[PPT] - Computing Stable Demers Cartograms GD 2019 September 17, 2019 PowerPoint Presentation

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Osnabr¨ uck University Markus Chimani TU Eindhoven Max Sondag Wouter Meulemans

Computing Stable Demers Cartograms

GD 2019 · September 17, 2019 TU Vienna University of Arizona Tampere University Soeren Nickel Martin N¨

llenburg

Stephen Kobourov Jaakko Peltonen Stephen Kobourov University of Arizona Stephen Kobourov Stephen Kobourov

[Decidedly not to scale]

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Soeren Nickel, Max Sondag, Wouter Meulemans, Markus Chimani, Stephen Kobourov, Jaakko Peltonen, and Martin N¨

llenburg · Computing Stable Demers Cartograms

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Cartograms

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Soeren Nickel, Max Sondag, Wouter Meulemans, Markus Chimani, Stephen Kobourov, Jaakko Peltonen, and Martin N¨

llenburg · Computing Stable Demers Cartograms

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Cartograms

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Soeren Nickel, Max Sondag, Wouter Meulemans, Markus Chimani, Stephen Kobourov, Jaakko Peltonen, and Martin N¨

llenburg · Computing Stable Demers Cartograms

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Cartograms

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Soeren Nickel, Max Sondag, Wouter Meulemans, Markus Chimani, Stephen Kobourov, Jaakko Peltonen, and Martin N¨

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Cartograms

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Soeren Nickel, Max Sondag, Wouter Meulemans, Markus Chimani, Stephen Kobourov, Jaakko Peltonen, and Martin N¨

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Cartograms

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Soeren Nickel, Max Sondag, Wouter Meulemans, Markus Chimani, Stephen Kobourov, Jaakko Peltonen, and Martin N¨

llenburg · Computing Stable Demers Cartograms

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Cartograms

Dorling Cartogram [Dorling, 1996]

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Soeren Nickel, Max Sondag, Wouter Meulemans, Markus Chimani, Stephen Kobourov, Jaakko Peltonen, and Martin N¨

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Cartograms

Regions: non-overlapping circles

Dorling Cartogram [Dorling, 1996]

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Soeren Nickel, Max Sondag, Wouter Meulemans, Markus Chimani, Stephen Kobourov, Jaakko Peltonen, and Martin N¨

llenburg · Computing Stable Demers Cartograms

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Cartograms

Regions: non-overlapping circles Size proportional to data

Dorling Cartogram [Dorling, 1996]

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Soeren Nickel, Max Sondag, Wouter Meulemans, Markus Chimani, Stephen Kobourov, Jaakko Peltonen, and Martin N¨

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Cartograms

Regions: non-overlapping circles Size proportional to data Contact if regions are neighbours Disk contact representation

Dorling Cartogram [Dorling, 1996]

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Soeren Nickel, Max Sondag, Wouter Meulemans, Markus Chimani, Stephen Kobourov, Jaakko Peltonen, and Martin N¨

llenburg · Computing Stable Demers Cartograms

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Cartograms

Size proportional to data Contact if regions are neighbours Regions: non-overlapping squares Square contact representation

Dorling Cartogram [Dorling, 1996]

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w

w i t h 1 % S q u a r e s ! ! !

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Soeren Nickel, Max Sondag, Wouter Meulemans, Markus Chimani, Stephen Kobourov, Jaakko Peltonen, and Martin N¨

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Cartograms

Demers Cartograms [Demers et al., 2002] Dorling Cartogram [Dorling, 1996]

N

w

w i t h 1 % S q u a r e s ! ! !

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Soeren Nickel, Max Sondag, Wouter Meulemans, Markus Chimani, Stephen Kobourov, Jaakko Peltonen, and Martin N¨

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Cartograms

Demers Cartograms [Demers et al., 2002] Dorling Cartogram [Dorling, 1996]

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w

w i t h 1 % S q u a r e s ! ! ! Advantages: Comparability Exact size of regions Dense packing

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Soeren Nickel, Max Sondag, Wouter Meulemans, Markus Chimani, Stephen Kobourov, Jaakko Peltonen, and Martin N¨

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Quality of Cartograms [Nusrat & Kobourov, 2016]

What is important for a “good cartogram”?

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Soeren Nickel, Max Sondag, Wouter Meulemans, Markus Chimani, Stephen Kobourov, Jaakko Peltonen, and Martin N¨

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Quality of Cartograms [Nusrat & Kobourov, 2016]

What is important for a “good cartogram”?

1. Cartographic accuracy:

Relative region sizes should be close to the data values

SLIDE 16

Soeren Nickel, Max Sondag, Wouter Meulemans, Markus Chimani, Stephen Kobourov, Jaakko Peltonen, and Martin N¨

llenburg · Computing Stable Demers Cartograms

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Quality of Cartograms [Nusrat & Kobourov, 2016]

What is important for a “good cartogram”?

