euclidean steiner forest colored non-crossing · philipp kindermann lg theoretische informatik...
TRANSCRIPT
![Page 1: Euclidean Steiner Forest Colored Non-Crossing · Philipp Kindermann LG Theoretische Informatik FernUniversit at in Hagen Joint work with Sergey Bereg, Krzysztof Fleszar, Sergey Pupyrev,](https://reader033.vdokument.com/reader033/viewer/2022060704/60709f9f2bbd466d621802c1/html5/thumbnails/1.jpg)
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
![Page 2: Euclidean Steiner Forest Colored Non-Crossing · Philipp Kindermann LG Theoretische Informatik FernUniversit at in Hagen Joint work with Sergey Bereg, Krzysztof Fleszar, Sergey Pupyrev,](https://reader033.vdokument.com/reader033/viewer/2022060704/60709f9f2bbd466d621802c1/html5/thumbnails/2.jpg)
Colored Steiner Forest
![Page 3: Euclidean Steiner Forest Colored Non-Crossing · Philipp Kindermann LG Theoretische Informatik FernUniversit at in Hagen Joint work with Sergey Bereg, Krzysztof Fleszar, Sergey Pupyrev,](https://reader033.vdokument.com/reader033/viewer/2022060704/60709f9f2bbd466d621802c1/html5/thumbnails/3.jpg)
Colored Steiner Forestc©Google Maps
![Page 4: Euclidean Steiner Forest Colored Non-Crossing · Philipp Kindermann LG Theoretische Informatik FernUniversit at in Hagen Joint work with Sergey Bereg, Krzysztof Fleszar, Sergey Pupyrev,](https://reader033.vdokument.com/reader033/viewer/2022060704/60709f9f2bbd466d621802c1/html5/thumbnails/4.jpg)
Colored Steiner Forestc©Google Maps
[Icon of Me So Ramen by Moxy Games, LLC]
![Page 5: Euclidean Steiner Forest Colored Non-Crossing · Philipp Kindermann LG Theoretische Informatik FernUniversit at in Hagen Joint work with Sergey Bereg, Krzysztof Fleszar, Sergey Pupyrev,](https://reader033.vdokument.com/reader033/viewer/2022060704/60709f9f2bbd466d621802c1/html5/thumbnails/5.jpg)
Colored Steiner Forestc©Google Maps
[Icon of Me So Ramen by Moxy Games, LLC] [Aha-Soft, via seaicons.com]
![Page 6: Euclidean Steiner Forest Colored Non-Crossing · Philipp Kindermann LG Theoretische Informatik FernUniversit at in Hagen Joint work with Sergey Bereg, Krzysztof Fleszar, Sergey Pupyrev,](https://reader033.vdokument.com/reader033/viewer/2022060704/60709f9f2bbd466d621802c1/html5/thumbnails/6.jpg)
Colored Steiner Forestc©Google Maps
[Icon of Me So Ramen by Moxy Games, LLC] [Aha-Soft, via seaicons.com] c©Leeners, LLC
![Page 7: Euclidean Steiner Forest Colored Non-Crossing · Philipp Kindermann LG Theoretische Informatik FernUniversit at in Hagen Joint work with Sergey Bereg, Krzysztof Fleszar, Sergey Pupyrev,](https://reader033.vdokument.com/reader033/viewer/2022060704/60709f9f2bbd466d621802c1/html5/thumbnails/7.jpg)
Colored Steiner ForestEuler diagrams [Simonetto Auber Archambault, CGF’09]c©Google Maps
[Icon of Me So Ramen by Moxy Games, LLC] [Aha-Soft, via seaicons.com] c©Leeners, LLC
![Page 8: Euclidean Steiner Forest Colored Non-Crossing · Philipp Kindermann LG Theoretische Informatik FernUniversit at in Hagen Joint work with Sergey Bereg, Krzysztof Fleszar, Sergey Pupyrev,](https://reader033.vdokument.com/reader033/viewer/2022060704/60709f9f2bbd466d621802c1/html5/thumbnails/8.jpg)
Colored Steiner ForestBubbleSets [Collins Penn Carpendale, TVCG’09]c©Google Maps
[Icon of Me So Ramen by Moxy Games, LLC] [Aha-Soft, via seaicons.com] c©Leeners, LLC
![Page 9: Euclidean Steiner Forest Colored Non-Crossing · Philipp Kindermann LG Theoretische Informatik FernUniversit at in Hagen Joint work with Sergey Bereg, Krzysztof Fleszar, Sergey Pupyrev,](https://reader033.vdokument.com/reader033/viewer/2022060704/60709f9f2bbd466d621802c1/html5/thumbnails/9.jpg)
Colored Steiner ForestLineSets [Alper Riche Ramos Czerwinski, TVCG’11]c©Google Maps
[Icon of Me So Ramen by Moxy Games, LLC] [Aha-Soft, via seaicons.com] c©Leeners, LLC
![Page 10: Euclidean Steiner Forest Colored Non-Crossing · Philipp Kindermann LG Theoretische Informatik FernUniversit at in Hagen Joint work with Sergey Bereg, Krzysztof Fleszar, Sergey Pupyrev,](https://reader033.vdokument.com/reader033/viewer/2022060704/60709f9f2bbd466d621802c1/html5/thumbnails/10.jpg)
Colored Steiner ForestKelpFusion [Meulemans Riche Speckmann Alper Dwyer, TVCG’13]c©Google Maps
[Icon of Me So Ramen by Moxy Games, LLC] [Aha-Soft, via seaicons.com] c©Leeners, LLC
![Page 11: Euclidean Steiner Forest Colored Non-Crossing · Philipp Kindermann LG Theoretische Informatik FernUniversit at in Hagen Joint work with Sergey Bereg, Krzysztof Fleszar, Sergey Pupyrev,](https://reader033.vdokument.com/reader033/viewer/2022060704/60709f9f2bbd466d621802c1/html5/thumbnails/11.jpg)
Colored Steiner ForestGMap (Graph-to-Map) [Hu Gansner Kobourov, CGA’10]c©Google Maps
[Icon of Me So Ramen by Moxy Games, LLC] [Aha-Soft, via seaicons.com] c©Leeners, LLC
![Page 12: Euclidean Steiner Forest Colored Non-Crossing · Philipp Kindermann LG Theoretische Informatik FernUniversit at in Hagen Joint work with Sergey Bereg, Krzysztof Fleszar, Sergey Pupyrev,](https://reader033.vdokument.com/reader033/viewer/2022060704/60709f9f2bbd466d621802c1/html5/thumbnails/12.jpg)
Colored Steiner Foresta better solution [Efrat Hu Kobourov Pupyrev, GD’14]c©Google Maps
[Icon of Me So Ramen by Moxy Games, LLC] [Aha-Soft, via seaicons.com] c©Leeners, LLC
![Page 13: Euclidean Steiner Forest Colored Non-Crossing · Philipp Kindermann LG Theoretische Informatik FernUniversit at in Hagen Joint work with Sergey Bereg, Krzysztof Fleszar, Sergey Pupyrev,](https://reader033.vdokument.com/reader033/viewer/2022060704/60709f9f2bbd466d621802c1/html5/thumbnails/13.jpg)
Colored Steiner Foresta better solution
all regions arecontiguous anddisjoint
[Efrat Hu Kobourov Pupyrev, GD’14]c©Google Maps
[Icon of Me So Ramen by Moxy Games, LLC] [Aha-Soft, via seaicons.com] c©Leeners, LLC
![Page 14: Euclidean Steiner Forest Colored Non-Crossing · Philipp Kindermann LG Theoretische Informatik FernUniversit at in Hagen Joint work with Sergey Bereg, Krzysztof Fleszar, Sergey Pupyrev,](https://reader033.vdokument.com/reader033/viewer/2022060704/60709f9f2bbd466d621802c1/html5/thumbnails/14.jpg)
