d aniel gerbner - 上海交通大学数学系d aniel gerbner r enyi institute joint work with viola...
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![Page 1: D aniel Gerbner - 上海交通大学数学系D aniel Gerbner R enyi Institute Joint work with Viola M esz aros, D om ot or P alv olgyi, Alexey Pokrovskiy and Gunter Rote D aniel Gerbner](https://reader033.vdocument.in/reader033/viewer/2022052409/60a0036b35193b63873aa4b3/html5/thumbnails/1.jpg)
Advantage in the discrete Voronoi game
Advantage in the discrete Voronoi game
Daniel Gerbner
Renyi Institute
Joint work with Viola Meszaros, Domotor Palvolgyi, AlexeyPokrovskiy and Gunter Rote
Daniel Gerbner Advantage in the discrete Voronoi game
![Page 2: D aniel Gerbner - 上海交通大学数学系D aniel Gerbner R enyi Institute Joint work with Viola M esz aros, D om ot or P alv olgyi, Alexey Pokrovskiy and Gunter Rote D aniel Gerbner](https://reader033.vdocument.in/reader033/viewer/2022052409/60a0036b35193b63873aa4b3/html5/thumbnails/2.jpg)
Advantage in the discrete Voronoi game Daniel Gerbner
What is the Voronoi game?
Applet by Jens Anuth.
Two players, First and Second claim points alternating for trounds.
At end area is divided, each point goes to closest claimed.
Daniel Gerbner Advantage in the discrete Voronoi game
![Page 3: D aniel Gerbner - 上海交通大学数学系D aniel Gerbner R enyi Institute Joint work with Viola M esz aros, D om ot or P alv olgyi, Alexey Pokrovskiy and Gunter Rote D aniel Gerbner](https://reader033.vdocument.in/reader033/viewer/2022052409/60a0036b35193b63873aa4b3/html5/thumbnails/3.jpg)
Advantage in the discrete Voronoi game Daniel Gerbner
What is the Voronoi game?
Applet by Jens Anuth.
Two players, First and Second claim points alternating for trounds.
At end area is divided, each point goes to closest claimed.
Daniel Gerbner Advantage in the discrete Voronoi game
![Page 4: D aniel Gerbner - 上海交通大学数学系D aniel Gerbner R enyi Institute Joint work with Viola M esz aros, D om ot or P alv olgyi, Alexey Pokrovskiy and Gunter Rote D aniel Gerbner](https://reader033.vdocument.in/reader033/viewer/2022052409/60a0036b35193b63873aa4b3/html5/thumbnails/4.jpg)
Advantage in the discrete Voronoi game Daniel Gerbner
What is the Voronoi game?
Applet by Jens Anuth.
Two players, First and Second claim points alternating for trounds.
At end area is divided, each point goes to closest claimed.
Daniel Gerbner Advantage in the discrete Voronoi game
![Page 5: D aniel Gerbner - 上海交通大学数学系D aniel Gerbner R enyi Institute Joint work with Viola M esz aros, D om ot or P alv olgyi, Alexey Pokrovskiy and Gunter Rote D aniel Gerbner](https://reader033.vdocument.in/reader033/viewer/2022052409/60a0036b35193b63873aa4b3/html5/thumbnails/5.jpg)
Advantage in the discrete Voronoi game Daniel Gerbner
What is the Voronoi game?
Applet by Jens Anuth.
Two players, First and Second claim points alternating for trounds.
At end area is divided, each point goes to closest claimed.
Daniel Gerbner Advantage in the discrete Voronoi game
![Page 6: D aniel Gerbner - 上海交通大学数学系D aniel Gerbner R enyi Institute Joint work with Viola M esz aros, D om ot or P alv olgyi, Alexey Pokrovskiy and Gunter Rote D aniel Gerbner](https://reader033.vdocument.in/reader033/viewer/2022052409/60a0036b35193b63873aa4b3/html5/thumbnails/6.jpg)
Advantage in the discrete Voronoi game Daniel Gerbner
Competitive facility location problem
Two chains of supermarkets build shops in a city.
The customers always go to the nearest shop.
Daniel Gerbner Advantage in the discrete Voronoi game
![Page 7: D aniel Gerbner - 上海交通大学数学系D aniel Gerbner R enyi Institute Joint work with Viola M esz aros, D om ot or P alv olgyi, Alexey Pokrovskiy and Gunter Rote D aniel Gerbner](https://reader033.vdocument.in/reader033/viewer/2022052409/60a0036b35193b63873aa4b3/html5/thumbnails/7.jpg)
Advantage in the discrete Voronoi game Daniel Gerbner
Competitive facility location problem
Two chains of supermarkets build shops in a city.
The customers always go to the nearest shop.
Daniel Gerbner Advantage in the discrete Voronoi game
![Page 8: D aniel Gerbner - 上海交通大学数学系D aniel Gerbner R enyi Institute Joint work with Viola M esz aros, D om ot or P alv olgyi, Alexey Pokrovskiy and Gunter Rote D aniel Gerbner](https://reader033.vdocument.in/reader033/viewer/2022052409/60a0036b35193b63873aa4b3/html5/thumbnails/8.jpg)
Advantage in the discrete Voronoi game Daniel Gerbner
Competitive facility location problem
Two chains of supermarkets build shops in a city.
The customers always go to the nearest shop.
Daniel Gerbner Advantage in the discrete Voronoi game
![Page 9: D aniel Gerbner - 上海交通大学数学系D aniel Gerbner R enyi Institute Joint work with Viola M esz aros, D om ot or P alv olgyi, Alexey Pokrovskiy and Gunter Rote D aniel Gerbner](https://reader033.vdocument.in/reader033/viewer/2022052409/60a0036b35193b63873aa4b3/html5/thumbnails/9.jpg)
Advantage in the discrete Voronoi game Daniel Gerbner
What is the discrete Voronoi game?
Same on a graph!
Players can claim only vertices and vertices are divided at end.
So the above game ended in a draw.
Theorem (Kiyomi, Saitoh, Uehara)
Game on path is a draw unless odd vertices and t = 1.
Moreover, even then First wins with only one.
Daniel Gerbner Advantage in the discrete Voronoi game
![Page 10: D aniel Gerbner - 上海交通大学数学系D aniel Gerbner R enyi Institute Joint work with Viola M esz aros, D om ot or P alv olgyi, Alexey Pokrovskiy and Gunter Rote D aniel Gerbner](https://reader033.vdocument.in/reader033/viewer/2022052409/60a0036b35193b63873aa4b3/html5/thumbnails/10.jpg)
Advantage in the discrete Voronoi game Daniel Gerbner
What is the discrete Voronoi game?
Same on a graph!
Players can claim only vertices and vertices are divided at end.
So the above game ended in a draw.
Theorem (Kiyomi, Saitoh, Uehara)
Game on path is a draw unless odd vertices and t = 1.
Moreover, even then First wins with only one.