1. Cartographic accuracy:

Relative region sizes should be close to the data values

SLIDE 17

Soeren Nickel, Max Sondag, Wouter Meulemans, Markus Chimani, Stephen Kobourov, Jaakko Peltonen, and Martin N¨

llenburg · Computing Stable Demers Cartograms

3/10

Quality of Cartograms [Nusrat & Kobourov, 2016]

What is important for a “good cartogram”?

1. Cartographic accuracy:
2. Shape deformation:

Relative region sizes should be close to the data values Each region should resemble its geographic shape

SLIDE 18

Soeren Nickel, Max Sondag, Wouter Meulemans, Markus Chimani, Stephen Kobourov, Jaakko Peltonen, and Martin N¨

llenburg · Computing Stable Demers Cartograms

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Quality of Cartograms [Nusrat & Kobourov, 2016]

What is important for a “good cartogram”?

1. Cartographic accuracy:

Egypt

Source: pappubahry.com/misc/rectangles/

2. Shape deformation:

Relative region sizes should be close to the data values Each region should resemble its geographic shape

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Soeren Nickel, Max Sondag, Wouter Meulemans, Markus Chimani, Stephen Kobourov, Jaakko Peltonen, and Martin N¨

llenburg · Computing Stable Demers Cartograms

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Quality of Cartograms [Nusrat & Kobourov, 2016]

What is important for a “good cartogram”?

1. Cartographic accuracy:

Egypt Italy

Source: pappubahry.com/misc/rectangles/

2. Shape deformation:

Relative region sizes should be close to the data values Each region should resemble its geographic shape

SLIDE 20

Soeren Nickel, Max Sondag, Wouter Meulemans, Markus Chimani, Stephen Kobourov, Jaakko Peltonen, and Martin N¨

llenburg · Computing Stable Demers Cartograms

3/10

Quality of Cartograms [Nusrat & Kobourov, 2016]

What is important for a “good cartogram”?

1. Cartographic accuracy:
2. Shape deformation:

Relative region sizes should be close to the data values Each region should resemble its geographic shape

SLIDE 21

Soeren Nickel, Max Sondag, Wouter Meulemans, Markus Chimani, Stephen Kobourov, Jaakko Peltonen, and Martin N¨

llenburg · Computing Stable Demers Cartograms

3/10

Quality of Cartograms [Nusrat & Kobourov, 2016]

What is important for a “good cartogram”?

1. Cartographic accuracy:
2. Shape deformation:
3. Preservation of relative directions:

Relative region sizes should be close to the data values Each region should resemble its geographic shape Spatial relations (north-south/east-west) should be maintained

SLIDE 22

Soeren Nickel, Max Sondag, Wouter Meulemans, Markus Chimani, Stephen Kobourov, Jaakko Peltonen, and Martin N¨

llenburg · Computing Stable Demers Cartograms

3/10

Quality of Cartograms [Nusrat & Kobourov, 2016]

What is important for a “good cartogram”?

1. Cartographic accuracy:
2. Shape deformation:
3. Preservation of relative directions:

Relative region sizes should be close to the data values Each region should resemble its geographic shape Spatial relations (north-south/east-west) should be maintained

IRL GBR NLD BEL FRA AUT DEU DNK CZE POL GBR DNK FRA NLD BEL DEU AUT CZE POL IRL IRL GBR NLD BEL FRA DNK DEU POL CZE AUT

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Soeren Nickel, Max Sondag, Wouter Meulemans, Markus Chimani, Stephen Kobourov, Jaakko Peltonen, and Martin N¨

llenburg · Computing Stable Demers Cartograms

3/10

Quality of Cartograms [Nusrat & Kobourov, 2016]

What is important for a “good cartogram”?

1. Cartographic accuracy:
2. Shape deformation:
3. Preservation of relative directions:

Relative region sizes should be close to the data values Each region should resemble its geographic shape Spatial relations (north-south/east-west) should be maintained

separation constraints

IRL GBR NLD BEL FRA AUT DEU DNK CZE POL GBR DNK FRA NLD BEL DEU AUT CZE POL IRL IRL GBR NLD BEL FRA DNK DEU POL CZE AUT

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Soeren Nickel, Max Sondag, Wouter Meulemans, Markus Chimani, Stephen Kobourov, Jaakko Peltonen, and Martin N¨

llenburg · Computing Stable Demers Cartograms

3/10

Quality of Cartograms [Nusrat & Kobourov, 2016]

What is important for a “good cartogram”?