Colored Steiner Forest
![Page 15: Euclidean Steiner Forest Colored Non-Crossing · Philipp Kindermann LG Theoretische Informatik FernUniversit at in Hagen Joint work with Sergey Bereg, Krzysztof Fleszar, Sergey Pupyrev,](https://reader033.vdokument.com/reader033/viewer/2022060704/60709f9f2bbd466d621802c1/html5/thumbnails/15.jpg)
Colored Steiner Forest
![Page 16: Euclidean Steiner Forest Colored Non-Crossing · Philipp Kindermann LG Theoretische Informatik FernUniversit at in Hagen Joint work with Sergey Bereg, Krzysztof Fleszar, Sergey Pupyrev,](https://reader033.vdokument.com/reader033/viewer/2022060704/60709f9f2bbd466d621802c1/html5/thumbnails/16.jpg)
Colored Steiner Forest
n points, k colors
![Page 17: Euclidean Steiner Forest Colored Non-Crossing · Philipp Kindermann LG Theoretische Informatik FernUniversit at in Hagen Joint work with Sergey Bereg, Krzysztof Fleszar, Sergey Pupyrev,](https://reader033.vdokument.com/reader033/viewer/2022060704/60709f9f2bbd466d621802c1/html5/thumbnails/17.jpg)
Colored Steiner Forest
n points, k colors
→ k Steiner Trees
![Page 18: Euclidean Steiner Forest Colored Non-Crossing · Philipp Kindermann LG Theoretische Informatik FernUniversit at in Hagen Joint work with Sergey Bereg, Krzysztof Fleszar, Sergey Pupyrev,](https://reader033.vdokument.com/reader033/viewer/2022060704/60709f9f2bbd466d621802c1/html5/thumbnails/18.jpg)
Colored Steiner Forest
n points, k colors
→ k Steiner Treesunion planar
![Page 19: Euclidean Steiner Forest Colored Non-Crossing · Philipp Kindermann LG Theoretische Informatik FernUniversit at in Hagen Joint work with Sergey Bereg, Krzysztof Fleszar, Sergey Pupyrev,](https://reader033.vdokument.com/reader033/viewer/2022060704/60709f9f2bbd466d621802c1/html5/thumbnails/19.jpg)
Colored Steiner Forest
n points, k colors
→ k Steiner Treesunion planar
![Page 20: Euclidean Steiner Forest Colored Non-Crossing · Philipp Kindermann LG Theoretische Informatik FernUniversit at in Hagen Joint work with Sergey Bereg, Krzysztof Fleszar, Sergey Pupyrev,](https://reader033.vdokument.com/reader033/viewer/2022060704/60709f9f2bbd466d621802c1/html5/thumbnails/20.jpg)
Colored Steiner Forest
n points, k colors
→ k Steiner Treesunion planar
![Page 21: Euclidean Steiner Forest Colored Non-Crossing · Philipp Kindermann LG Theoretische Informatik FernUniversit at in Hagen Joint work with Sergey Bereg, Krzysztof Fleszar, Sergey Pupyrev,](https://reader033.vdokument.com/reader033/viewer/2022060704/60709f9f2bbd466d621802c1/html5/thumbnails/21.jpg)
Bad Examples
![Page 22: Euclidean Steiner Forest Colored Non-Crossing · Philipp Kindermann LG Theoretische Informatik FernUniversit at in Hagen Joint work with Sergey Bereg, Krzysztof Fleszar, Sergey Pupyrev,](https://reader033.vdokument.com/reader033/viewer/2022060704/60709f9f2bbd466d621802c1/html5/thumbnails/22.jpg)
Bad Examples
![Page 23: Euclidean Steiner Forest Colored Non-Crossing · Philipp Kindermann LG Theoretische Informatik FernUniversit at in Hagen Joint work with Sergey Bereg, Krzysztof Fleszar, Sergey Pupyrev,](https://reader033.vdokument.com/reader033/viewer/2022060704/60709f9f2bbd466d621802c1/html5/thumbnails/23.jpg)
Bad Examples
![Page 24: Euclidean Steiner Forest Colored Non-Crossing · Philipp Kindermann LG Theoretische Informatik FernUniversit at in Hagen Joint work with Sergey Bereg, Krzysztof Fleszar, Sergey Pupyrev,](https://reader033.vdokument.com/reader033/viewer/2022060704/60709f9f2bbd466d621802c1/html5/thumbnails/24.jpg)
Bad Examples
![Page 25: Euclidean Steiner Forest Colored Non-Crossing · Philipp Kindermann LG Theoretische Informatik FernUniversit at in Hagen Joint work with Sergey Bereg, Krzysztof Fleszar, Sergey Pupyrev,](https://reader033.vdokument.com/reader033/viewer/2022060704/60709f9f2bbd466d621802c1/html5/thumbnails/25.jpg)
Bad Examples
![Page 26: Euclidean Steiner Forest Colored Non-Crossing · Philipp Kindermann LG Theoretische Informatik FernUniversit at in Hagen Joint work with Sergey Bereg, Krzysztof Fleszar, Sergey Pupyrev,](https://reader033.vdokument.com/reader033/viewer/2022060704/60709f9f2bbd466d621802c1/html5/thumbnails/26.jpg)
Bad Examples
![Page 27: Euclidean Steiner Forest Colored Non-Crossing · Philipp Kindermann LG Theoretische Informatik FernUniversit at in Hagen Joint work with Sergey Bereg, Krzysztof Fleszar, Sergey Pupyrev,](https://reader033.vdokument.com/reader033/viewer/2022060704/60709f9f2bbd466d621802c1/html5/thumbnails/27.jpg)
Bad Examples
![Page 28: Euclidean Steiner Forest Colored Non-Crossing · Philipp Kindermann LG Theoretische Informatik FernUniversit at in Hagen Joint work with Sergey Bereg, Krzysztof Fleszar, Sergey Pupyrev,](https://reader033.vdokument.com/reader033/viewer/2022060704/60709f9f2bbd466d621802c1/html5/thumbnails/28.jpg)
Known Results
1-CESF (= Euclidean Steiner Tree)
![Page 29: Euclidean Steiner Forest Colored Non-Crossing · Philipp Kindermann LG Theoretische Informatik FernUniversit at in Hagen Joint work with Sergey Bereg, Krzysztof Fleszar, Sergey Pupyrev,](https://reader033.vdokument.com/reader033/viewer/2022060704/60709f9f2bbd466d621802c1/html5/thumbnails/29.jpg)
Known Results
1-CESF (= Euclidean Steiner Tree)• is NP-hard [Garey Johnson, 1979]
![Page 30: Euclidean Steiner Forest Colored Non-Crossing · Philipp Kindermann LG Theoretische Informatik FernUniversit at in Hagen Joint work with Sergey Bereg, Krzysztof Fleszar, Sergey Pupyrev,](https://reader033.vdokument.com/reader033/viewer/2022060704/60709f9f2bbd466d621802c1/html5/thumbnails/30.jpg)
Known Results
1-CESF (= Euclidean Steiner Tree)• is NP-hard [Garey Johnson, 1979]• admits a PTAS [Arora, JACM’98][Mitchell, SICOMP’99]