Daniel Gerbner Advantage in the discrete Voronoi game
![Page 11: D aniel Gerbner - 上海交通大学数学系D aniel Gerbner R enyi Institute Joint work with Viola M esz aros, D om ot or P alv olgyi, Alexey Pokrovskiy and Gunter Rote D aniel Gerbner](https://reader033.vdocument.in/reader033/viewer/2022052409/60a0036b35193b63873aa4b3/html5/thumbnails/11.jpg)
Advantage in the discrete Voronoi game Daniel Gerbner
What is the discrete Voronoi game?
Same on a graph!
Players can claim only vertices and vertices are divided at end.
So the above game ended in a draw.
Theorem (Kiyomi, Saitoh, Uehara)
Game on path is a draw unless odd vertices and t = 1.
Moreover, even then First wins with only one.
Daniel Gerbner Advantage in the discrete Voronoi game
![Page 12: D aniel Gerbner - 上海交通大学数学系D aniel Gerbner R enyi Institute Joint work with Viola M esz aros, D om ot or P alv olgyi, Alexey Pokrovskiy and Gunter Rote D aniel Gerbner](https://reader033.vdocument.in/reader033/viewer/2022052409/60a0036b35193b63873aa4b3/html5/thumbnails/12.jpg)
Advantage in the discrete Voronoi game Daniel Gerbner
What is the discrete Voronoi game?
Same on a graph!
Players can claim only vertices and vertices are divided at end.
So the above game ended in a draw.
Theorem (Kiyomi, Saitoh, Uehara)
Game on path is a draw unless odd vertices and t = 1.
Moreover, even then First wins with only one.
Daniel Gerbner Advantage in the discrete Voronoi game
![Page 13: D aniel Gerbner - 上海交通大学数学系D aniel Gerbner R enyi Institute Joint work with Viola M esz aros, D om ot or P alv olgyi, Alexey Pokrovskiy and Gunter Rote D aniel Gerbner](https://reader033.vdocument.in/reader033/viewer/2022052409/60a0036b35193b63873aa4b3/html5/thumbnails/13.jpg)
Advantage in the discrete Voronoi game Daniel Gerbner
What is the discrete Voronoi game?
Same on a graph!
Players can claim only vertices and vertices are divided at end.
So the above game ended in a draw.
Theorem (Kiyomi, Saitoh, Uehara)
Game on path is a draw unless odd vertices and t = 1.
Moreover, even then First wins with only one.
Daniel Gerbner Advantage in the discrete Voronoi game
![Page 14: D aniel Gerbner - 上海交通大学数学系D aniel Gerbner R enyi Institute Joint work with Viola M esz aros, D om ot or P alv olgyi, Alexey Pokrovskiy and Gunter Rote D aniel Gerbner](https://reader033.vdocument.in/reader033/viewer/2022052409/60a0036b35193b63873aa4b3/html5/thumbnails/14.jpg)
Advantage in the discrete Voronoi game Daniel Gerbner
What is the discrete Voronoi game?
Same on a graph!
Players can claim only vertices and vertices are divided at end.
So the above game ended in a draw.
Theorem (Kiyomi, Saitoh, Uehara)
Game on path is a draw unless odd vertices and t = 1.
Moreover, even then First wins with only one.
Daniel Gerbner Advantage in the discrete Voronoi game
![Page 15: D aniel Gerbner - 上海交通大学数学系D aniel Gerbner R enyi Institute Joint work with Viola M esz aros, D om ot or P alv olgyi, Alexey Pokrovskiy and Gunter Rote D aniel Gerbner](https://reader033.vdocument.in/reader033/viewer/2022052409/60a0036b35193b63873aa4b3/html5/thumbnails/15.jpg)
Advantage in the discrete Voronoi game Daniel Gerbner
What is the discrete Voronoi game?
Same on a graph!
Players can claim only vertices and vertices are divided at end.
So the above game ended in a draw.
Theorem (Kiyomi, Saitoh, Uehara)
Game on path is a draw unless odd vertices and t = 1.
Moreover, even then First wins with only one.
Daniel Gerbner Advantage in the discrete Voronoi game
![Page 16: D aniel Gerbner - 上海交通大学数学系D aniel Gerbner R enyi Institute Joint work with Viola M esz aros, D om ot or P alv olgyi, Alexey Pokrovskiy and Gunter Rote D aniel Gerbner](https://reader033.vdocument.in/reader033/viewer/2022052409/60a0036b35193b63873aa4b3/html5/thumbnails/16.jpg)
Advantage in the discrete Voronoi game Daniel Gerbner
What is the discrete Voronoi game?
Same on a graph!
Players can claim only vertices and vertices are divided at end.
So the above game ended in a draw.
Theorem (Kiyomi, Saitoh, Uehara)
Game on path is a draw unless odd vertices and t = 1.
Moreover, even then First wins with only one.
Daniel Gerbner Advantage in the discrete Voronoi game
![Page 17: D aniel Gerbner - 上海交通大学数学系D aniel Gerbner R enyi Institute Joint work with Viola M esz aros, D om ot or P alv olgyi, Alexey Pokrovskiy and Gunter Rote D aniel Gerbner](https://reader033.vdocument.in/reader033/viewer/2022052409/60a0036b35193b63873aa4b3/html5/thumbnails/17.jpg)
Advantage in the discrete Voronoi game Daniel Gerbner
What is the discrete Voronoi game?
Same on a graph!
Players can claim only vertices and vertices are divided at end.
So the above game ended in a draw.
Theorem (Kiyomi, Saitoh, Uehara)
Game on path is a draw unless odd vertices and t = 1.
Moreover, even then First wins with only one.
Daniel Gerbner Advantage in the discrete Voronoi game
![Page 18: D aniel Gerbner - 上海交通大学数学系D aniel Gerbner R enyi Institute Joint work with Viola M esz aros, D om ot or P alv olgyi, Alexey Pokrovskiy and Gunter Rote D aniel Gerbner](https://reader033.vdocument.in/reader033/viewer/2022052409/60a0036b35193b63873aa4b3/html5/thumbnails/18.jpg)
Advantage in the discrete Voronoi game Daniel Gerbner
What is the discrete Voronoi game?
Same on a graph!
Players can claim only vertices and vertices are divided at end.
So the above game ended in a draw.
Theorem (Kiyomi, Saitoh, Uehara)
Game on path is a draw unless odd vertices and t = 1.
Moreover, even then First wins with only one.
Daniel Gerbner Advantage in the discrete Voronoi game
![Page 19: D aniel Gerbner - 上海交通大学数学系D aniel Gerbner R enyi Institute Joint work with Viola M esz aros, D om ot or P alv olgyi, Alexey Pokrovskiy and Gunter Rote D aniel Gerbner](https://reader033.vdocument.in/reader033/viewer/2022052409/60a0036b35193b63873aa4b3/html5/thumbnails/19.jpg)
Advantage in the discrete Voronoi game Daniel Gerbner
What is the discrete Voronoi game?
Same on a graph!
Players can claim only vertices and vertices are divided at end.