1. Cartographic accuracy:
2. Shape deformation:
3. Preservation of relative directions:

Relative region sizes should be close to the data values Each region should resemble its geographic shape Spatial relations (north-south/east-west) should be maintained

separation constraints Next slide

IRL GBR NLD BEL FRA AUT DEU DNK CZE POL GBR DNK FRA NLD BEL DEU AUT CZE POL IRL IRL GBR NLD BEL FRA DNK DEU POL CZE AUT

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Soeren Nickel, Max Sondag, Wouter Meulemans, Markus Chimani, Stephen Kobourov, Jaakko Peltonen, and Martin N¨

llenburg · Computing Stable Demers Cartograms

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Quality of Cartograms [Nusrat & Kobourov, 2016]

What is important for a “good cartogram”?

1. Cartographic accuracy:
2. Shape deformation:
4. Topological accuracy:
3. Preservation of relative directions:

Relative region sizes should be close to the data values Each region should resemble its geographic shape Spatial relations (north-south/east-west) should be maintained Geographically adjacent regions should be adjacent in the cartogram, and vice versa

separation constraints Next slide

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Soeren Nickel, Max Sondag, Wouter Meulemans, Markus Chimani, Stephen Kobourov, Jaakko Peltonen, and Martin N¨

llenburg · Computing Stable Demers Cartograms

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Quality of Cartograms [Nusrat & Kobourov, 2016]

What is important for a “good cartogram”?

1. Cartographic accuracy:
2. Shape deformation:
4. Topological accuracy:
3. Preservation of relative directions:

Relative region sizes should be close to the data values Each region should resemble its geographic shape Spatial relations (north-south/east-west) should be maintained Geographically adjacent regions should be adjacent in the cartogram, and vice versa

separation constraints Next slide

MI WI MN IL IA IN OH KY SD NE ND WV MO WY MT PA VA CO TN NY KS UT OK NM ID AZ AR NV TX OR WA NC NJ MS CA MD AL CT GA LA MA VT RI DE NH SC FL ME MI WI MN IL IA IN OH KY SD NE ND WV MO WY MT PA VA CO TN NY KS UT OK NM ID AZ AR NV TX OR WA NC NJ MS CA MD AL CT GA LA MA VT RI DE NH SC FL ME

SLIDE 27

Soeren Nickel, Max Sondag, Wouter Meulemans, Markus Chimani, Stephen Kobourov, Jaakko Peltonen, and Martin N¨

llenburg · Computing Stable Demers Cartograms

3/10

Quality of Cartograms [Nusrat & Kobourov, 2016]

What is important for a “good cartogram”?

1. Cartographic accuracy:
2. Shape deformation:
4. Topological accuracy:
3. Preservation of relative directions:

Relative region sizes should be close to the data values Each region should resemble its geographic shape Spatial relations (north-south/east-west) should be maintained Geographically adjacent regions should be adjacent in the cartogram, and vice versa

separation constraints main optimization goal! Next slide

MI WI MN IL IA IN OH KY SD NE ND WV MO WY MT PA VA CO TN NY KS UT OK NM ID AZ AR NV TX OR WA NC NJ MS CA MD AL CT GA LA MA VT RI DE NH SC FL ME MI WI MN IL IA IN OH KY SD NE ND WV MO WY MT PA VA CO TN NY KS UT OK NM ID AZ AR NV TX OR WA NC NJ MS CA MD AL CT GA LA MA VT RI DE NH SC FL ME

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Soeren Nickel, Max Sondag, Wouter Meulemans, Markus Chimani, Stephen Kobourov, Jaakko Peltonen, and Martin N¨

llenburg · Computing Stable Demers Cartograms

3/10

Quality of Cartograms [Nusrat & Kobourov, 2016]

What is important for a “good cartogram”?

1. Cartographic accuracy:
2. Shape deformation:
4. Topological accuracy:
3. Preservation of relative directions:
5. Spatial deformation:

Relative region sizes should be close to the data values Each region should resemble its geographic shape Spatial relations (north-south/east-west) should be maintained Geographically adjacent regions should be adjacent in the cartogram, and vice versa Regions should be placed close to their geographic location

separation constraints main optimization goal! Next slide

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Soeren Nickel, Max Sondag, Wouter Meulemans, Markus Chimani, Stephen Kobourov, Jaakko Peltonen, and Martin N¨

llenburg · Computing Stable Demers Cartograms

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Quality of Cartograms [Nusrat & Kobourov, 2016]

What is important for a “good cartogram”?