![Page 31: Euclidean Steiner Forest Colored Non-Crossing · Philipp Kindermann LG Theoretische Informatik FernUniversit at in Hagen Joint work with Sergey Bereg, Krzysztof Fleszar, Sergey Pupyrev,](https://reader033.vdokument.com/reader033/viewer/2022060704/60709f9f2bbd466d621802c1/html5/thumbnails/31.jpg)
Known Results
1-CESF (= Euclidean Steiner Tree)• is NP-hard [Garey Johnson, 1979]• admits a PTAS [Arora, JACM’98][Mitchell, SICOMP’99]
n/2-CESF (= Euclidean Matching)
![Page 32: Euclidean Steiner Forest Colored Non-Crossing · Philipp Kindermann LG Theoretische Informatik FernUniversit at in Hagen Joint work with Sergey Bereg, Krzysztof Fleszar, Sergey Pupyrev,](https://reader033.vdokument.com/reader033/viewer/2022060704/60709f9f2bbd466d621802c1/html5/thumbnails/32.jpg)
Known Results
1-CESF (= Euclidean Steiner Tree)• is NP-hard [Garey Johnson, 1979]• admits a PTAS [Arora, JACM’98][Mitchell, SICOMP’99]
n/2-CESF (= Euclidean Matching)• is NP-hard [Bastert Fekete, TR’98]
![Page 33: Euclidean Steiner Forest Colored Non-Crossing · Philipp Kindermann LG Theoretische Informatik FernUniversit at in Hagen Joint work with Sergey Bereg, Krzysztof Fleszar, Sergey Pupyrev,](https://reader033.vdokument.com/reader033/viewer/2022060704/60709f9f2bbd466d621802c1/html5/thumbnails/33.jpg)
Known Results
1-CESF (= Euclidean Steiner Tree)• is NP-hard [Garey Johnson, 1979]• admits a PTAS [Arora, JACM’98][Mitchell, SICOMP’99]
n/2-CESF (= Euclidean Matching)• is NP-hard [Bastert Fekete, TR’98]• O(n log
√n)-approx. [Chan Hoffmann Kiazyk Lubiw, GD’13]
![Page 34: Euclidean Steiner Forest Colored Non-Crossing · Philipp Kindermann LG Theoretische Informatik FernUniversit at in Hagen Joint work with Sergey Bereg, Krzysztof Fleszar, Sergey Pupyrev,](https://reader033.vdokument.com/reader033/viewer/2022060704/60709f9f2bbd466d621802c1/html5/thumbnails/34.jpg)
Known Results
1-CESF (= Euclidean Steiner Tree)• is NP-hard [Garey Johnson, 1979]• admits a PTAS [Arora, JACM’98][Mitchell, SICOMP’99]
n/2-CESF (= Euclidean Matching)• is NP-hard [Bastert Fekete, TR’98]• O(n log
√n)-approx. [Chan Hoffmann Kiazyk Lubiw, GD’13]
k-CESF
![Page 35: Euclidean Steiner Forest Colored Non-Crossing · Philipp Kindermann LG Theoretische Informatik FernUniversit at in Hagen Joint work with Sergey Bereg, Krzysztof Fleszar, Sergey Pupyrev,](https://reader033.vdokument.com/reader033/viewer/2022060704/60709f9f2bbd466d621802c1/html5/thumbnails/35.jpg)
Known Results
1-CESF (= Euclidean Steiner Tree)• is NP-hard [Garey Johnson, 1979]• admits a PTAS [Arora, JACM’98][Mitchell, SICOMP’99]
n/2-CESF (= Euclidean Matching)• is NP-hard [Bastert Fekete, TR’98]• O(n log
√n)-approx. [Chan Hoffmann Kiazyk Lubiw, GD’13]
k-CESF• has a kρ-approximation [Efrat Hu Kobourov Pupyrev, GD’14]
![Page 36: Euclidean Steiner Forest Colored Non-Crossing · Philipp Kindermann LG Theoretische Informatik FernUniversit at in Hagen Joint work with Sergey Bereg, Krzysztof Fleszar, Sergey Pupyrev,](https://reader033.vdokument.com/reader033/viewer/2022060704/60709f9f2bbd466d621802c1/html5/thumbnails/36.jpg)
Known Results
1-CESF (= Euclidean Steiner Tree)• is NP-hard [Garey Johnson, 1979]• admits a PTAS [Arora, JACM’98][Mitchell, SICOMP’99]
n/2-CESF (= Euclidean Matching)• is NP-hard [Bastert Fekete, TR’98]• O(n log
√n)-approx. [Chan Hoffmann Kiazyk Lubiw, GD’13]
k-CESF• has a kρ-approximation [Efrat Hu Kobourov Pupyrev, GD’14]
Steiner ratio
![Page 37: Euclidean Steiner Forest Colored Non-Crossing · Philipp Kindermann LG Theoretische Informatik FernUniversit at in Hagen Joint work with Sergey Bereg, Krzysztof Fleszar, Sergey Pupyrev,](https://reader033.vdokument.com/reader033/viewer/2022060704/60709f9f2bbd466d621802c1/html5/thumbnails/37.jpg)
Known Results
1-CESF (= Euclidean Steiner Tree)• is NP-hard [Garey Johnson, 1979]• admits a PTAS [Arora, JACM’98][Mitchell, SICOMP’99]
n/2-CESF (= Euclidean Matching)• is NP-hard [Bastert Fekete, TR’98]• O(n log
√n)-approx. [Chan Hoffmann Kiazyk Lubiw, GD’13]
k-CESF• has a kρ-approximation [Efrat Hu Kobourov Pupyrev, GD’14]
Steiner ratio
ρ ≤ 1.21 [Chung Graham, ANYAS’85]
![Page 38: Euclidean Steiner Forest Colored Non-Crossing · Philipp Kindermann LG Theoretische Informatik FernUniversit at in Hagen Joint work with Sergey Bereg, Krzysztof Fleszar, Sergey Pupyrev,](https://reader033.vdokument.com/reader033/viewer/2022060704/60709f9f2bbd466d621802c1/html5/thumbnails/38.jpg)
2-CESF
![Page 39: Euclidean Steiner Forest Colored Non-Crossing · Philipp Kindermann LG Theoretische Informatik FernUniversit at in Hagen Joint work with Sergey Bereg, Krzysztof Fleszar, Sergey Pupyrev,](https://reader033.vdokument.com/reader033/viewer/2022060704/60709f9f2bbd466d621802c1/html5/thumbnails/39.jpg)
2-CESF
![Page 40: Euclidean Steiner Forest Colored Non-Crossing · Philipp Kindermann LG Theoretische Informatik FernUniversit at in Hagen Joint work with Sergey Bereg, Krzysztof Fleszar, Sergey Pupyrev,](https://reader033.vdokument.com/reader033/viewer/2022060704/60709f9f2bbd466d621802c1/html5/thumbnails/40.jpg)
2-CESF
![Page 41: Euclidean Steiner Forest Colored Non-Crossing · Philipp Kindermann LG Theoretische Informatik FernUniversit at in Hagen Joint work with Sergey Bereg, Krzysztof Fleszar, Sergey Pupyrev,](https://reader033.vdokument.com/reader033/viewer/2022060704/60709f9f2bbd466d621802c1/html5/thumbnails/41.jpg)
2-CESF
![Page 42: Euclidean Steiner Forest Colored Non-Crossing · Philipp Kindermann LG Theoretische Informatik FernUniversit at in Hagen Joint work with Sergey Bereg, Krzysztof Fleszar, Sergey Pupyrev,](https://reader033.vdokument.com/reader033/viewer/2022060704/60709f9f2bbd466d621802c1/html5/thumbnails/42.jpg)
2-CESF
![Page 43: Euclidean Steiner Forest Colored Non-Crossing · Philipp Kindermann LG Theoretische Informatik FernUniversit at in Hagen Joint work with Sergey Bereg, Krzysztof Fleszar, Sergey Pupyrev,](https://reader033.vdokument.com/reader033/viewer/2022060704/60709f9f2bbd466d621802c1/html5/thumbnails/43.jpg)
2-CESF