So the above game ended in a draw.
Theorem (Kiyomi, Saitoh, Uehara)
Game on path is a draw unless odd vertices and t = 1.Moreover, even then First wins with only one.
Daniel Gerbner Advantage in the discrete Voronoi game
![Page 20: D aniel Gerbner - 上海交通大学数学系D aniel Gerbner R enyi Institute Joint work with Viola M esz aros, D om ot or P alv olgyi, Alexey Pokrovskiy and Gunter Rote D aniel Gerbner](https://reader033.vdocument.in/reader033/viewer/2022052409/60a0036b35193b63873aa4b3/html5/thumbnails/20.jpg)
Advantage in the discrete Voronoi game Daniel Gerbner
What percentage can each player get?
Definition
VR(G , t) =# Closer to First + 1
2# Tied
n
Goal: Bound VR(G , t) for certain graph(family)
VR(path, t) ≥ 12 and → 1
2
VR(star , t)→ 1
Daniel Gerbner Advantage in the discrete Voronoi game
![Page 21: D aniel Gerbner - 上海交通大学数学系D aniel Gerbner R enyi Institute Joint work with Viola M esz aros, D om ot or P alv olgyi, Alexey Pokrovskiy and Gunter Rote D aniel Gerbner](https://reader033.vdocument.in/reader033/viewer/2022052409/60a0036b35193b63873aa4b3/html5/thumbnails/21.jpg)
Advantage in the discrete Voronoi game Daniel Gerbner
What percentage can each player get?
Definition
VR(G , t) =# Closer to First + 1
2# Tied
n
Goal: Bound VR(G , t) for certain graph(family)
VR(path, t) ≥ 12 and → 1
2
VR(star , t)→ 1
Daniel Gerbner Advantage in the discrete Voronoi game
![Page 22: D aniel Gerbner - 上海交通大学数学系D aniel Gerbner R enyi Institute Joint work with Viola M esz aros, D om ot or P alv olgyi, Alexey Pokrovskiy and Gunter Rote D aniel Gerbner](https://reader033.vdocument.in/reader033/viewer/2022052409/60a0036b35193b63873aa4b3/html5/thumbnails/22.jpg)
Advantage in the discrete Voronoi game Daniel Gerbner
What percentage can each player get?
Definition
VR(G , t) =# Closer to First + 1
2# Tied
n
Goal: Bound VR(G , t) for certain graph(family)
VR(path, t) ≥ 12 and → 1
2
VR(star , t)→ 1
Daniel Gerbner Advantage in the discrete Voronoi game
![Page 23: D aniel Gerbner - 上海交通大学数学系D aniel Gerbner R enyi Institute Joint work with Viola M esz aros, D om ot or P alv olgyi, Alexey Pokrovskiy and Gunter Rote D aniel Gerbner](https://reader033.vdocument.in/reader033/viewer/2022052409/60a0036b35193b63873aa4b3/html5/thumbnails/23.jpg)
Advantage in the discrete Voronoi game Daniel Gerbner
What percentage can each player get?
Definition
VR(G , t) =# Closer to First + 1
2# Tied
n
Goal: Bound VR(G , t) for certain graph(family)
VR(path, t) ≥ 12 and → 1
2
VR(star , t)→ 1
Daniel Gerbner Advantage in the discrete Voronoi game
![Page 24: D aniel Gerbner - 上海交通大学数学系D aniel Gerbner R enyi Institute Joint work with Viola M esz aros, D om ot or P alv olgyi, Alexey Pokrovskiy and Gunter Rote D aniel Gerbner](https://reader033.vdocument.in/reader033/viewer/2022052409/60a0036b35193b63873aa4b3/html5/thumbnails/24.jpg)
Advantage in the discrete Voronoi game Daniel Gerbner
What percentage can each player get?
Definition
VR(G , t) =# Closer to First + 1
2# Tied
n
Goal: Bound VR(G , t) for certain graph(family)
VR(path, t) ≥ 12 and → 1
2
VR(star , t)→ 1
Daniel Gerbner Advantage in the discrete Voronoi game
![Page 25: D aniel Gerbner - 上海交通大学数学系D aniel Gerbner R enyi Institute Joint work with Viola M esz aros, D om ot or P alv olgyi, Alexey Pokrovskiy and Gunter Rote D aniel Gerbner](https://reader033.vdocument.in/reader033/viewer/2022052409/60a0036b35193b63873aa4b3/html5/thumbnails/25.jpg)
Advantage in the discrete Voronoi game Daniel Gerbner
What percentage can each player get?
Definition
VR(G , t) =# Closer to First + 1
2# Tied
n
Goal: Bound VR(G , t) for certain graph(family)
VR(path, t) ≥ 12 and → 1
2
VR(star , t)→ 1
Daniel Gerbner Advantage in the discrete Voronoi game
![Page 26: D aniel Gerbner - 上海交通大学数学系D aniel Gerbner R enyi Institute Joint work with Viola M esz aros, D om ot or P alv olgyi, Alexey Pokrovskiy and Gunter Rote D aniel Gerbner](https://reader033.vdocument.in/reader033/viewer/2022052409/60a0036b35193b63873aa4b3/html5/thumbnails/26.jpg)
Advantage in the discrete Voronoi game Daniel Gerbner
What percentage can each player get?
Definition
VR(G , t) =# Closer to First + 1
2# Tied
n
Goal: Bound VR(G , t) for certain graph(family)
VR(path, t) ≥ 12 and → 1
2
VR(star , t)→ 1
Daniel Gerbner Advantage in the discrete Voronoi game
![Page 27: D aniel Gerbner - 上海交通大学数学系D aniel Gerbner R enyi Institute Joint work with Viola M esz aros, D om ot or P alv olgyi, Alexey Pokrovskiy and Gunter Rote D aniel Gerbner](https://reader033.vdocument.in/reader033/viewer/2022052409/60a0036b35193b63873aa4b3/html5/thumbnails/27.jpg)
Advantage in the discrete Voronoi game Daniel Gerbner
What percentage can each player get?
Definition
VR(G , t) =# Closer to First + 1
2# Tied
n
Goal: Bound VR(G , t) for certain graph(family)
VR(path, t) ≥ 12 and → 1
2
VR(star , t)→ 1
Daniel Gerbner Advantage in the discrete Voronoi game
![Page 28: D aniel Gerbner - 上海交通大学数学系D aniel Gerbner R enyi Institute Joint work with Viola M esz aros, D om ot or P alv olgyi, Alexey Pokrovskiy and Gunter Rote D aniel Gerbner](https://reader033.vdocument.in/reader033/viewer/2022052409/60a0036b35193b63873aa4b3/html5/thumbnails/28.jpg)
Advantage in the discrete Voronoi game Daniel Gerbner
What percentage can each player get?