1. Cartographic accuracy:
2. Shape deformation:
4. Topological accuracy:
3. Preservation of relative directions:
5. Spatial deformation:

Relative region sizes should be close to the data values Each region should resemble its geographic shape Spatial relations (north-south/east-west) should be maintained Geographically adjacent regions should be adjacent in the cartogram, and vice versa Regions should be placed close to their geographic location

separation constraints main optimization goal! alternative optimization goal! Next slide

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Soeren Nickel, Max Sondag, Wouter Meulemans, Markus Chimani, Stephen Kobourov, Jaakko Peltonen, and Martin N¨

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Formal Setting

Given: a set of regions R with their adjacencies and the cen- troids, the size of every region r ∈ R and two sets H, V of

rdered region pairs (separation constraints)

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Soeren Nickel, Max Sondag, Wouter Meulemans, Markus Chimani, Stephen Kobourov, Jaakko Peltonen, and Martin N¨

llenburg · Computing Stable Demers Cartograms

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Formal Setting

r′ must be above r (r, r′) ∈ V Given: a set of regions R with their adjacencies and the cen- troids, the size of every region r ∈ R and two sets H, V of

rdered region pairs (separation constraints)

r′ must be to the right of r (r, r′) ∈ H

SLIDE 32

Soeren Nickel, Max Sondag, Wouter Meulemans, Markus Chimani, Stephen Kobourov, Jaakko Peltonen, and Martin N¨

llenburg · Computing Stable Demers Cartograms

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Formal Setting

Place regions according to constraints r′ must be above r (r, r′) ∈ V r Given: a set of regions R with their adjacencies and the cen- troids, the size of every region r ∈ R and two sets H, V of

rdered region pairs (separation constraints)

r′ must be to the right of r (r, r′) ∈ H

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Soeren Nickel, Max Sondag, Wouter Meulemans, Markus Chimani, Stephen Kobourov, Jaakko Peltonen, and Martin N¨

llenburg · Computing Stable Demers Cartograms

4/10

Formal Setting

Place regions according to constraints r′ must be above r r′ must be placed right of r (r, r′) ∈ V r r′ Given: a set of regions R with their adjacencies and the cen- troids, the size of every region r ∈ R and two sets H, V of

rdered region pairs (separation constraints)

r′ must be to the right of r (r, r′) ∈ H

SLIDE 34

Soeren Nickel, Max Sondag, Wouter Meulemans, Markus Chimani, Stephen Kobourov, Jaakko Peltonen, and Martin N¨

llenburg · Computing Stable Demers Cartograms

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Formal Setting

Place regions according to constraints r′ must be above r r′ must be placed right of r Adjacent regions should touch. This might have to be violated. (r, r′) ∈ V r r′ Given: a set of regions R with their adjacencies and the cen- troids, the size of every region r ∈ R and two sets H, V of

rdered region pairs (separation constraints)

r′ must be to the right of r (r, r′) ∈ H

SLIDE 35

Soeren Nickel, Max Sondag, Wouter Meulemans, Markus Chimani, Stephen Kobourov, Jaakko Peltonen, and Martin N¨

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Approach to Compute Cartograms

We use a linear program to create optimized cartograms

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Soeren Nickel, Max Sondag, Wouter Meulemans, Markus Chimani, Stephen Kobourov, Jaakko Peltonen, and Martin N¨

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Approach to Compute Cartograms

Formulation of desired qualities and required properties as linear (in-)equalities We use a linear program to create optimized cartograms

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Soeren Nickel, Max Sondag, Wouter Meulemans, Markus Chimani, Stephen Kobourov, Jaakko Peltonen, and Martin N¨

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Approach to Compute Cartograms

Formulation of desired qualities and required properties as linear (in-)equalities Existing solvers for linear programs (IBM ILOG cplex) Poly-Time! Why use an LP? We use a linear program to create optimized cartograms

SLIDE 38

Soeren Nickel, Max Sondag, Wouter Meulemans, Markus Chimani, Stephen Kobourov, Jaakko Peltonen, and Martin N¨

llenburg · Computing Stable Demers Cartograms

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Approach to Compute Cartograms

Formulation of desired qualities and required properties as linear (in-)equalities Existing solvers for linear programs (IBM ILOG cplex) Poly-Time! Why use an LP? What should the LP do? Encode placement of regions Fulfill separation constraints Minimize objective function (topological error or spatial deformation) We use a linear program to create optimized cartograms Existing solvers for linear programs (IBM ILOG cplex) Poly-Time!