![Page 44: Euclidean Steiner Forest Colored Non-Crossing · Philipp Kindermann LG Theoretische Informatik FernUniversit at in Hagen Joint work with Sergey Bereg, Krzysztof Fleszar, Sergey Pupyrev,](https://reader033.vdokument.com/reader033/viewer/2022060704/60709f9f2bbd466d621802c1/html5/thumbnails/44.jpg)
2-CESF
![Page 45: Euclidean Steiner Forest Colored Non-Crossing · Philipp Kindermann LG Theoretische Informatik FernUniversit at in Hagen Joint work with Sergey Bereg, Krzysztof Fleszar, Sergey Pupyrev,](https://reader033.vdokument.com/reader033/viewer/2022060704/60709f9f2bbd466d621802c1/html5/thumbnails/45.jpg)
Rounding to the Grid
![Page 46: Euclidean Steiner Forest Colored Non-Crossing · Philipp Kindermann LG Theoretische Informatik FernUniversit at in Hagen Joint work with Sergey Bereg, Krzysztof Fleszar, Sergey Pupyrev,](https://reader033.vdokument.com/reader033/viewer/2022060704/60709f9f2bbd466d621802c1/html5/thumbnails/46.jpg)
Rounding to the Grid
• L0 diameter of smallestbounding box• 3√
2n/ε ≤ L ≤ 6√
2n/ε• (L× L)-grid• granularity L0/L
L0
![Page 47: Euclidean Steiner Forest Colored Non-Crossing · Philipp Kindermann LG Theoretische Informatik FernUniversit at in Hagen Joint work with Sergey Bereg, Krzysztof Fleszar, Sergey Pupyrev,](https://reader033.vdokument.com/reader033/viewer/2022060704/60709f9f2bbd466d621802c1/html5/thumbnails/47.jpg)
Rounding to the Grid
• L0 diameter of smallestbounding box• 3√
2n/ε ≤ L ≤ 6√
2n/ε• (L× L)-grid• granularity L0/L
L0
![Page 48: Euclidean Steiner Forest Colored Non-Crossing · Philipp Kindermann LG Theoretische Informatik FernUniversit at in Hagen Joint work with Sergey Bereg, Krzysztof Fleszar, Sergey Pupyrev,](https://reader033.vdokument.com/reader033/viewer/2022060704/60709f9f2bbd466d621802c1/html5/thumbnails/48.jpg)
Rounding to the Grid
• L0 diameter of smallestbounding box• 3√
2n/ε ≤ L ≤ 6√
2n/ε• (L× L)-grid• granularity L0/L
![Page 49: Euclidean Steiner Forest Colored Non-Crossing · Philipp Kindermann LG Theoretische Informatik FernUniversit at in Hagen Joint work with Sergey Bereg, Krzysztof Fleszar, Sergey Pupyrev,](https://reader033.vdokument.com/reader033/viewer/2022060704/60709f9f2bbd466d621802c1/html5/thumbnails/49.jpg)
Rounding to the Grid
• L0 diameter of smallestbounding box• 3√
2n/ε ≤ L ≤ 6√
2n/ε• (L× L)-grid• granularity L0/L
![Page 50: Euclidean Steiner Forest Colored Non-Crossing · Philipp Kindermann LG Theoretische Informatik FernUniversit at in Hagen Joint work with Sergey Bereg, Krzysztof Fleszar, Sergey Pupyrev,](https://reader033.vdokument.com/reader033/viewer/2022060704/60709f9f2bbd466d621802c1/html5/thumbnails/50.jpg)
Rounding to the Grid
• L0 diameter of smallestbounding box• 3√
2n/ε ≤ L ≤ 6√
2n/ε• (L× L)-grid• granularity L0/L 1
![Page 51: Euclidean Steiner Forest Colored Non-Crossing · Philipp Kindermann LG Theoretische Informatik FernUniversit at in Hagen Joint work with Sergey Bereg, Krzysztof Fleszar, Sergey Pupyrev,](https://reader033.vdokument.com/reader033/viewer/2022060704/60709f9f2bbd466d621802c1/html5/thumbnails/51.jpg)
Rounding to the Grid
• 3√
2n/ε ≤ L ≤ 6√
2n/ε• (L× L)-grid• granularity 1
![Page 52: Euclidean Steiner Forest Colored Non-Crossing · Philipp Kindermann LG Theoretische Informatik FernUniversit at in Hagen Joint work with Sergey Bereg, Krzysztof Fleszar, Sergey Pupyrev,](https://reader033.vdokument.com/reader033/viewer/2022060704/60709f9f2bbd466d621802c1/html5/thumbnails/52.jpg)
Rounding to the Grid
→ (even,even)
→ (odd,odd)
• 3√
2n/ε ≤ L ≤ 6√
2n/ε• (L× L)-grid• granularity 1
![Page 53: Euclidean Steiner Forest Colored Non-Crossing · Philipp Kindermann LG Theoretische Informatik FernUniversit at in Hagen Joint work with Sergey Bereg, Krzysztof Fleszar, Sergey Pupyrev,](https://reader033.vdokument.com/reader033/viewer/2022060704/60709f9f2bbd466d621802c1/html5/thumbnails/53.jpg)
Rounding to the Grid
→ (even,even)
→ (odd,odd)
• 3√
2n/ε ≤ L ≤ 6√
2n/ε• (L× L)-grid• granularity 1
![Page 54: Euclidean Steiner Forest Colored Non-Crossing · Philipp Kindermann LG Theoretische Informatik FernUniversit at in Hagen Joint work with Sergey Bereg, Krzysztof Fleszar, Sergey Pupyrev,](https://reader033.vdokument.com/reader033/viewer/2022060704/60709f9f2bbd466d621802c1/html5/thumbnails/54.jpg)
Going Back
• 3√
2n/ε ≤ L ≤ 6√
2n/ε• (L× L)-grid• granularity 1
![Page 55: Euclidean Steiner Forest Colored Non-Crossing · Philipp Kindermann LG Theoretische Informatik FernUniversit at in Hagen Joint work with Sergey Bereg, Krzysztof Fleszar, Sergey Pupyrev,](https://reader033.vdokument.com/reader033/viewer/2022060704/60709f9f2bbd466d621802c1/html5/thumbnails/55.jpg)
Going Back
• 3√
2n/ε ≤ L ≤ 6√
2n/ε• (L× L)-grid• granularity 1
![Page 56: Euclidean Steiner Forest Colored Non-Crossing · Philipp Kindermann LG Theoretische Informatik FernUniversit at in Hagen Joint work with Sergey Bereg, Krzysztof Fleszar, Sergey Pupyrev,](https://reader033.vdokument.com/reader033/viewer/2022060704/60709f9f2bbd466d621802c1/html5/thumbnails/56.jpg)
Going Back
• 3√
2n/ε ≤ L ≤ 6√
2n/ε• (L× L)-grid• granularity 1
![Page 57: Euclidean Steiner Forest Colored Non-Crossing · Philipp Kindermann LG Theoretische Informatik FernUniversit at in Hagen Joint work with Sergey Bereg, Krzysztof Fleszar, Sergey Pupyrev,](https://reader033.vdokument.com/reader033/viewer/2022060704/60709f9f2bbd466d621802c1/html5/thumbnails/57.jpg)
Going Back
• 3√
2n/ε ≤ L ≤ 6√
2n/ε• (L× L)-grid• granularity 1
![Page 58: Euclidean Steiner Forest Colored Non-Crossing · Philipp Kindermann LG Theoretische Informatik FernUniversit at in Hagen Joint work with Sergey Bereg, Krzysztof Fleszar, Sergey Pupyrev,](https://reader033.vdokument.com/reader033/viewer/2022060704/60709f9f2bbd466d621802c1/html5/thumbnails/58.jpg)
Going Back
• 3√
2n/ε ≤ L ≤ 6√
2n/ε• (L× L)-grid• granularity 1
![Page 59: Euclidean Steiner Forest Colored Non-Crossing · Philipp Kindermann LG Theoretische Informatik FernUniversit at in Hagen Joint work with Sergey Bereg, Krzysztof Fleszar, Sergey Pupyrev,](https://reader033.vdokument.com/reader033/viewer/2022060704/60709f9f2bbd466d621802c1/html5/thumbnails/59.jpg)