Definition
VR(G , t) =# Closer to First + 1
2# Tied
n
Goal: Bound VR(G , t) for certain graph(family)
VR(path, t) ≥ 12 and → 1
2
VR(star , t)→ 1
Daniel Gerbner Advantage in the discrete Voronoi game
![Page 29: D aniel Gerbner - 上海交通大学数学系D aniel Gerbner R enyi Institute Joint work with Viola M esz aros, D om ot or P alv olgyi, Alexey Pokrovskiy and Gunter Rote D aniel Gerbner](https://reader033.vdocument.in/reader033/viewer/2022052409/60a0036b35193b63873aa4b3/html5/thumbnails/29.jpg)
Advantage in the discrete Voronoi game Daniel Gerbner
What percentage can each player get?
Definition
VR(G , t) =# Closer to First + 1
2# Tied
n
Goal: Bound VR(G , t) for certain graph(family)
VR(path, t) ≥ 12 and → 1
2
VR(star , t)→ 1
Daniel Gerbner Advantage in the discrete Voronoi game
![Page 30: D aniel Gerbner - 上海交通大学数学系D aniel Gerbner R enyi Institute Joint work with Viola M esz aros, D om ot or P alv olgyi, Alexey Pokrovskiy and Gunter Rote D aniel Gerbner](https://reader033.vdocument.in/reader033/viewer/2022052409/60a0036b35193b63873aa4b3/html5/thumbnails/30.jpg)
Advantage in the discrete Voronoi game Daniel Gerbner
Can the second player win?
Daniel Gerbner Advantage in the discrete Voronoi game
![Page 31: D aniel Gerbner - 上海交通大学数学系D aniel Gerbner R enyi Institute Joint work with Viola M esz aros, D om ot or P alv olgyi, Alexey Pokrovskiy and Gunter Rote D aniel Gerbner](https://reader033.vdocument.in/reader033/viewer/2022052409/60a0036b35193b63873aa4b3/html5/thumbnails/31.jpg)
Advantage in the discrete Voronoi game Daniel Gerbner
Can the second player win?
Daniel Gerbner Advantage in the discrete Voronoi game
![Page 32: D aniel Gerbner - 上海交通大学数学系D aniel Gerbner R enyi Institute Joint work with Viola M esz aros, D om ot or P alv olgyi, Alexey Pokrovskiy and Gunter Rote D aniel Gerbner](https://reader033.vdocument.in/reader033/viewer/2022052409/60a0036b35193b63873aa4b3/html5/thumbnails/32.jpg)
Advantage in the discrete Voronoi game Daniel Gerbner
Can the second player win?
Daniel Gerbner Advantage in the discrete Voronoi game
![Page 33: D aniel Gerbner - 上海交通大学数学系D aniel Gerbner R enyi Institute Joint work with Viola M esz aros, D om ot or P alv olgyi, Alexey Pokrovskiy and Gunter Rote D aniel Gerbner](https://reader033.vdocument.in/reader033/viewer/2022052409/60a0036b35193b63873aa4b3/html5/thumbnails/33.jpg)
Advantage in the discrete Voronoi game Daniel Gerbner
Can the second player win?
Daniel Gerbner Advantage in the discrete Voronoi game
![Page 34: D aniel Gerbner - 上海交通大学数学系D aniel Gerbner R enyi Institute Joint work with Viola M esz aros, D om ot or P alv olgyi, Alexey Pokrovskiy and Gunter Rote D aniel Gerbner](https://reader033.vdocument.in/reader033/viewer/2022052409/60a0036b35193b63873aa4b3/html5/thumbnails/34.jpg)
Advantage in the discrete Voronoi game Daniel Gerbner
Can the second player win?
Daniel Gerbner Advantage in the discrete Voronoi game
![Page 35: D aniel Gerbner - 上海交通大学数学系D aniel Gerbner R enyi Institute Joint work with Viola M esz aros, D om ot or P alv olgyi, Alexey Pokrovskiy and Gunter Rote D aniel Gerbner](https://reader033.vdocument.in/reader033/viewer/2022052409/60a0036b35193b63873aa4b3/html5/thumbnails/35.jpg)
Advantage in the discrete Voronoi game Daniel Gerbner
Can the second player win?
Daniel Gerbner Advantage in the discrete Voronoi game
![Page 36: D aniel Gerbner - 上海交通大学数学系D aniel Gerbner R enyi Institute Joint work with Viola M esz aros, D om ot or P alv olgyi, Alexey Pokrovskiy and Gunter Rote D aniel Gerbner](https://reader033.vdocument.in/reader033/viewer/2022052409/60a0036b35193b63873aa4b3/html5/thumbnails/36.jpg)
Advantage in the discrete Voronoi game Daniel Gerbner
Can the second player win?
Daniel Gerbner Advantage in the discrete Voronoi game
![Page 37: D aniel Gerbner - 上海交通大学数学系D aniel Gerbner R enyi Institute Joint work with Viola M esz aros, D om ot or P alv olgyi, Alexey Pokrovskiy and Gunter Rote D aniel Gerbner](https://reader033.vdocument.in/reader033/viewer/2022052409/60a0036b35193b63873aa4b3/html5/thumbnails/37.jpg)
Advantage in the discrete Voronoi game Daniel Gerbner
Questions
VR(G , t) < ε?
What if t = 1?
VR(T , t) < 12 for a tree?
ClaimVR(T , 1) ≥ 1
2 for trees.
Proof.First takes center of tree.
(A vertex which cuts the tree intoconnected components of order at most n/2.)
Daniel Gerbner Advantage in the discrete Voronoi game
![Page 38: D aniel Gerbner - 上海交通大学数学系D aniel Gerbner R enyi Institute Joint work with Viola M esz aros, D om ot or P alv olgyi, Alexey Pokrovskiy and Gunter Rote D aniel Gerbner](https://reader033.vdocument.in/reader033/viewer/2022052409/60a0036b35193b63873aa4b3/html5/thumbnails/38.jpg)
Advantage in the discrete Voronoi game Daniel Gerbner
Questions
VR(G , t) < ε?
What if t = 1?
VR(T , t) < 12 for a tree?
ClaimVR(T , 1) ≥ 1
2 for trees.
Proof.First takes center of tree. (A vertex which cuts the tree intoconnected components of order at most n/2.)
Daniel Gerbner Advantage in the discrete Voronoi game
![Page 39: D aniel Gerbner - 上海交通大学数学系D aniel Gerbner R enyi Institute Joint work with Viola M esz aros, D om ot or P alv olgyi, Alexey Pokrovskiy and Gunter Rote D aniel Gerbner](https://reader033.vdocument.in/reader033/viewer/2022052409/60a0036b35193b63873aa4b3/html5/thumbnails/39.jpg)
Advantage in the discrete Voronoi game Daniel Gerbner
Questions
VR(G , t) < ε?
What if t = 1?
VR(T , t) < 12 for a tree?
ClaimVR(T , 1) ≥ 1
2 for trees.
Proof.First takes center of tree. (A vertex which cuts the tree intoconnected components of order at most n/2.)