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Soeren Nickel, Max Sondag, Wouter Meulemans, Markus Chimani, Stephen Kobourov, Jaakko Peltonen, and Martin N¨

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Approach to Compute Cartograms

r r′

Introduce centroid position variables:

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Soeren Nickel, Max Sondag, Wouter Meulemans, Markus Chimani, Stephen Kobourov, Jaakko Peltonen, and Martin N¨

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Approach to Compute Cartograms

r r′

P Introduce centroid position variables: xr, yr

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Soeren Nickel, Max Sondag, Wouter Meulemans, Markus Chimani, Stephen Kobourov, Jaakko Peltonen, and Martin N¨

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Approach to Compute Cartograms

r r′

P Q Introduce centroid position variables: xr′, yr′ xr, yr

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Soeren Nickel, Max Sondag, Wouter Meulemans, Markus Chimani, Stephen Kobourov, Jaakko Peltonen, and Martin N¨

llenburg · Computing Stable Demers Cartograms

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Approach to Compute Cartograms

r r′

P Q Introduce centroid position variables: xr′, yr′ Guarantee non-overlap: xr, yr

r r′

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Soeren Nickel, Max Sondag, Wouter Meulemans, Markus Chimani, Stephen Kobourov, Jaakko Peltonen, and Martin N¨

llenburg · Computing Stable Demers Cartograms

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Approach to Compute Cartograms

r r′

P Q Introduce centroid position variables: xr′, yr′ Guarantee non-overlap: xr, yr xr′ − xr ≥ w+w′

2

r r′

w w′

SLIDE 44

Soeren Nickel, Max Sondag, Wouter Meulemans, Markus Chimani, Stephen Kobourov, Jaakko Peltonen, and Martin N¨

llenburg · Computing Stable Demers Cartograms

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Approach to Compute Cartograms

r r′

P Q Introduce centroid position variables: xr′, yr′ Guarantee non-overlap: xr, yr xr′ − xr ≥ w+w′

2

yr′ − yr ≥ w+w′

2

r r′

w w′ Choice dependent on separation constraint

SLIDE 45

Soeren Nickel, Max Sondag, Wouter Meulemans, Markus Chimani, Stephen Kobourov, Jaakko Peltonen, and Martin N¨

llenburg · Computing Stable Demers Cartograms

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Approach to Compute Cartograms

r r′

P Q Introduce centroid position variables: xr′, yr′ Guarantee non-overlap: xr, yr xr′ − xr ≥ w+w′

2

r r′

ε +ε

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Soeren Nickel, Max Sondag, Wouter Meulemans, Markus Chimani, Stephen Kobourov, Jaakko Peltonen, and Martin N¨

llenburg · Computing Stable Demers Cartograms

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Approach to Compute Cartograms

r r′

P Q Introduce centroid position variables: xr′, yr′ Guarantee non-overlap: xr, yr xr′ − xr ≥ w+w′

2

r r′

ε +ε Minimize L1-distance between adjacent regions:

4. Topological accuracy:

Geographically adjacent regions should be adjacent in the cartogram

main optimization goal!

SLIDE 47

Soeren Nickel, Max Sondag, Wouter Meulemans, Markus Chimani, Stephen Kobourov, Jaakko Peltonen, and Martin N¨

llenburg · Computing Stable Demers Cartograms

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Approach to Compute Cartograms

r r′

P Q Introduce centroid position variables: xr′, yr′ Guarantee non-overlap: xr, yr xr′ − xr ≥ w+w′

2

r r′

ε +ε Minimize L1-distance between adjacent regions:

min

{r,r′}∈E

hr,r′ + vr,r′

SLIDE 48

Soeren Nickel, Max Sondag, Wouter Meulemans, Markus Chimani, Stephen Kobourov, Jaakko Peltonen, and Martin N¨

llenburg · Computing Stable Demers Cartograms

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Approach to Compute Cartograms

r r′

P Q Introduce centroid position variables: xr′, yr′ Guarantee non-overlap: xr, yr xr′ − xr ≥ w+w′

2

r r′

ε +ε Minimize L1-distance between adjacent regions: hr,r′ ≥ (xr′ − xr) − w+w′

2

min

{r,r′}∈E

hr,r′ + vr,r′

SLIDE 49

Soeren Nickel, Max Sondag, Wouter Meulemans, Markus Chimani, Stephen Kobourov, Jaakko Peltonen, and Martin N¨

llenburg · Computing Stable Demers Cartograms

5/10

Approach to Compute Cartograms

r r′

P Q Introduce centroid position variables: xr′, yr′ Guarantee non-overlap: xr, yr xr′ − xr ≥ w+w′

2

r r′

ε +ε Minimize L1-distance between adjacent regions: hr,r′ ≥ (xr′ − xr) − w+w′

2

vr,r′ ≥ max{(yr − yr′) − w+w′

2

, (yr′ − yr) − w+w′

2

)} min

{r,r′}∈E

hr,r′ + vr,r′ r r r′ r′

SLIDE 50

Soeren Nickel, Max Sondag, Wouter Meulemans, Markus Chimani, Stephen Kobourov, Jaakko Peltonen, and Martin N¨

llenburg · Computing Stable Demers Cartograms

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Approach to Compute Cartograms

r r′

P Q Introduce centroid position variables: xr′, yr′ Guarantee non-overlap: xr, yr xr′ − xr ≥ w+w′