Going Back
• 3√
2n/ε ≤ L ≤ 6√
2n/ε• (L× L)-grid• granularity 1
![Page 60: Euclidean Steiner Forest Colored Non-Crossing · Philipp Kindermann LG Theoretische Informatik FernUniversit at in Hagen Joint work with Sergey Bereg, Krzysztof Fleszar, Sergey Pupyrev,](https://reader033.vdokument.com/reader033/viewer/2022060704/60709f9f2bbd466d621802c1/html5/thumbnails/60.jpg)
Going Back
• 3√
2n/ε ≤ L ≤ 6√
2n/ε• (L× L)-grid• granularity 1
![Page 61: Euclidean Steiner Forest Colored Non-Crossing · Philipp Kindermann LG Theoretische Informatik FernUniversit at in Hagen Joint work with Sergey Bereg, Krzysztof Fleszar, Sergey Pupyrev,](https://reader033.vdokument.com/reader033/viewer/2022060704/60709f9f2bbd466d621802c1/html5/thumbnails/61.jpg)
Going Back
• 3√
2n/ε ≤ L ≤ 6√
2n/ε• (L× L)-grid• granularity 1
![Page 62: Euclidean Steiner Forest Colored Non-Crossing · Philipp Kindermann LG Theoretische Informatik FernUniversit at in Hagen Joint work with Sergey Bereg, Krzysztof Fleszar, Sergey Pupyrev,](https://reader033.vdokument.com/reader033/viewer/2022060704/60709f9f2bbd466d621802c1/html5/thumbnails/62.jpg)
Going Back
• 3√
2n/ε ≤ L ≤ 6√
2n/ε• (L× L)-grid• granularity 1
![Page 63: Euclidean Steiner Forest Colored Non-Crossing · Philipp Kindermann LG Theoretische Informatik FernUniversit at in Hagen Joint work with Sergey Bereg, Krzysztof Fleszar, Sergey Pupyrev,](https://reader033.vdokument.com/reader033/viewer/2022060704/60709f9f2bbd466d621802c1/html5/thumbnails/63.jpg)
Going Back
• 3√
2n/ε ≤ L ≤ 6√
2n/ε• (L× L)-grid• granularity 1
![Page 64: Euclidean Steiner Forest Colored Non-Crossing · Philipp Kindermann LG Theoretische Informatik FernUniversit at in Hagen Joint work with Sergey Bereg, Krzysztof Fleszar, Sergey Pupyrev,](https://reader033.vdokument.com/reader033/viewer/2022060704/60709f9f2bbd466d621802c1/html5/thumbnails/64.jpg)
Going Back
≤√
2
• 3√
2n/ε ≤ L ≤ 6√
2n/ε• (L× L)-grid• granularity 1
![Page 65: Euclidean Steiner Forest Colored Non-Crossing · Philipp Kindermann LG Theoretische Informatik FernUniversit at in Hagen Joint work with Sergey Bereg, Krzysztof Fleszar, Sergey Pupyrev,](https://reader033.vdokument.com/reader033/viewer/2022060704/60709f9f2bbd466d621802c1/html5/thumbnails/65.jpg)
Going Back
≤√
2
≤ 2√
2
• 3√
2n/ε ≤ L ≤ 6√
2n/ε• (L× L)-grid• granularity 1
![Page 66: Euclidean Steiner Forest Colored Non-Crossing · Philipp Kindermann LG Theoretische Informatik FernUniversit at in Hagen Joint work with Sergey Bereg, Krzysztof Fleszar, Sergey Pupyrev,](https://reader033.vdokument.com/reader033/viewer/2022060704/60709f9f2bbd466d621802c1/html5/thumbnails/66.jpg)
Going Back
≤√
2
≤ 2√
2
• 3√
2n/ε ≤ L ≤ 6√
2n/ε• (L× L)-grid• granularity 1
in total ≤ 3√
2n ≤ εL ≤ εOPT
![Page 67: Euclidean Steiner Forest Colored Non-Crossing · Philipp Kindermann LG Theoretische Informatik FernUniversit at in Hagen Joint work with Sergey Bereg, Krzysztof Fleszar, Sergey Pupyrev,](https://reader033.vdokument.com/reader033/viewer/2022060704/60709f9f2bbd466d621802c1/html5/thumbnails/67.jpg)
Going Back
≤√
2
≤ 2√
2
• 3√
2n/ε ≤ L ≤ 6√
2n/ε• (L× L)-grid• granularity 1
in total ≤ 3√
2n ≤ εL ≤ εOPT
![Page 68: Euclidean Steiner Forest Colored Non-Crossing · Philipp Kindermann LG Theoretische Informatik FernUniversit at in Hagen Joint work with Sergey Bereg, Krzysztof Fleszar, Sergey Pupyrev,](https://reader033.vdokument.com/reader033/viewer/2022060704/60709f9f2bbd466d621802c1/html5/thumbnails/68.jpg)
Going Back
≤√
2
≤ 2√
2
• 3√
2n/ε ≤ L ≤ 6√
2n/ε• (L× L)-grid• granularity 1
in total ≤ 3√
2n ≤ εL ≤ εOPT
![Page 69: Euclidean Steiner Forest Colored Non-Crossing · Philipp Kindermann LG Theoretische Informatik FernUniversit at in Hagen Joint work with Sergey Bereg, Krzysztof Fleszar, Sergey Pupyrev,](https://reader033.vdokument.com/reader033/viewer/2022060704/60709f9f2bbd466d621802c1/html5/thumbnails/69.jpg)
Going Back
≤√
2
≤ 2√
2
• 3√
2n/ε ≤ L ≤ 6√
2n/ε• (L× L)-grid• granularity 1
in total ≤ 3√
2n ≤ εL ≤ εOPT
2-CESF instance I → rounded instance I ∗ → solution LI
|LI | ≤ (1 + ε)OPTI∗ ≤ (1 + ε)2OPTI
![Page 70: Euclidean Steiner Forest Colored Non-Crossing · Philipp Kindermann LG Theoretische Informatik FernUniversit at in Hagen Joint work with Sergey Bereg, Krzysztof Fleszar, Sergey Pupyrev,](https://reader033.vdokument.com/reader033/viewer/2022060704/60709f9f2bbd466d621802c1/html5/thumbnails/70.jpg)
Quadtree Placement
![Page 71: Euclidean Steiner Forest Colored Non-Crossing · Philipp Kindermann LG Theoretische Informatik FernUniversit at in Hagen Joint work with Sergey Bereg, Krzysztof Fleszar, Sergey Pupyrev,](https://reader033.vdokument.com/reader033/viewer/2022060704/60709f9f2bbd466d621802c1/html5/thumbnails/71.jpg)
Quadtree Placement
level 0
![Page 72: Euclidean Steiner Forest Colored Non-Crossing · Philipp Kindermann LG Theoretische Informatik FernUniversit at in Hagen Joint work with Sergey Bereg, Krzysztof Fleszar, Sergey Pupyrev,](https://reader033.vdokument.com/reader033/viewer/2022060704/60709f9f2bbd466d621802c1/html5/thumbnails/72.jpg)
Quadtree Placement
level 0
level 1
![Page 73: Euclidean Steiner Forest Colored Non-Crossing · Philipp Kindermann LG Theoretische Informatik FernUniversit at in Hagen Joint work with Sergey Bereg, Krzysztof Fleszar, Sergey Pupyrev,](https://reader033.vdokument.com/reader033/viewer/2022060704/60709f9f2bbd466d621802c1/html5/thumbnails/73.jpg)
Quadtree Placement
. . . . . .
level 0
level 1
level 2. . .
![Page 74: Euclidean Steiner Forest Colored Non-Crossing · Philipp Kindermann LG Theoretische Informatik FernUniversit at in Hagen Joint work with Sergey Bereg, Krzysztof Fleszar, Sergey Pupyrev,](https://reader033.vdokument.com/reader033/viewer/2022060704/60709f9f2bbd466d621802c1/html5/thumbnails/74.jpg)
Quadtree Placement
. . . . . .
level 0
level 1
level 2. . .
...
level log L
......
...