Daniel Gerbner Advantage in the discrete Voronoi game
![Page 40: D aniel Gerbner - 上海交通大学数学系D aniel Gerbner R enyi Institute Joint work with Viola M esz aros, D om ot or P alv olgyi, Alexey Pokrovskiy and Gunter Rote D aniel Gerbner](https://reader033.vdocument.in/reader033/viewer/2022052409/60a0036b35193b63873aa4b3/html5/thumbnails/40.jpg)
Advantage in the discrete Voronoi game Daniel Gerbner
Questions
VR(G , t) < ε?
What if t = 1?
VR(T , t) < 12 for a tree?
ClaimVR(T , 1) ≥ 1
2 for trees.
Proof.First takes center of tree. (A vertex which cuts the tree intoconnected components of order at most n/2.)
Daniel Gerbner Advantage in the discrete Voronoi game
![Page 41: D aniel Gerbner - 上海交通大学数学系D aniel Gerbner R enyi Institute Joint work with Viola M esz aros, D om ot or P alv olgyi, Alexey Pokrovskiy and Gunter Rote D aniel Gerbner](https://reader033.vdocument.in/reader033/viewer/2022052409/60a0036b35193b63873aa4b3/html5/thumbnails/41.jpg)
Advantage in the discrete Voronoi game Daniel Gerbner
Questions
VR(G , t) < ε?
What if t = 1?
VR(T , t) < 12 for a tree?
ClaimVR(T , 1) ≥ 1
2 for trees.
Proof.First takes center of tree. (A vertex which cuts the tree intoconnected components of order at most n/2.)
Daniel Gerbner Advantage in the discrete Voronoi game
![Page 42: D aniel Gerbner - 上海交通大学数学系D aniel Gerbner R enyi Institute Joint work with Viola M esz aros, D om ot or P alv olgyi, Alexey Pokrovskiy and Gunter Rote D aniel Gerbner](https://reader033.vdocument.in/reader033/viewer/2022052409/60a0036b35193b63873aa4b3/html5/thumbnails/42.jpg)
Advantage in the discrete Voronoi game Daniel Gerbner
Questions
VR(G , t) < ε?
What if t = 1?
VR(T , t) < 12 for a tree?
ClaimVR(T , 1) ≥ 1
2 for trees.
Proof.First takes center of tree. (A vertex which cuts the tree intoconnected components of order at most n/2.)
Daniel Gerbner Advantage in the discrete Voronoi game
![Page 43: D aniel Gerbner - 上海交通大学数学系D aniel Gerbner R enyi Institute Joint work with Viola M esz aros, D om ot or P alv olgyi, Alexey Pokrovskiy and Gunter Rote D aniel Gerbner](https://reader033.vdocument.in/reader033/viewer/2022052409/60a0036b35193b63873aa4b3/html5/thumbnails/43.jpg)
Advantage in the discrete Voronoi game Daniel Gerbner
Trees
For any t ≥ 2 there exists a tree T with VG (T , t) < 0.34:
x
=
NNNN
2 4 8kN
c
NNNN
xlegshead
h
Basic idea: most of the legs are controlled by the player whoclaims the last vertex there; the head is controlled by the playerwho claims h.
Optimal for t = 2: VG (T , 2) > 1/3 for any tree T .
Daniel Gerbner Advantage in the discrete Voronoi game
![Page 44: D aniel Gerbner - 上海交通大学数学系D aniel Gerbner R enyi Institute Joint work with Viola M esz aros, D om ot or P alv olgyi, Alexey Pokrovskiy and Gunter Rote D aniel Gerbner](https://reader033.vdocument.in/reader033/viewer/2022052409/60a0036b35193b63873aa4b3/html5/thumbnails/44.jpg)
Advantage in the discrete Voronoi game Daniel Gerbner
Trees
For any t ≥ 2 there exists a tree T with VG (T , t) < 0.34:
x
=
NNNN
2 4 8kN
c
NNNN
xlegshead
h
Basic idea: most of the legs are controlled by the player whoclaims the last vertex there; the head is controlled by the playerwho claims h.
Optimal for t = 2: VG (T , 2) > 1/3 for any tree T .
Daniel Gerbner Advantage in the discrete Voronoi game
![Page 45: D aniel Gerbner - 上海交通大学数学系D aniel Gerbner R enyi Institute Joint work with Viola M esz aros, D om ot or P alv olgyi, Alexey Pokrovskiy and Gunter Rote D aniel Gerbner](https://reader033.vdocument.in/reader033/viewer/2022052409/60a0036b35193b63873aa4b3/html5/thumbnails/45.jpg)
Advantage in the discrete Voronoi game Daniel Gerbner
Trees
For any t ≥ 2 there exists a tree T with VG (T , t) < 0.34:
x
=
NNNN
2 4 8kN
c
NNNN
xlegshead
h
Basic idea: most of the legs are controlled by the player whoclaims the last vertex there; the head is controlled by the playerwho claims h.
Optimal for t = 2: VG (T , 2) > 1/3 for any tree T .
Daniel Gerbner Advantage in the discrete Voronoi game
![Page 46: D aniel Gerbner - 上海交通大学数学系D aniel Gerbner R enyi Institute Joint work with Viola M esz aros, D om ot or P alv olgyi, Alexey Pokrovskiy and Gunter Rote D aniel Gerbner](https://reader033.vdocument.in/reader033/viewer/2022052409/60a0036b35193b63873aa4b3/html5/thumbnails/46.jpg)
Advantage in the discrete Voronoi game Daniel Gerbner
Trees
For any t ≥ 2 there exists a tree T with VG (T , t) < 0.34:
x
=
NNNN
2 4 8kN
c
NNNN
xlegshead
h
Basic idea: most of the legs are controlled by the player whoclaims the last vertex there; the head is controlled by the playerwho claims h.
Optimal for t = 2: VG (T , 2) > 1/3 for any tree T .
Daniel Gerbner Advantage in the discrete Voronoi game
![Page 47: D aniel Gerbner - 上海交通大学数学系D aniel Gerbner R enyi Institute Joint work with Viola M esz aros, D om ot or P alv olgyi, Alexey Pokrovskiy and Gunter Rote D aniel Gerbner](https://reader033.vdocument.in/reader033/viewer/2022052409/60a0036b35193b63873aa4b3/html5/thumbnails/47.jpg)
Advantage in the discrete Voronoi game Daniel Gerbner
Strategy stealing
TheoremFor every graph and t we have
1
2VR(G , 1) ≤ VR(G , t) ≤ 1
2(VR(G , 1) + 1).
Proof of Lower bound.First starts with best pick in one round game vthen plays Second’s strategy(ignoring own first pick).Cannot make last pick v ′ of Second’s strategybut v is there.If he could now change v to v ′, he could notcontrol more than 1− VR(G , 1) new vertices,as a one round game with v and v ′ shows.