2

r r′

ε +ε Minimize L1-distance between adjacent regions: hr,r′ ≥ (xr′ − xr) − w+w′

2

vr,r′ ≥ max{(yr − yr′) − w+w′

2

, (yr′ − yr) − w+w′

2

)} vr,r′, hr,r′ ≥ 0

r r′

min

{r,r′}∈E

hr,r′ + vr,r′

SLIDE 51

Soeren Nickel, Max Sondag, Wouter Meulemans, Markus Chimani, Stephen Kobourov, Jaakko Peltonen, and Martin N¨

llenburg · Computing Stable Demers Cartograms

5/10

Approach to Compute Cartograms

r r′

P Q Introduce centroid position variables: xr′, yr′ Guarantee non-overlap: xr, yr xr′ − xr ≥ w+w′

2

r r′

ε +ε Minimize L1-distance between adjacent regions: hr,r′ ≥ (xr′ − xr) − w+w′

2

vr,r′ ≥ max{(yr − yr′) − w+w′

2

, (yr′ − yr) − w+w′

2

)} vr,r′, hr,r′ ≥ 0 min

{r,r′}∈E

hr,r′ + vr,r′ If r, r′ are touching then hr,r′ + vr,r′ = 0

SLIDE 52

Soeren Nickel, Max Sondag, Wouter Meulemans, Markus Chimani, Stephen Kobourov, Jaakko Peltonen, and Martin N¨

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Linear Program—Other Constraints

Keep the slope between regions

δslope

3. Preservation of relative directions

SLIDE 53

Soeren Nickel, Max Sondag, Wouter Meulemans, Markus Chimani, Stephen Kobourov, Jaakko Peltonen, and Martin N¨

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Linear Program—Other Constraints

Keep the slope between regions Minimize distance to input position

δslope

DEU FRA

3. Preservation of relative directions
5. Spatial deformation

SLIDE 54

Soeren Nickel, Max Sondag, Wouter Meulemans, Markus Chimani, Stephen Kobourov, Jaakko Peltonen, and Martin N¨

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Linear Program—Other Constraints

Keep the slope between regions Minimize distance to input position An ILP is able minimize the actual number of lost adjacencies significantly higher runtimes / ILP is NP-hard

δslope

DEU FRA

3. Preservation of relative directions
5. Spatial deformation

SLIDE 55

Soeren Nickel, Max Sondag, Wouter Meulemans, Markus Chimani, Stephen Kobourov, Jaakko Peltonen, and Martin N¨

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Computing Cartogram Sequences

Data over time / different data on the same map

SLIDE 56

Soeren Nickel, Max Sondag, Wouter Meulemans, Markus Chimani, Stephen Kobourov, Jaakko Peltonen, and Martin N¨

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Computing Cartogram Sequences

Data over time / different data on the same map

SLIDE 57

Soeren Nickel, Max Sondag, Wouter Meulemans, Markus Chimani, Stephen Kobourov, Jaakko Peltonen, and Martin N¨

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Computing Cartogram Sequences

Data over time / different data on the same map Small changes in the data Small changes in the cartogram

SLIDE 58

Soeren Nickel, Max Sondag, Wouter Meulemans, Markus Chimani, Stephen Kobourov, Jaakko Peltonen, and Martin N¨

llenburg · Computing Stable Demers Cartograms

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Computing Cartogram Sequences

Small changes in the data Small changes in the cartogram

Stability

SLIDE 59

Soeren Nickel, Max Sondag, Wouter Meulemans, Markus Chimani, Stephen Kobourov, Jaakko Peltonen, and Martin N¨

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Computing Cartogram Sequences

Small changes in the data Small changes in the cartogram

Stability

How do we achieve stability?

SLIDE 60

Soeren Nickel, Max Sondag, Wouter Meulemans, Markus Chimani, Stephen Kobourov, Jaakko Peltonen, and Martin N¨

llenburg · Computing Stable Demers Cartograms

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Computing Cartogram Sequences

Small changes in the data Small changes in the cartogram

Stability

Cartogram A Cartogram B How do we achieve stability?

SLIDE 61

Soeren Nickel, Max Sondag, Wouter Meulemans, Markus Chimani, Stephen Kobourov, Jaakko Peltonen, and Martin N¨

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Computing Cartogram Sequences

Overlay A&B Small changes in the data Small changes in the cartogram

Stability

Cartogram A Cartogram B How do we achieve stability?

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Computing Cartogram Sequences

Overlay A&B Small changes in the data Small changes in the cartogram

Stability

Cartogram A Cartogram B How do we achieve stability?