![Page 75: Euclidean Steiner Forest Colored Non-Crossing · Philipp Kindermann LG Theoretische Informatik FernUniversit at in Hagen Joint work with Sergey Bereg, Krzysztof Fleszar, Sergey Pupyrev,](https://reader033.vdokument.com/reader033/viewer/2022060704/60709f9f2bbd466d621802c1/html5/thumbnails/75.jpg)
Quadtree Placement
• m = 4 log(L)/ε
![Page 76: Euclidean Steiner Forest Colored Non-Crossing · Philipp Kindermann LG Theoretische Informatik FernUniversit at in Hagen Joint work with Sergey Bereg, Krzysztof Fleszar, Sergey Pupyrev,](https://reader033.vdokument.com/reader033/viewer/2022060704/60709f9f2bbd466d621802c1/html5/thumbnails/76.jpg)
Quadtree Placement
• m = 4 log(L)/ε
• portals on level-i-line withdistance L/(2im)
m portals
m portals
![Page 77: Euclidean Steiner Forest Colored Non-Crossing · Philipp Kindermann LG Theoretische Informatik FernUniversit at in Hagen Joint work with Sergey Bereg, Krzysztof Fleszar, Sergey Pupyrev,](https://reader033.vdokument.com/reader033/viewer/2022060704/60709f9f2bbd466d621802c1/html5/thumbnails/77.jpg)
Quadtree Placement
• m = 4 log(L)/ε
• portals on level-i-line withdistance L/(2im)
m portals
m portals• level-i-square has at most
4m portals on its margin
![Page 78: Euclidean Steiner Forest Colored Non-Crossing · Philipp Kindermann LG Theoretische Informatik FernUniversit at in Hagen Joint work with Sergey Bereg, Krzysztof Fleszar, Sergey Pupyrev,](https://reader033.vdokument.com/reader033/viewer/2022060704/60709f9f2bbd466d621802c1/html5/thumbnails/78.jpg)
Quadtree Placement
• m = 4 log(L)/ε
• portals on level-i-line withdistance L/(2im)
• level-i-square has at most4m portals on its margin
• place origin uniformly atrandom
![Page 79: Euclidean Steiner Forest Colored Non-Crossing · Philipp Kindermann LG Theoretische Informatik FernUniversit at in Hagen Joint work with Sergey Bereg, Krzysztof Fleszar, Sergey Pupyrev,](https://reader033.vdokument.com/reader033/viewer/2022060704/60709f9f2bbd466d621802c1/html5/thumbnails/79.jpg)
Quadtree Placement
• m = 4 log(L)/ε
• portals on level-i-line withdistance L/(2im)
• level-i-square has at most4m portals on its margin
• place origin uniformly atrandom
![Page 80: Euclidean Steiner Forest Colored Non-Crossing · Philipp Kindermann LG Theoretische Informatik FernUniversit at in Hagen Joint work with Sergey Bereg, Krzysztof Fleszar, Sergey Pupyrev,](https://reader033.vdokument.com/reader033/viewer/2022060704/60709f9f2bbd466d621802c1/html5/thumbnails/80.jpg)
Quadtree Placement
• m = 4 log(L)/ε
• portals on level-i-line withdistance L/(2im)
• level-i-square has at most4m portals on its margin
• place origin uniformly atrandom
portal-respecting solution:crosses grid lines only at portals
![Page 81: Euclidean Steiner Forest Colored Non-Crossing · Philipp Kindermann LG Theoretische Informatik FernUniversit at in Hagen Joint work with Sergey Bereg, Krzysztof Fleszar, Sergey Pupyrev,](https://reader033.vdokument.com/reader033/viewer/2022060704/60709f9f2bbd466d621802c1/html5/thumbnails/81.jpg)
Quadtree Placement
• m = 4 log(L)/ε
• portals on level-i-line withdistance L/(2im)
• level-i-square has at most4m portals on its margin
• place origin uniformly atrandom
portal-respecting solution:crosses grid lines only at portals
![Page 82: Euclidean Steiner Forest Colored Non-Crossing · Philipp Kindermann LG Theoretische Informatik FernUniversit at in Hagen Joint work with Sergey Bereg, Krzysztof Fleszar, Sergey Pupyrev,](https://reader033.vdokument.com/reader033/viewer/2022060704/60709f9f2bbd466d621802c1/html5/thumbnails/82.jpg)
Quadtree Placement
• m = 4 log(L)/ε
• portals on level-i-line withdistance L/(2im)
• level-i-square has at most4m portals on its margin
• place origin uniformly atrandom
portal-respecting solution:crosses grid lines only at portals
![Page 83: Euclidean Steiner Forest Colored Non-Crossing · Philipp Kindermann LG Theoretische Informatik FernUniversit at in Hagen Joint work with Sergey Bereg, Krzysztof Fleszar, Sergey Pupyrev,](https://reader033.vdokument.com/reader033/viewer/2022060704/60709f9f2bbd466d621802c1/html5/thumbnails/83.jpg)
Quadtree Placement
• m = 4 log(L)/ε
• portals on level-i-line withdistance L/(2im)
• level-i-square has at most4m portals on its margin
• place origin uniformly atrandom
portal-respecting solution:crosses grid lines only at portals
line ` crosses drawing t(`) times;
expected length increase: ≤ εt(`)
4
![Page 84: Euclidean Steiner Forest Colored Non-Crossing · Philipp Kindermann LG Theoretische Informatik FernUniversit at in Hagen Joint work with Sergey Bereg, Krzysztof Fleszar, Sergey Pupyrev,](https://reader033.vdokument.com/reader033/viewer/2022060704/60709f9f2bbd466d621802c1/html5/thumbnails/84.jpg)
Quadtree Placement
• m = 4 log(L)/ε
• portals on level-i-line withdistance L/(2im)
• level-i-square has at most4m portals on its margin
• place origin uniformly atrandom
portal-respecting solution:crosses grid lines only at portals
line ` crosses drawing t(`) times;
expected length increase: ≤ εt(`)
4
2-CESF instance I → portal-respecting solution L|L| ≤ (1 + ε)3OPTI
![Page 85: Euclidean Steiner Forest Colored Non-Crossing · Philipp Kindermann LG Theoretische Informatik FernUniversit at in Hagen Joint work with Sergey Bereg, Krzysztof Fleszar, Sergey Pupyrev,](https://reader033.vdokument.com/reader033/viewer/2022060704/60709f9f2bbd466d621802c1/html5/thumbnails/85.jpg)
3-Light Solution
3-light solution: each portal is crossed at most 3 times
![Page 86: Euclidean Steiner Forest Colored Non-Crossing · Philipp Kindermann LG Theoretische Informatik FernUniversit at in Hagen Joint work with Sergey Bereg, Krzysztof Fleszar, Sergey Pupyrev,](https://reader033.vdokument.com/reader033/viewer/2022060704/60709f9f2bbd466d621802c1/html5/thumbnails/86.jpg)
3-Light Solution
3-light solution: each portal is crossed at most 3 times
![Page 87: Euclidean Steiner Forest Colored Non-Crossing · Philipp Kindermann LG Theoretische Informatik FernUniversit at in Hagen Joint work with Sergey Bereg, Krzysztof Fleszar, Sergey Pupyrev,](https://reader033.vdokument.com/reader033/viewer/2022060704/60709f9f2bbd466d621802c1/html5/thumbnails/87.jpg)
3-Light Solution
3-light solution: each portal is crossed at most 3 times
![Page 88: Euclidean Steiner Forest Colored Non-Crossing · Philipp Kindermann LG Theoretische Informatik FernUniversit at in Hagen Joint work with Sergey Bereg, Krzysztof Fleszar, Sergey Pupyrev,](https://reader033.vdokument.com/reader033/viewer/2022060704/60709f9f2bbd466d621802c1/html5/thumbnails/88.jpg)
3-Light Solution
3-light solution: each portal is crossed at most 3 times
![Page 89: Euclidean Steiner Forest Colored Non-Crossing · Philipp Kindermann LG Theoretische Informatik FernUniversit at in Hagen Joint work with Sergey Bereg, Krzysztof Fleszar, Sergey Pupyrev,](https://reader033.vdokument.com/reader033/viewer/2022060704/60709f9f2bbd466d621802c1/html5/thumbnails/89.jpg)
3-Light Solution
3-light solution: each portal is crossed at most 3 times
![Page 90: Euclidean Steiner Forest Colored Non-Crossing · Philipp Kindermann LG Theoretische Informatik FernUniversit at in Hagen Joint work with Sergey Bereg, Krzysztof Fleszar, Sergey Pupyrev,](https://reader033.vdokument.com/reader033/viewer/2022060704/60709f9f2bbd466d621802c1/html5/thumbnails/90.jpg)
3-Light Solution
3-light solution: each portal is crossed at most 3 times
![Page 91: Euclidean Steiner Forest Colored Non-Crossing · Philipp Kindermann LG Theoretische Informatik FernUniversit at in Hagen Joint work with Sergey Bereg, Krzysztof Fleszar, Sergey Pupyrev,](https://reader033.vdokument.com/reader033/viewer/2022060704/60709f9f2bbd466d621802c1/html5/thumbnails/91.jpg)
3-Light Solution
3-light solution: each portal is crossed at most 3 times
![Page 92: Euclidean Steiner Forest Colored Non-Crossing · Philipp Kindermann LG Theoretische Informatik FernUniversit at in Hagen Joint work with Sergey Bereg, Krzysztof Fleszar, Sergey Pupyrev,](https://reader033.vdokument.com/reader033/viewer/2022060704/60709f9f2bbd466d621802c1/html5/thumbnails/92.jpg)
3-Light Solution
3-light solution: each portal is crossed at most 3 times
![Page 93: Euclidean Steiner Forest Colored Non-Crossing · Philipp Kindermann LG Theoretische Informatik FernUniversit at in Hagen Joint work with Sergey Bereg, Krzysztof Fleszar, Sergey Pupyrev,](https://reader033.vdokument.com/reader033/viewer/2022060704/60709f9f2bbd466d621802c1/html5/thumbnails/93.jpg)
3-Light Solution
3-light solution: each portal is crossed at most 3 times
width= 0
![Page 94: Euclidean Steiner Forest Colored Non-Crossing · Philipp Kindermann LG Theoretische Informatik FernUniversit at in Hagen Joint work with Sergey Bereg, Krzysztof Fleszar, Sergey Pupyrev,](https://reader033.vdokument.com/reader033/viewer/2022060704/60709f9f2bbd466d621802c1/html5/thumbnails/94.jpg)
3-Light Solution
3-light solution: each portal is crossed at most 3 times
width= 0
2-CESF instance I → portal-respecting 3-light solution L∗|L∗| ≤ (1 + ε)3OPTI
![Page 95: Euclidean Steiner Forest Colored Non-Crossing · Philipp Kindermann LG Theoretische Informatik FernUniversit at in Hagen Joint work with Sergey Bereg, Krzysztof Fleszar, Sergey Pupyrev,](https://reader033.vdokument.com/reader033/viewer/2022060704/60709f9f2bbd466d621802c1/html5/thumbnails/95.jpg)
3-Light Solution
3-light solution: each portal is crossed at most 3 times
width= 0
2-CESF instance I → portal-respecting 3-light solution L∗|L∗| ≤ (1 + ε)3OPTI ≤ (1 + ε′)OPTI
![Page 96: Euclidean Steiner Forest Colored Non-Crossing · Philipp Kindermann LG Theoretische Informatik FernUniversit at in Hagen Joint work with Sergey Bereg, Krzysztof Fleszar, Sergey Pupyrev,](https://reader033.vdokument.com/reader033/viewer/2022060704/60709f9f2bbd466d621802c1/html5/thumbnails/96.jpg)
Putting Things Together
Use a dynamic program!
