Euclidean Steiner Forest Colored Non-Crossing · Philipp Kindermann LG Theoretische Informatik...

Colored Non-Crossing

Euclidean Steiner Forest

Philipp KindermannLG Theoretische Informatik

FernUniversitat in Hagen

Joint work withSergey Bereg, Krzysztof Fleszar, Sergey Pupyrev,

Joachim Spoerhase & Alexander Wolff

Colored Steiner Forest

Colored Steiner Forestc©Google Maps


[Icon of Me So Ramen by Moxy Games, LLC]


[Icon of Me So Ramen by Moxy Games, LLC] [Aha-Soft, via seaicons.com]


[Icon of Me So Ramen by Moxy Games, LLC] [Aha-Soft, via seaicons.com] c©Leeners, LLC

Colored Steiner ForestEuler diagrams [Simonetto Auber Archambault, CGF’09]c©Google Maps


Colored Steiner ForestBubbleSets [Collins Penn Carpendale, TVCG’09]c©Google Maps


Colored Steiner ForestLineSets [Alper Riche Ramos Czerwinski, TVCG’11]c©Google Maps


Colored Steiner ForestKelpFusion [Meulemans Riche Speckmann Alper Dwyer, TVCG’13]c©Google Maps


Colored Steiner ForestGMap (Graph-to-Map) [Hu Gansner Kobourov, CGA’10]c©Google Maps


Colored Steiner Foresta better solution [Efrat Hu Kobourov Pupyrev, GD’14]c©Google Maps


Colored Steiner Foresta better solution

all regions arecontiguous anddisjoint

[Efrat Hu Kobourov Pupyrev, GD’14]c©Google Maps



n points, k colors


n points, k colors

→ k Steiner Trees


n points, k colors

→ k Steiner Treesunion planar

Bad Examples

Known Results

1-CESF (= Euclidean Steiner Tree)

Known Results

1-CESF (= Euclidean Steiner Tree)• is NP-hard [Garey Johnson, 1979]

Known Results

1-CESF (= Euclidean Steiner Tree)• is NP-hard [Garey Johnson, 1979]• admits a PTAS [Arora, JACM’98][Mitchell, SICOMP’99]

Known Results


n/2-CESF (= Euclidean Matching)

Known Results


n/2-CESF (= Euclidean Matching)• is NP-hard [Bastert Fekete, TR’98]

Known Results


n/2-CESF (= Euclidean Matching)• is NP-hard [Bastert Fekete, TR’98]• O(n log

√n)-approx. [Chan Hoffmann Kiazyk Lubiw, GD’13]

Known Results




k-CESF

Known Results




k-CESF• has a kρ-approximation [Efrat Hu Kobourov Pupyrev, GD’14]

Known Results





Steiner ratio

Known Results





Steiner ratio

ρ ≤ 1.21 [Chung Graham, ANYAS’85]

2-CESF

Rounding to the Grid


• L0 diameter of smallestbounding box• 3√

2n/ε ≤ L ≤ 6√

2n/ε• (L× L)-grid• granularity L0/L

L0



2n/ε ≤ L ≤ 6√

2n/ε• (L× L)-grid• granularity L0/L



2n/ε ≤ L ≤ 6√

2n/ε• (L× L)-grid• granularity L0/L 1


• 3√

2n/ε ≤ L ≤ 6√

2n/ε• (L× L)-grid• granularity 1


→ (even,even)

→ (odd,odd)

• 3√

2n/ε ≤ L ≤ 6√


Going Back

• 3√

2n/ε ≤ L ≤ 6√


Going Back

≤√

2

• 3√

2n/ε ≤ L ≤ 6√


Going Back

≤√

2

≤ 2√

2

• 3√

2n/ε ≤ L ≤ 6√


Going Back

≤√

2

≤ 2√

2

• 3√

2n/ε ≤ L ≤ 6√


in total ≤ 3√

2n ≤ εL ≤ εOPT

Going Back

≤√

2

≤ 2√

2

• 3√

2n/ε ≤ L ≤ 6√


in total ≤ 3√

2n ≤ εL ≤ εOPT

2-CESF instance I → rounded instance I ∗ → solution LI

|LI | ≤ (1 + ε)OPTI∗ ≤ (1 + ε)2OPTI

Quadtree Placement

Quadtree Placement

level 0

Quadtree Placement

level 0

level 1

Quadtree Placement

. . . . . .

level 0

level 1

level 2. . .

Quadtree Placement

. . . . . .

level 0

level 1

level 2. . .

...

level log L

......

...

Quadtree Placement

• m = 4 log(L)/ε

Quadtree Placement

• m = 4 log(L)/ε

• portals on level-i-line withdistance L/(2im)

m portals

m portals

Quadtree Placement

• m = 4 log(L)/ε


m portals

m portals• level-i-square has at most

4m portals on its margin

Quadtree Placement

• m = 4 log(L)/ε


• level-i-square has at most4m portals on its margin

• place origin uniformly atrandom

Quadtree Placement

• m = 4 log(L)/ε




portal-respecting solution:crosses grid lines only at portals

Quadtree Placement

• m = 4 log(L)/ε





line ` crosses drawing t(`) times;

expected length increase: ≤ εt(`)

4

Quadtree Placement

• m = 4 log(L)/ε





line ` crosses drawing t(`) times;

expected length increase: ≤ εt(`)

4

2-CESF instance I → portal-respecting solution L|L| ≤ (1 + ε)3OPTI

3-Light Solution

3-light solution: each portal is crossed at most 3 times

3-Light Solution


width= 0

3-Light Solution


width= 0

2-CESF instance I → portal-respecting 3-light solution L∗|L∗| ≤ (1 + ε)3OPTI

3-Light Solution


width= 0

2-CESF instance I → portal-respecting 3-light solution L∗|L∗| ≤ (1 + ε)3OPTI ≤ (1 + ε′)OPTI

Putting Things Together

Use a dynamic program!


Use a dynamic program! A subproblem consists of:


Use a dynamic program! A subproblem consists of:• a square of the quadtree


Use a dynamic program! A subproblem consists of:• a square of the quadtree• up to three red and blue points on each portal


Use a dynamic program! A subproblem consists of:• a square of the quadtree• up to three red and blue points on each portal• non-crossing partition of the points into sets of same color



crossing non-crossing




O(n2)




O(n2) 2O(log n/ε) = nO(1/ε)





CO(log n/ε) = nO(1/ε)



Base case: unit square





Base case: unit square• portals (and points) only in corners





Base case: unit square• portals (and points) only in corners• solve with PTAS for EST






Composite squares:






Composite squares:• divide into squares (acc. to quadtree)






Composite squares:• divide into squares (acc. to quadtree)• solve each combination of nO(1/ε)

compatible subproblems






Composite squares:• divide into squares (acc. to quadtree)• solve each combination of nO(1/ε)

compatible subproblems

2-CESF admits a PTAS.



3-CESF

3-CESF

3-CESF admits a (5/3 + ε)-approximation algorithm.

k-CESF

k-CESF

• split into 2 groups

k-CESF

• split into 2 groups• use PTAS for the groups

k-CESF

• split into 2 groups• use PTAS for the groups• Construct trees fromthis “super-tree”

k-CESF

• split into 2 groups• use PTAS for the groups• Construct trees fromthis “super-tree”

k-CESF admits an

• (k + ε)-approximation algorithm is k is odd• (k − 1 + ε)-approximation algorithm is k is even

Euclidean Steiner Forest Colored Non-Crossing · Philipp Kindermann LG Theoretische Informatik...

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