< VR(G , 1)/2
> 1− VR(G , 1)/2
�
Daniel Gerbner Advantage in the discrete Voronoi game
![Page 48: D aniel Gerbner - 上海交通大学数学系D aniel Gerbner R enyi Institute Joint work with Viola M esz aros, D om ot or P alv olgyi, Alexey Pokrovskiy and Gunter Rote D aniel Gerbner](https://reader033.vdocument.in/reader033/viewer/2022052409/60a0036b35193b63873aa4b3/html5/thumbnails/48.jpg)
Advantage in the discrete Voronoi game Daniel Gerbner
Strategy stealing
TheoremFor every graph and t we have
1
2VR(G , 1) ≤ VR(G , t) ≤ 1
2(VR(G , 1) + 1).
Proof of Lower bound.First starts with best pick in one round game vthen plays Second’s strategy(ignoring own first pick).Cannot make last pick v ′ of Second’s strategybut v is there.If he could now change v to v ′, he could notcontrol more than 1− VR(G , 1) new vertices,as a one round game with v and v ′ shows.
< VR(G , 1)/2
> 1− VR(G , 1)/2
�
Daniel Gerbner Advantage in the discrete Voronoi game
![Page 49: D aniel Gerbner - 上海交通大学数学系D aniel Gerbner R enyi Institute Joint work with Viola M esz aros, D om ot or P alv olgyi, Alexey Pokrovskiy and Gunter Rote D aniel Gerbner](https://reader033.vdocument.in/reader033/viewer/2022052409/60a0036b35193b63873aa4b3/html5/thumbnails/49.jpg)
Advantage in the discrete Voronoi game Daniel Gerbner
Strategy stealing
TheoremFor every graph and t we have
1
2VR(G , 1) ≤ VR(G , t) ≤ 1
2(VR(G , 1) + 1).
Proof of Lower bound.
First starts with best pick in one round game vthen plays Second’s strategy(ignoring own first pick).Cannot make last pick v ′ of Second’s strategybut v is there.If he could now change v to v ′, he could notcontrol more than 1− VR(G , 1) new vertices,as a one round game with v and v ′ shows.
< VR(G , 1)/2
> 1− VR(G , 1)/2
�
Daniel Gerbner Advantage in the discrete Voronoi game
![Page 50: D aniel Gerbner - 上海交通大学数学系D aniel Gerbner R enyi Institute Joint work with Viola M esz aros, D om ot or P alv olgyi, Alexey Pokrovskiy and Gunter Rote D aniel Gerbner](https://reader033.vdocument.in/reader033/viewer/2022052409/60a0036b35193b63873aa4b3/html5/thumbnails/50.jpg)
Advantage in the discrete Voronoi game Daniel Gerbner
Strategy stealing
TheoremFor every graph and t we have
1
2VR(G , 1) ≤ VR(G , t) ≤ 1
2(VR(G , 1) + 1).
Proof of Lower bound.First starts with best pick in one round game v
then plays Second’s strategy(ignoring own first pick).Cannot make last pick v ′ of Second’s strategybut v is there.If he could now change v to v ′, he could notcontrol more than 1− VR(G , 1) new vertices,as a one round game with v and v ′ shows.
< VR(G , 1)/2
> 1− VR(G , 1)/2
�
Daniel Gerbner Advantage in the discrete Voronoi game
![Page 51: D aniel Gerbner - 上海交通大学数学系D aniel Gerbner R enyi Institute Joint work with Viola M esz aros, D om ot or P alv olgyi, Alexey Pokrovskiy and Gunter Rote D aniel Gerbner](https://reader033.vdocument.in/reader033/viewer/2022052409/60a0036b35193b63873aa4b3/html5/thumbnails/51.jpg)
Advantage in the discrete Voronoi game Daniel Gerbner
Strategy stealing
TheoremFor every graph and t we have
1
2VR(G , 1) ≤ VR(G , t) ≤ 1
2(VR(G , 1) + 1).
Proof of Lower bound.First starts with best pick in one round game vthen plays Second’s strategy
(ignoring own first pick).Cannot make last pick v ′ of Second’s strategybut v is there.If he could now change v to v ′, he could notcontrol more than 1− VR(G , 1) new vertices,as a one round game with v and v ′ shows.
< VR(G , 1)/2
> 1− VR(G , 1)/2
�
Daniel Gerbner Advantage in the discrete Voronoi game
![Page 52: D aniel Gerbner - 上海交通大学数学系D aniel Gerbner R enyi Institute Joint work with Viola M esz aros, D om ot or P alv olgyi, Alexey Pokrovskiy and Gunter Rote D aniel Gerbner](https://reader033.vdocument.in/reader033/viewer/2022052409/60a0036b35193b63873aa4b3/html5/thumbnails/52.jpg)
Advantage in the discrete Voronoi game Daniel Gerbner
Strategy stealing
TheoremFor every graph and t we have
1
2VR(G , 1) ≤ VR(G , t) ≤ 1
2(VR(G , 1) + 1).
Proof of Lower bound.First starts with best pick in one round game vthen plays Second’s strategy(ignoring own first pick).
Cannot make last pick v ′ of Second’s strategybut v is there.If he could now change v to v ′, he could notcontrol more than 1− VR(G , 1) new vertices,as a one round game with v and v ′ shows.
< VR(G , 1)/2
> 1− VR(G , 1)/2
�
Daniel Gerbner Advantage in the discrete Voronoi game
![Page 53: D aniel Gerbner - 上海交通大学数学系D aniel Gerbner R enyi Institute Joint work with Viola M esz aros, D om ot or P alv olgyi, Alexey Pokrovskiy and Gunter Rote D aniel Gerbner](https://reader033.vdocument.in/reader033/viewer/2022052409/60a0036b35193b63873aa4b3/html5/thumbnails/53.jpg)
Advantage in the discrete Voronoi game Daniel Gerbner
Strategy stealing
TheoremFor every graph and t we have
1
2VR(G , 1) ≤ VR(G , t) ≤ 1
2(VR(G , 1) + 1).
Proof of Lower bound.First starts with best pick in one round game vthen plays Second’s strategy(ignoring own first pick).Cannot make last pick v ′ of Second’s strategy
but v is there.If he could now change v to v ′, he could notcontrol more than 1− VR(G , 1) new vertices,as a one round game with v and v ′ shows.
< VR(G , 1)/2
> 1− VR(G , 1)/2
�
Daniel Gerbner Advantage in the discrete Voronoi game
![Page 54: D aniel Gerbner - 上海交通大学数学系D aniel Gerbner R enyi Institute Joint work with Viola M esz aros, D om ot or P alv olgyi, Alexey Pokrovskiy and Gunter Rote D aniel Gerbner](https://reader033.vdocument.in/reader033/viewer/2022052409/60a0036b35193b63873aa4b3/html5/thumbnails/54.jpg)
Advantage in the discrete Voronoi game Daniel Gerbner
Strategy stealing
TheoremFor every graph and t we have
1
2VR(G , 1) ≤ VR(G , t) ≤ 1
2(VR(G , 1) + 1).
Proof of Lower bound.First starts with best pick in one round game vthen plays Second’s strategy(ignoring own first pick).Cannot make last pick v ′ of Second’s strategybut v is there.