![Page 97: Euclidean Steiner Forest Colored Non-Crossing · Philipp Kindermann LG Theoretische Informatik FernUniversit at in Hagen Joint work with Sergey Bereg, Krzysztof Fleszar, Sergey Pupyrev,](https://reader033.vdokument.com/reader033/viewer/2022060704/60709f9f2bbd466d621802c1/html5/thumbnails/97.jpg)
Putting Things Together
Use a dynamic program! A subproblem consists of:
![Page 98: Euclidean Steiner Forest Colored Non-Crossing · Philipp Kindermann LG Theoretische Informatik FernUniversit at in Hagen Joint work with Sergey Bereg, Krzysztof Fleszar, Sergey Pupyrev,](https://reader033.vdokument.com/reader033/viewer/2022060704/60709f9f2bbd466d621802c1/html5/thumbnails/98.jpg)
Putting Things Together
Use a dynamic program! A subproblem consists of:• a square of the quadtree
![Page 99: Euclidean Steiner Forest Colored Non-Crossing · Philipp Kindermann LG Theoretische Informatik FernUniversit at in Hagen Joint work with Sergey Bereg, Krzysztof Fleszar, Sergey Pupyrev,](https://reader033.vdokument.com/reader033/viewer/2022060704/60709f9f2bbd466d621802c1/html5/thumbnails/99.jpg)
Putting Things Together
Use a dynamic program! A subproblem consists of:• a square of the quadtree• up to three red and blue points on each portal
![Page 100: Euclidean Steiner Forest Colored Non-Crossing · Philipp Kindermann LG Theoretische Informatik FernUniversit at in Hagen Joint work with Sergey Bereg, Krzysztof Fleszar, Sergey Pupyrev,](https://reader033.vdokument.com/reader033/viewer/2022060704/60709f9f2bbd466d621802c1/html5/thumbnails/100.jpg)
Putting Things Together
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
![Page 101: Euclidean Steiner Forest Colored Non-Crossing · Philipp Kindermann LG Theoretische Informatik FernUniversit at in Hagen Joint work with Sergey Bereg, Krzysztof Fleszar, Sergey Pupyrev,](https://reader033.vdokument.com/reader033/viewer/2022060704/60709f9f2bbd466d621802c1/html5/thumbnails/101.jpg)
Putting Things Together
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
![Page 102: Euclidean Steiner Forest Colored Non-Crossing · Philipp Kindermann LG Theoretische Informatik FernUniversit at in Hagen Joint work with Sergey Bereg, Krzysztof Fleszar, Sergey Pupyrev,](https://reader033.vdokument.com/reader033/viewer/2022060704/60709f9f2bbd466d621802c1/html5/thumbnails/102.jpg)
Putting Things Together
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)
![Page 103: Euclidean Steiner Forest Colored Non-Crossing · Philipp Kindermann LG Theoretische Informatik FernUniversit at in Hagen Joint work with Sergey Bereg, Krzysztof Fleszar, Sergey Pupyrev,](https://reader033.vdokument.com/reader033/viewer/2022060704/60709f9f2bbd466d621802c1/html5/thumbnails/103.jpg)
Putting Things Together
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) 2O(log n/ε) = nO(1/ε)
![Page 104: Euclidean Steiner Forest Colored Non-Crossing · Philipp Kindermann LG Theoretische Informatik FernUniversit at in Hagen Joint work with Sergey Bereg, Krzysztof Fleszar, Sergey Pupyrev,](https://reader033.vdokument.com/reader033/viewer/2022060704/60709f9f2bbd466d621802c1/html5/thumbnails/104.jpg)
Putting Things Together
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) 2O(log n/ε) = nO(1/ε)
CO(log n/ε) = nO(1/ε)
![Page 105: Euclidean Steiner Forest Colored Non-Crossing · Philipp Kindermann LG Theoretische Informatik FernUniversit at in Hagen Joint work with Sergey Bereg, Krzysztof Fleszar, Sergey Pupyrev,](https://reader033.vdokument.com/reader033/viewer/2022060704/60709f9f2bbd466d621802c1/html5/thumbnails/105.jpg)
Putting Things Together
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
Base case: unit square
O(n2) 2O(log n/ε) = nO(1/ε)
CO(log n/ε) = nO(1/ε)
![Page 106: Euclidean Steiner Forest Colored Non-Crossing · Philipp Kindermann LG Theoretische Informatik FernUniversit at in Hagen Joint work with Sergey Bereg, Krzysztof Fleszar, Sergey Pupyrev,](https://reader033.vdokument.com/reader033/viewer/2022060704/60709f9f2bbd466d621802c1/html5/thumbnails/106.jpg)
Putting Things Together
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
Base case: unit square• portals (and points) only in corners
O(n2) 2O(log n/ε) = nO(1/ε)
CO(log n/ε) = nO(1/ε)
![Page 107: Euclidean Steiner Forest Colored Non-Crossing · Philipp Kindermann LG Theoretische Informatik FernUniversit at in Hagen Joint work with Sergey Bereg, Krzysztof Fleszar, Sergey Pupyrev,](https://reader033.vdokument.com/reader033/viewer/2022060704/60709f9f2bbd466d621802c1/html5/thumbnails/107.jpg)
Putting Things Together
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
Base case: unit square• portals (and points) only in corners• solve with PTAS for EST
O(n2) 2O(log n/ε) = nO(1/ε)
CO(log n/ε) = nO(1/ε)
![Page 108: Euclidean Steiner Forest Colored Non-Crossing · Philipp Kindermann LG Theoretische Informatik FernUniversit at in Hagen Joint work with Sergey Bereg, Krzysztof Fleszar, Sergey Pupyrev,](https://reader033.vdokument.com/reader033/viewer/2022060704/60709f9f2bbd466d621802c1/html5/thumbnails/108.jpg)
Putting Things Together
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
Base case: unit square• portals (and points) only in corners• solve with PTAS for EST
Composite squares:
O(n2) 2O(log n/ε) = nO(1/ε)
CO(log n/ε) = nO(1/ε)
![Page 109: Euclidean Steiner Forest Colored Non-Crossing · Philipp Kindermann LG Theoretische Informatik FernUniversit at in Hagen Joint work with Sergey Bereg, Krzysztof Fleszar, Sergey Pupyrev,](https://reader033.vdokument.com/reader033/viewer/2022060704/60709f9f2bbd466d621802c1/html5/thumbnails/109.jpg)
Putting Things Together
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
Base case: unit square• portals (and points) only in corners• solve with PTAS for EST
Composite squares:• divide into squares (acc. to quadtree)
O(n2) 2O(log n/ε) = nO(1/ε)
CO(log n/ε) = nO(1/ε)
![Page 110: Euclidean Steiner Forest Colored Non-Crossing · Philipp Kindermann LG Theoretische Informatik FernUniversit at in Hagen Joint work with Sergey Bereg, Krzysztof Fleszar, Sergey Pupyrev,](https://reader033.vdokument.com/reader033/viewer/2022060704/60709f9f2bbd466d621802c1/html5/thumbnails/110.jpg)
Putting Things Together
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
Base case: unit square• portals (and points) only in corners• solve with PTAS for EST
Composite squares:• divide into squares (acc. to quadtree)• solve each combination of nO(1/ε)
compatible subproblems
O(n2) 2O(log n/ε) = nO(1/ε)
CO(log n/ε) = nO(1/ε)
![Page 111: Euclidean Steiner Forest Colored Non-Crossing · Philipp Kindermann LG Theoretische Informatik FernUniversit at in Hagen Joint work with Sergey Bereg, Krzysztof Fleszar, Sergey Pupyrev,](https://reader033.vdokument.com/reader033/viewer/2022060704/60709f9f2bbd466d621802c1/html5/thumbnails/111.jpg)
Putting Things Together
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
Base case: unit square• portals (and points) only in corners• solve with PTAS for EST
Composite squares:• divide into squares (acc. to quadtree)• solve each combination of nO(1/ε)
compatible subproblems
2-CESF admits a PTAS.