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Computing Cartogram Sequences

Overlay A&B Small changes in the data Small changes in the cartogram

Stability

Cartogram A Cartogram B How do we achieve stability? Minimize distance of one region to itself in different cartograms

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Computing Cartogram Sequences

Small changes in the data Small changes in the cartogram

Stability Stability Models

How do we achieve stability? Minimize distance of one region to itself in different cartograms

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Computing Cartogram Sequences

Small changes in the data Small changes in the cartogram

Stability Stability Models

How do we achieve stability? Minimize distance of one region to itself in different cartograms

2 3 4 5 6 7 1

Star

2 3 4 5 6 7 1

Star

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Soeren Nickel, Max Sondag, Wouter Meulemans, Markus Chimani, Stephen Kobourov, Jaakko Peltonen, and Martin N¨

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Computing Cartogram Sequences

Small changes in the data Small changes in the cartogram

Stability Stability Models

How do we achieve stability? Minimize distance of one region to itself in different cartograms

2 3 4 5 6 7 1 1 2 3 4 5 6 7

Star Complete

2 3 4 5 6 7 1 1 2 3 4 5 6 7

Star Complete

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Soeren Nickel, Max Sondag, Wouter Meulemans, Markus Chimani, Stephen Kobourov, Jaakko Peltonen, and Martin N¨

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Computing Cartogram Sequences

Small changes in the data Small changes in the cartogram

Stability Stability Models

How do we achieve stability? Minimize distance of one region to itself in different cartograms

2 3 4 5 6 7 1 1 2 3 4 5 6 7 1 2 3 4 5 6 7 1 2 3 4 5 6 7

Star Complete Successive Iterative

2 3 4 5 6 7 1 1 2 3 4 5 6 7 1 2 3 4 5 6 7 1 2 3 4 5 6 7

Star Complete Successive Iterative

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Experiments

Optimization Constraints Force-Directed (FRC) (FRC) (FRC) (FRC) (FRC) (FRC) (FRC) (FRC) (FRC) (FRC) (FRC) (FRC) (FRC) (FRC) (FRC) (FRC) (FRC) ILP (CNT) (CNT) (CNT) (CNT) (CNT) (CNT) (CNT) (CNT) (CNT) (CNT) (CNT) (CNT) (CNT) (CNT) (CNT) (CNT) (CNT) Distance & Slope (TOP) (TOP) (TOP) (TOP) (TOP) (TOP) (TOP) (TOP) (TOP) (TOP) (TOP) (TOP) (TOP) (TOP) (TOP) (TOP) (TOP) Origin Displacement (ORG) (ORG) (ORG) (ORG) (ORG) (ORG) (ORG) (ORG) (ORG) (ORG) (ORG) (ORG) (ORG) (ORG) (ORG) (ORG) (ORG) Topology Topology Topology Topology Topology Topology Topology Topology Topology Topology Topology Topology Topology Topology Topology Topology Topology Origin Origin Origin Origin Origin Origin Origin Origin Origin Origin Origin Origin Origin Origin Origin Origin Origin

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Experiments

Optimization Constraints

0.0 0.2 0.4 0.6 0.8 1.0

# lost adjacencies

Force-Directed (FRC) (FRC) (FRC) (FRC) (FRC) (FRC) (FRC) (FRC) (FRC) (FRC) (FRC) (FRC) (FRC) (FRC) (FRC) (FRC) (FRC) ILP (CNT) (CNT) (CNT) (CNT) (CNT) (CNT) (CNT) (CNT) (CNT) (CNT) (CNT) (CNT) (CNT) (CNT) (CNT) (CNT) (CNT) Distance & Slope (TOP) (TOP) (TOP) (TOP) (TOP) (TOP) (TOP) (TOP) (TOP) (TOP) (TOP) (TOP) (TOP) (TOP) (TOP) (TOP) (TOP) Origin Displacement (ORG) (ORG) (ORG) (ORG) (ORG) (ORG) (ORG) (ORG) (ORG) (ORG) (ORG) (ORG) (ORG) (ORG) (ORG) (ORG) (ORG)