If he could now change v to v ′, he could notcontrol more than 1− VR(G , 1) new vertices,as a one round game with v and v ′ shows.
< VR(G , 1)/2
> 1− VR(G , 1)/2
�
Daniel Gerbner Advantage in the discrete Voronoi game
![Page 55: D aniel Gerbner - 上海交通大学数学系D aniel Gerbner R enyi Institute Joint work with Viola M esz aros, D om ot or P alv olgyi, Alexey Pokrovskiy and Gunter Rote D aniel Gerbner](https://reader033.vdocument.in/reader033/viewer/2022052409/60a0036b35193b63873aa4b3/html5/thumbnails/55.jpg)
Advantage in the discrete Voronoi game Daniel Gerbner
Strategy stealing
TheoremFor every graph and t we have
1
2VR(G , 1) ≤ VR(G , t) ≤ 1
2(VR(G , 1) + 1).
Proof of Lower bound.First starts with best pick in one round game vthen plays Second’s strategy(ignoring own first pick).Cannot make last pick v ′ of Second’s strategybut v is there.If he could now change v to v ′, he could notcontrol more than 1− VR(G , 1) new vertices,as a one round game with v and v ′ shows.
< VR(G , 1)/2
> 1− VR(G , 1)/2
�
Daniel Gerbner Advantage in the discrete Voronoi game
![Page 56: D aniel Gerbner - 上海交通大学数学系D aniel Gerbner R enyi Institute Joint work with Viola M esz aros, D om ot or P alv olgyi, Alexey Pokrovskiy and Gunter Rote D aniel Gerbner](https://reader033.vdocument.in/reader033/viewer/2022052409/60a0036b35193b63873aa4b3/html5/thumbnails/56.jpg)
Advantage in the discrete Voronoi game Daniel Gerbner
Strategy stealing
TheoremFor every graph and t we have
1
2VR(G , 1) ≤ VR(G , t) ≤ 1
2(VR(G , 1) + 1).
Proof of Lower bound.First starts with best pick in one round game vthen plays Second’s strategy(ignoring own first pick).Cannot make last pick v ′ of Second’s strategybut v is there.If he could now change v to v ′, he could notcontrol more than 1− VR(G , 1) new vertices,as a one round game with v and v ′ shows.
VR(G , 1)
< VR(G , 1)/2
> 1− VR(G , 1)/2
�
Daniel Gerbner Advantage in the discrete Voronoi game
![Page 57: D aniel Gerbner - 上海交通大学数学系D aniel Gerbner R enyi Institute Joint work with Viola M esz aros, D om ot or P alv olgyi, Alexey Pokrovskiy and Gunter Rote D aniel Gerbner](https://reader033.vdocument.in/reader033/viewer/2022052409/60a0036b35193b63873aa4b3/html5/thumbnails/57.jpg)
Advantage in the discrete Voronoi game Daniel Gerbner
Strategy stealing
TheoremFor every graph and t we have
1
2VR(G , 1) ≤ VR(G , t) ≤ 1
2(VR(G , 1) + 1).
Proof of Lower bound.First starts with best pick in one round game vthen plays Second’s strategy(ignoring own first pick).Cannot make last pick v ′ of Second’s strategybut v is there.If he could now change v to v ′, he could notcontrol more than 1− VR(G , 1) new vertices,as a one round game with v and v ′ shows.
< VR(G , 1)/2
> 1− VR(G , 1)/2
�
Daniel Gerbner Advantage in the discrete Voronoi game
![Page 58: D aniel Gerbner - 上海交通大学数学系D aniel Gerbner R enyi Institute Joint work with Viola M esz aros, D om ot or P alv olgyi, Alexey Pokrovskiy and Gunter Rote D aniel Gerbner](https://reader033.vdocument.in/reader033/viewer/2022052409/60a0036b35193b63873aa4b3/html5/thumbnails/58.jpg)
Advantage in the discrete Voronoi game Daniel Gerbner
Strategy stealing
TheoremFor every graph and t we have
1
2VR(G , 1) ≤ VR(G , t) ≤ 1
2(VR(G , 1) + 1).
Proof of Lower bound.First starts with best pick in one round game vthen plays Second’s strategy(ignoring own first pick).Cannot make last pick v ′ of Second’s strategybut v is there.If he could now change v to v ′, he could notcontrol more than 1− VR(G , 1) new vertices,as a one round game with v and v ′ shows.
< VR(G , 1)/2
> 1− VR(G , 1)/2
�
Daniel Gerbner Advantage in the discrete Voronoi game
![Page 59: D aniel Gerbner - 上海交通大学数学系D aniel Gerbner R enyi Institute Joint work with Viola M esz aros, D om ot or P alv olgyi, Alexey Pokrovskiy and Gunter Rote D aniel Gerbner](https://reader033.vdocument.in/reader033/viewer/2022052409/60a0036b35193b63873aa4b3/html5/thumbnails/59.jpg)
Advantage in the discrete Voronoi game Daniel Gerbner
Strategy stealing
TheoremFor every graph and t we have
1
2VR(G , 1) ≤ VR(G , t) ≤ 1
2(VR(G , 1) + 1).
Proof of Lower bound.First starts with best pick in one round game vthen plays Second’s strategy(ignoring own first pick).Cannot make last pick v ′ of Second’s strategybut v is there.If he could now change v to v ′, he could notcontrol more than 1− VR(G , 1) new vertices,as a one round game with v and v ′ shows.
< VR(G , 1)/2
> 1− VR(G , 1)/2
> 1− VR(G , 1)
�
Daniel Gerbner Advantage in the discrete Voronoi game
![Page 60: D aniel Gerbner - 上海交通大学数学系D aniel Gerbner R enyi Institute Joint work with Viola M esz aros, D om ot or P alv olgyi, Alexey Pokrovskiy and Gunter Rote D aniel Gerbner](https://reader033.vdocument.in/reader033/viewer/2022052409/60a0036b35193b63873aa4b3/html5/thumbnails/60.jpg)
Advantage in the discrete Voronoi game Daniel Gerbner
Trees
TheoremFor every graph and t we have 1
2VR(G , 1) ≤ VR(G , t).
Corollary
For every tree and t we have VR(T , t) ≥ 14 .
TheoremFor every tree we have VR(T , 2) > 1
3 .
Where is truth for t > 2 between 14 and 1
3 ?
Daniel Gerbner Advantage in the discrete Voronoi game
![Page 61: D aniel Gerbner - 上海交通大学数学系D aniel Gerbner R enyi Institute Joint work with Viola M esz aros, D om ot or P alv olgyi, Alexey Pokrovskiy and Gunter Rote D aniel Gerbner](https://reader033.vdocument.in/reader033/viewer/2022052409/60a0036b35193b63873aa4b3/html5/thumbnails/61.jpg)
Advantage in the discrete Voronoi game Daniel Gerbner
Trees
TheoremFor every graph and t we have 1
2VR(G , 1) ≤ VR(G , t).