O(n2) 2O(log n/ε) = nO(1/ε)
CO(log n/ε) = nO(1/ε)
![Page 112: Euclidean Steiner Forest Colored Non-Crossing · Philipp Kindermann LG Theoretische Informatik FernUniversit at in Hagen Joint work with Sergey Bereg, Krzysztof Fleszar, Sergey Pupyrev,](https://reader033.vdokument.com/reader033/viewer/2022060704/60709f9f2bbd466d621802c1/html5/thumbnails/112.jpg)
3-CESF
![Page 113: Euclidean Steiner Forest Colored Non-Crossing · Philipp Kindermann LG Theoretische Informatik FernUniversit at in Hagen Joint work with Sergey Bereg, Krzysztof Fleszar, Sergey Pupyrev,](https://reader033.vdokument.com/reader033/viewer/2022060704/60709f9f2bbd466d621802c1/html5/thumbnails/113.jpg)
3-CESF
![Page 114: Euclidean Steiner Forest Colored Non-Crossing · Philipp Kindermann LG Theoretische Informatik FernUniversit at in Hagen Joint work with Sergey Bereg, Krzysztof Fleszar, Sergey Pupyrev,](https://reader033.vdokument.com/reader033/viewer/2022060704/60709f9f2bbd466d621802c1/html5/thumbnails/114.jpg)
3-CESF
![Page 115: Euclidean Steiner Forest Colored Non-Crossing · Philipp Kindermann LG Theoretische Informatik FernUniversit at in Hagen Joint work with Sergey Bereg, Krzysztof Fleszar, Sergey Pupyrev,](https://reader033.vdokument.com/reader033/viewer/2022060704/60709f9f2bbd466d621802c1/html5/thumbnails/115.jpg)
3-CESF
3-CESF admits a (5/3 + ε)-approximation algorithm.
![Page 116: Euclidean Steiner Forest Colored Non-Crossing · Philipp Kindermann LG Theoretische Informatik FernUniversit at in Hagen Joint work with Sergey Bereg, Krzysztof Fleszar, Sergey Pupyrev,](https://reader033.vdokument.com/reader033/viewer/2022060704/60709f9f2bbd466d621802c1/html5/thumbnails/116.jpg)
k-CESF
![Page 117: Euclidean Steiner Forest Colored Non-Crossing · Philipp Kindermann LG Theoretische Informatik FernUniversit at in Hagen Joint work with Sergey Bereg, Krzysztof Fleszar, Sergey Pupyrev,](https://reader033.vdokument.com/reader033/viewer/2022060704/60709f9f2bbd466d621802c1/html5/thumbnails/117.jpg)
k-CESF
• split into 2 groups
![Page 118: Euclidean Steiner Forest Colored Non-Crossing · Philipp Kindermann LG Theoretische Informatik FernUniversit at in Hagen Joint work with Sergey Bereg, Krzysztof Fleszar, Sergey Pupyrev,](https://reader033.vdokument.com/reader033/viewer/2022060704/60709f9f2bbd466d621802c1/html5/thumbnails/118.jpg)
k-CESF
• split into 2 groups• use PTAS for the groups
![Page 119: Euclidean Steiner Forest Colored Non-Crossing · Philipp Kindermann LG Theoretische Informatik FernUniversit at in Hagen Joint work with Sergey Bereg, Krzysztof Fleszar, Sergey Pupyrev,](https://reader033.vdokument.com/reader033/viewer/2022060704/60709f9f2bbd466d621802c1/html5/thumbnails/119.jpg)
k-CESF
• split into 2 groups• use PTAS for the groups
![Page 120: Euclidean Steiner Forest Colored Non-Crossing · Philipp Kindermann LG Theoretische Informatik FernUniversit at in Hagen Joint work with Sergey Bereg, Krzysztof Fleszar, Sergey Pupyrev,](https://reader033.vdokument.com/reader033/viewer/2022060704/60709f9f2bbd466d621802c1/html5/thumbnails/120.jpg)
k-CESF
• split into 2 groups• use PTAS for the groups• Construct trees fromthis “super-tree”
![Page 121: Euclidean Steiner Forest Colored Non-Crossing · Philipp Kindermann LG Theoretische Informatik FernUniversit at in Hagen Joint work with Sergey Bereg, Krzysztof Fleszar, Sergey Pupyrev,](https://reader033.vdokument.com/reader033/viewer/2022060704/60709f9f2bbd466d621802c1/html5/thumbnails/121.jpg)
k-CESF
• split into 2 groups• use PTAS for the groups• Construct trees fromthis “super-tree”
![Page 122: Euclidean Steiner Forest Colored Non-Crossing · Philipp Kindermann LG Theoretische Informatik FernUniversit at in Hagen Joint work with Sergey Bereg, Krzysztof Fleszar, Sergey Pupyrev,](https://reader033.vdokument.com/reader033/viewer/2022060704/60709f9f2bbd466d621802c1/html5/thumbnails/122.jpg)
k-CESF
• split into 2 groups• use PTAS for the groups• Construct trees fromthis “super-tree”
![Page 123: Euclidean Steiner Forest Colored Non-Crossing · Philipp Kindermann LG Theoretische Informatik FernUniversit at in Hagen Joint work with Sergey Bereg, Krzysztof Fleszar, Sergey Pupyrev,](https://reader033.vdokument.com/reader033/viewer/2022060704/60709f9f2bbd466d621802c1/html5/thumbnails/123.jpg)
k-CESF
• split into 2 groups• use PTAS for the groups• Construct trees fromthis “super-tree”
![Page 124: Euclidean Steiner Forest Colored Non-Crossing · Philipp Kindermann LG Theoretische Informatik FernUniversit at in Hagen Joint work with Sergey Bereg, Krzysztof Fleszar, Sergey Pupyrev,](https://reader033.vdokument.com/reader033/viewer/2022060704/60709f9f2bbd466d621802c1/html5/thumbnails/124.jpg)
k-CESF
• split into 2 groups• use PTAS for the groups• Construct trees fromthis “super-tree”
![Page 125: Euclidean Steiner Forest Colored Non-Crossing · Philipp Kindermann LG Theoretische Informatik FernUniversit at in Hagen Joint work with Sergey Bereg, Krzysztof Fleszar, Sergey Pupyrev,](https://reader033.vdokument.com/reader033/viewer/2022060704/60709f9f2bbd466d621802c1/html5/thumbnails/125.jpg)
k-CESF
• split into 2 groups• use PTAS for the groups• Construct trees fromthis “super-tree”
![Page 126: Euclidean Steiner Forest Colored Non-Crossing · Philipp Kindermann LG Theoretische Informatik FernUniversit at in Hagen Joint work with Sergey Bereg, Krzysztof Fleszar, Sergey Pupyrev,](https://reader033.vdokument.com/reader033/viewer/2022060704/60709f9f2bbd466d621802c1/html5/thumbnails/126.jpg)
k-CESF
• split into 2 groups• use PTAS for the groups• Construct trees fromthis “super-tree”
![Page 127: Euclidean Steiner Forest Colored Non-Crossing · Philipp Kindermann LG Theoretische Informatik FernUniversit at in Hagen Joint work with Sergey Bereg, Krzysztof Fleszar, Sergey Pupyrev,](https://reader033.vdokument.com/reader033/viewer/2022060704/60709f9f2bbd466d621802c1/html5/thumbnails/127.jpg)
k-CESF
• split into 2 groups• use PTAS for the groups• Construct trees fromthis “super-tree”
![Page 128: Euclidean Steiner Forest Colored Non-Crossing · Philipp Kindermann LG Theoretische Informatik FernUniversit at in Hagen Joint work with Sergey Bereg, Krzysztof Fleszar, Sergey Pupyrev,](https://reader033.vdokument.com/reader033/viewer/2022060704/60709f9f2bbd466d621802c1/html5/thumbnails/128.jpg)
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