: < 10 sec. [Kameda & Imai IEICE’03] ¨ Martin Nollenburg and Alexander Wolff 21 32 Drawing Metro Maps Our Model Our Solution NP-Hardness Rectilinear vs. octilinear drawing Reduction from planar 3-SAT Outline 1 Modeling the Metro Map Problem What is a metro map? Hard and soft constraints 2 Our Solution Mixed-integer programming formulation Experiments Labeling 3 NP-Hardness Rectilinear vs. octilinear drawing Reduction from planar 3-SAT ¨ Martin Nollenburg and Alexander Wolff 22 32 Drawing Metro Maps Our Model Our Solution NP-Hardness Rectilinear vs. octilinear drawing Reduction from planar 3-SAT Another Problem R ECTILINEAR G RAPH D RAWING Decision Problem Given 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? ¨ Martin Nollenburg and Alexander Wolff 23 32 Drawing Metro Maps Our Model Our Solution NP-Hardness Rectilinear vs. octilinear drawing Reduction from planar 3-SAT Another Problem R ECTILINEAR G RAPH D RAWING Decision Problem Given 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) R ECTILINEAR G RAPH D RAWING can be solved efficiently. ¨ Martin Nollenburg and Alexander Wolff 23 32 Drawing Metro Maps Our Model Our Solution NP-Hardness Rectilinear vs. octilinear drawing Reduction from planar 3-SAT Another Problem R ECTILINEAR G RAPH D RAWING Decision Problem Given 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) R ECTILINEAR G RAPH D RAWING can be solved efficiently. Proof. By reduction to a flow problem. ¨ Martin Nollenburg and Alexander Wolff 23 32 Drawing Metro Maps Our Model Our Solution NP-Hardness Rectilinear vs. octilinear drawing Reduction from planar 3-SAT Another Problem R ECTILINEAR G RAPH D RAWING Decision Problem Given 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) R ECTILINEAR G RAPH D RAWING can be solved efficiently. Proof. By reduction to a flow problem. ¨ Martin Nollenburg and Alexander Wolff 23 32 Drawing Metro Maps Our Model Our Solution NP-Hardness Rectilinear vs. octilinear drawing Reduction from planar 3-SAT Our Problem M ETRO M AP L AYOUT Decision Problem Given 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 (Nollenburg MSc’05) M ETRO M AP L AYOUT is NP-hard. Proof. Reduction from P LANAR 3-S AT to M ETRO M AP L AYOUT. ¨ Martin Nollenburg and Alexander Wolff 24 32 s Drawing Metro Maps Our Model Our Solution NP-Hardness Rectilinear vs. octilinear drawing Reduction from planar 3-SAT Outline of the Reduction x1 ∨ x3 ∨ x4 x1 ∨ x2 ∨ x3 x1 x2 x3 x4 x2 ∨ x3 ∨ x4 x1 ∨ x2 ∨ x4 Input: planar 3-SAT formula ϕ = (x1 ∨ x3 ∨ x4 ) ∧ (x1 ∨ x2 ∨ x3 ) ∧ . . . ¨ Martin Nollenburg and Alexander Wolff 25 32 Drawing Metro Maps Our Model Our Solution NP-Hardness Rectilinear vs. octilinear drawing Reduction from planar 3-SAT Outline of the Reduction x1 ∨ x3 ∨ x4 x1 ∨ x2 ∨ x3 x1 x2 x3 x4 x2 ∨ x3 ∨ x4 x1 ∨ x2 ∨ 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. ¨ Martin Nollenburg and Alexander Wolff 25 32 Drawing Metro Maps Our Model Our Solution NP-Hardness Rectilinear vs. octilinear drawing Reduction from planar 3-SAT Outline of the Reduction x1 ∨ x3 ∨ x4 x1 ∨ x2 ∨ x3 x1 x2 x3 x4 x2 ∨ x3 ∨ x4 x1 ∨ x2 ∨ 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. ¨ Martin Nollenburg and Alexander Wolff 25 32 Drawing Metro Maps Our Model Our Solution NP-Hardness Rectilinear vs. octilinear drawing Reduction from planar 3-SAT Variable Gadget .. . .. . .. . .. . .. . .. . ... ... ... .. . .. . .. . x = true ¨ Martin Nollenburg and Alexander Wolff 26 32 Drawing Metro Maps Our Model Our Solution NP-Hardness Rectilinear vs. octilinear drawing Reduction from planar 3-SAT Variable Gadget .. . .. . .. . .. . .. . .. . ... ... ... .. . .. . .. . x = false ¨ Martin Nollenburg and Alexander Wolff 26 32 Drawing Metro Maps Our Model Our Solution NP-Hardness Rectilinear vs. octilinear drawing Reduction from planar 3-SAT Outline of the Reduction x1 ∨ x3 ∨ x4 x1 ∨ x2 ∨ x3 x1 x2 x3 x4 x2 ∨ x3 ∨ x4 x1 ∨ x2 ∨ 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. ¨ Martin Nollenburg and Alexander Wolff 27 32 Drawing Metro Maps Our Model Our Solution NP-Hardness Rectilinear vs. octilinear drawing Reduction from planar 3-SAT Clause Gadget ¨ Martin Nollenburg and Alexander Wolff 28 32 Drawing Metro Maps Our Model Our Solution NP-Hardness Rectilinear vs. octilinear drawing Reduction from planar 3-SAT Clause Gadget ¨ Martin Nollenburg and Alexander Wolff 28 32 Drawing Metro Maps Our Model Our Solution NP-Hardness Rectilinear vs. octilinear drawing Reduction from planar 3-SAT Clause Gadget ¨ Martin Nollenburg and Alexander Wolff 28 32 Drawing Metro Maps Our Model Our Solution NP-Hardness Rectilinear vs. octilinear drawing Reduction from planar 3-SAT Clause Gadget ¨ Martin Nollenburg and Alexander Wolff 28 32 Drawing Metro Maps Our Model Our Solution NP-Hardness Rectilinear vs. octilinear drawing Reduction from planar 3-SAT Clause Gadget ¨ Martin Nollenburg and Alexander Wolff 28 32 Drawing Metro Maps Our Model Our Solution NP-Hardness Rectilinear vs. octilinear drawing Reduction from planar 3-SAT Clause Gadget ¨ Martin Nollenburg and Alexander Wolff 28 32 Drawing Metro Maps Our Model Our Solution NP-Hardness Rectilinear vs. octilinear drawing Reduction from planar 3-SAT Other applications O’REILLY 2003 OPEN SOURCE ROUTE MAP ® LINUX & KONSORTEN Perl Web Technologie & Gesellschaft Programmieren mit Perl DBI Beyond Contact SSH – Das umfassende Handbuch Programmieren von Grafiken mit Perl Writing Apache Modules with Perl and C Network Security with OpenSSL Linux Perl für SystemAdministration Technologie & Gesellschaft Peer-to-Peer sendmail Samba Mac OS X Writing GNU Emacs Extensions Programming Web Services with Perl Learning GNU Emacs PHP Kochbuch Mastering Perl/Tk Using csh & tcsh Practical MySQL – Programming PostgreSQL with GNU Einsatz & Programmierung Software Learning Perl on Win 32 Systems Einführung in Perl Mac OS X Programming Python Perl kurz & gut XML in a Nutshell Python Programming on Win32 Einführung in Python DocBook: The Definitive Guide Einführung in XML Perl & XML Jython Essentials Unix Web XML Python & XML Java™ Web, Graphics & Perl/Tk: Best of The Perl Journal XML Schema PHP kurz & gut www.oreilly.de Einführung in Perl für Bioinformatik Java™ & XML Linux – Wegweiser zur Installation & Konfiguration Linux Security Cookbook Learning Red Hat Linux Games, Diversions & Perl Culture: Best of The Perl Journal SAX2 Die Grübelei hat ein Ende! 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Formulated and implemented MIP. Our MIP can draw any kind of sketch “nicely”. Results comparable to manually designed maps. Reduced MIP size & runtime drastically. ¨ Martin Nollenburg and Alexander Wolff 32 32 Drawing Metro Maps Our Model Our Solution NP-Hardness Rectilinear vs. octilinear drawing Reduction from planar 3-SAT Summary (metro maps) M ETRO M AP L AYOUT is NP-hard. Formulated and implemented MIP. Our MIP can draw any kind of sketch “nicely”. Results comparable to manually designed maps. Reduced MIP size & runtime drastically. To Do rectangular stations multi-edges user interaction (e.g. fixing certain edge directions) ¨ Martin Nollenburg and Alexander Wolff 32 32 Drawing Metro Maps
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