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Experiments

Optimization Constraints

0.0 0.2 0.4 0.6 0.8 1.0 0.3 0.2 0.1 0.0

# lost adjacencies MDIS ˆ = distance to original location

Force-Directed (FRC) (FRC) (FRC) (FRC) (FRC) (FRC) (FRC) (FRC) (FRC) (FRC) (FRC) (FRC) (FRC) (FRC) (FRC) (FRC) (FRC) ILP (CNT) (CNT) (CNT) (CNT) (CNT) (CNT) (CNT) (CNT) (CNT) (CNT) (CNT) (CNT) (CNT) (CNT) (CNT) (CNT) (CNT) Distance & Slope (TOP) (TOP) (TOP) (TOP) (TOP) (TOP) (TOP) (TOP) (TOP) (TOP) (TOP) (TOP) (TOP) (TOP) (TOP) (TOP) (TOP) Origin Displacement (ORG) (ORG) (ORG) (ORG) (ORG) (ORG) (ORG) (ORG) (ORG) (ORG) (ORG) (ORG) (ORG) (ORG) (ORG) (ORG) (ORG) MDIS

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Experiments

Optimization Constraints

0.0 0.2 0.4 0.6 0.8 1.0 0.3 0.2 0.1 0.0

# lost adjacencies MDIS ˆ = distance to original location SDIS ˆ = distance to position of same region in different Cartograms

Force-Directed (FRC) (FRC) (FRC) (FRC) (FRC) (FRC) (FRC) (FRC) (FRC) (FRC) (FRC) (FRC) (FRC) (FRC) (FRC) (FRC) (FRC) ILP (CNT) (CNT) (CNT) (CNT) (CNT) (CNT) (CNT) (CNT) (CNT) (CNT) (CNT) (CNT) (CNT) (CNT) (CNT) (CNT) (CNT) Distance & Slope (TOP) (TOP) (TOP) (TOP) (TOP) (TOP) (TOP) (TOP) (TOP) (TOP) (TOP) (TOP) (TOP) (TOP) (TOP) (TOP) (TOP) Origin Displacement (ORG) (ORG) (ORG) (ORG) (ORG) (ORG) (ORG) (ORG) (ORG) (ORG) (ORG) (ORG) (ORG) (ORG) (ORG) (ORG) (ORG) SDIS MDIS

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Experiments

Optimization Constraints

0.0 0.2 0.4 0.6 0.8 1.0

MREL

0.3 0.2 0.1 0.0

# lost adjacencies MREL ˆ = position of regions relative to input MDIS ˆ = distance to original location SDIS ˆ = distance to position of same region in different Cartograms

Force-Directed (FRC) (FRC) (FRC) (FRC) (FRC) (FRC) (FRC) (FRC) (FRC) (FRC) (FRC) (FRC) (FRC) (FRC) (FRC) (FRC) (FRC) ILP (CNT) (CNT) (CNT) (CNT) (CNT) (CNT) (CNT) (CNT) (CNT) (CNT) (CNT) (CNT) (CNT) (CNT) (CNT) (CNT) (CNT) Distance & Slope (TOP) (TOP) (TOP) (TOP) (TOP) (TOP) (TOP) (TOP) (TOP) (TOP) (TOP) (TOP) (TOP) (TOP) (TOP) (TOP) (TOP) Origin Displacement (ORG) (ORG) (ORG) (ORG) (ORG) (ORG) (ORG) (ORG) (ORG) (ORG) (ORG) (ORG) (ORG) (ORG) (ORG) (ORG) (ORG) SDIS MDIS

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Experiments

Optimization Constraints

0.0 0.2 0.4 0.6 0.8 1.0

MREL

0.3 0.2 0.1 0.0

# lost adjacencies MREL ˆ = position of regions relative to input MDIS ˆ = distance to original location SDIS ˆ = distance to position of same region in different Cartograms SREL ˆ = position of regions relative to other cartograms

Force-Directed (FRC) (FRC) (FRC) (FRC) (FRC) (FRC) (FRC) (FRC) (FRC) (FRC) (FRC) (FRC) (FRC) (FRC) (FRC) (FRC) (FRC) ILP (CNT) (CNT) (CNT) (CNT) (CNT) (CNT) (CNT) (CNT) (CNT) (CNT) (CNT) (CNT) (CNT) (CNT) (CNT) (CNT) (CNT) Distance & Slope (TOP) (TOP) (TOP) (TOP) (TOP) (TOP) (TOP) (TOP) (TOP) (TOP) (TOP) (TOP) (TOP) (TOP) (TOP) (TOP) (TOP) Origin Displacement (ORG) (ORG) (ORG) (ORG) (ORG) (ORG) (ORG) (ORG) (ORG) (ORG) (ORG) (ORG) (ORG) (ORG) (ORG) (ORG) (ORG) SREL SDIS MDIS

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Incoming Example

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Incoming Example

In case of emergency

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Conclusion

Added complexity of the complete stability model does not seem to pay

ff compared to the successive model

General trade-off between the topological error and other metrics: TOP strikes a sensible balance ILP minimizes the topological error ORG scores high on all metrics except topological error Lost adjacencies can be visualized by low complexity orthogonal line Ensuring orthogonal separation constraints:

utperforms a basic force-directed approach