Corollary
For every tree and t we have VR(T , t) ≥ 14 .
TheoremFor every tree we have VR(T , 2) > 1
3 .
Where is truth for t > 2 between 14 and 1
3 ?
Daniel Gerbner Advantage in the discrete Voronoi game
![Page 62: D aniel Gerbner - 上海交通大学数学系D aniel Gerbner R enyi Institute Joint work with Viola M esz aros, D om ot or P alv olgyi, Alexey Pokrovskiy and Gunter Rote D aniel Gerbner](https://reader033.vdocument.in/reader033/viewer/2022052409/60a0036b35193b63873aa4b3/html5/thumbnails/62.jpg)
Advantage in the discrete Voronoi game Daniel Gerbner
Trees
TheoremFor every graph and t we have 1
2VR(G , 1) ≤ VR(G , t).
Corollary
For every tree and t we have VR(T , t) ≥ 14 .
TheoremFor every tree we have VR(T , 2) > 1
3 .
Where is truth for t > 2 between 14 and 1
3 ?
Daniel Gerbner Advantage in the discrete Voronoi game
![Page 63: D aniel Gerbner - 上海交通大学数学系D aniel Gerbner R enyi Institute Joint work with Viola M esz aros, D om ot or P alv olgyi, Alexey Pokrovskiy and Gunter Rote D aniel Gerbner](https://reader033.vdocument.in/reader033/viewer/2022052409/60a0036b35193b63873aa4b3/html5/thumbnails/63.jpg)
Advantage in the discrete Voronoi game Daniel Gerbner
Trees
TheoremFor every graph and t we have 1
2VR(G , 1) ≤ VR(G , t).
Corollary
For every tree and t we have VR(T , t) ≥ 14 .
TheoremFor every tree we have VR(T , 2) > 1
3 .
Where is truth for t > 2 between 14 and 1
3 ?
Daniel Gerbner Advantage in the discrete Voronoi game
![Page 64: D aniel Gerbner - 上海交通大学数学系D aniel Gerbner R enyi Institute Joint work with Viola M esz aros, D om ot or P alv olgyi, Alexey Pokrovskiy and Gunter Rote D aniel Gerbner](https://reader033.vdocument.in/reader033/viewer/2022052409/60a0036b35193b63873aa4b3/html5/thumbnails/64.jpg)
Advantage in the discrete Voronoi game Daniel Gerbner
VR(G , t) < ε
TheoremFor all t and ε there is G with VR(G , t) < ε.
Proof for t = 1 and semicontinuous case.Play on d-dimensional simplex with weights on vertices.
�
Daniel Gerbner Advantage in the discrete Voronoi game
![Page 65: D aniel Gerbner - 上海交通大学数学系D aniel Gerbner R enyi Institute Joint work with Viola M esz aros, D om ot or P alv olgyi, Alexey Pokrovskiy and Gunter Rote D aniel Gerbner](https://reader033.vdocument.in/reader033/viewer/2022052409/60a0036b35193b63873aa4b3/html5/thumbnails/65.jpg)
Advantage in the discrete Voronoi game Daniel Gerbner
VR(G , t) < ε
TheoremFor all t and ε there is G with VR(G , t) < ε.
Proof for t = 1 and semicontinuous case.Play on d-dimensional simplex with weights on vertices.
�
Daniel Gerbner Advantage in the discrete Voronoi game
![Page 66: D aniel Gerbner - 上海交通大学数学系D aniel Gerbner R enyi Institute Joint work with Viola M esz aros, D om ot or P alv olgyi, Alexey Pokrovskiy and Gunter Rote D aniel Gerbner](https://reader033.vdocument.in/reader033/viewer/2022052409/60a0036b35193b63873aa4b3/html5/thumbnails/66.jpg)
Advantage in the discrete Voronoi game Daniel Gerbner
VR(G , t) < ε
TheoremFor all t and ε there is G with VR(G , t) < ε.
Proof for t = 1 and semicontinuous case.
Play on d-dimensional simplex with weights on vertices.
�
Daniel Gerbner Advantage in the discrete Voronoi game
![Page 67: D aniel Gerbner - 上海交通大学数学系D aniel Gerbner R enyi Institute Joint work with Viola M esz aros, D om ot or P alv olgyi, Alexey Pokrovskiy and Gunter Rote D aniel Gerbner](https://reader033.vdocument.in/reader033/viewer/2022052409/60a0036b35193b63873aa4b3/html5/thumbnails/67.jpg)
Advantage in the discrete Voronoi game Daniel Gerbner
VR(G , t) < ε
TheoremFor all t and ε there is G with VR(G , t) < ε.
Proof for t = 1 and semicontinuous case.Play on d-dimensional simplex with weights on vertices.
�
Daniel Gerbner Advantage in the discrete Voronoi game
![Page 68: D aniel Gerbner - 上海交通大学数学系D aniel Gerbner R enyi Institute Joint work with Viola M esz aros, D om ot or P alv olgyi, Alexey Pokrovskiy and Gunter Rote D aniel Gerbner](https://reader033.vdocument.in/reader033/viewer/2022052409/60a0036b35193b63873aa4b3/html5/thumbnails/68.jpg)
Advantage in the discrete Voronoi game Daniel Gerbner
VR(G , t) < ε
TheoremFor all t and ε there is G with VR(G , t) < ε.
Proof for t = 1 and semicontinuous case.Play on d-dimensional simplex with weights on vertices.
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Daniel Gerbner Advantage in the discrete Voronoi game
![Page 69: D aniel Gerbner - 上海交通大学数学系D aniel Gerbner R enyi Institute Joint work with Viola M esz aros, D om ot or P alv olgyi, Alexey Pokrovskiy and Gunter Rote D aniel Gerbner](https://reader033.vdocument.in/reader033/viewer/2022052409/60a0036b35193b63873aa4b3/html5/thumbnails/69.jpg)
Advantage in the discrete Voronoi game Daniel Gerbner
VR(G , t) < ε
TheoremFor all t and ε there is G with VR(G , t) < ε.
Proof for t = 1 and semicontinuous case.Play on d-dimensional simplex with weights on vertices.
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Daniel Gerbner Advantage in the discrete Voronoi game
![Page 70: D aniel Gerbner - 上海交通大学数学系D aniel Gerbner R enyi Institute Joint work with Viola M esz aros, D om ot or P alv olgyi, Alexey Pokrovskiy and Gunter Rote D aniel Gerbner](https://reader033.vdocument.in/reader033/viewer/2022052409/60a0036b35193b63873aa4b3/html5/thumbnails/70.jpg)
Advantage in the discrete Voronoi game Daniel Gerbner
VR(G , t) < ε
TheoremFor all t and ε there is G with VR(G , t) < ε.
Proof for t = 1 and semicontinuous case.Play on d-dimensional simplex with weights on vertices.
�Daniel Gerbner Advantage in the discrete Voronoi game
![Page 71: D aniel Gerbner - 上海交通大学数学系D aniel Gerbner R enyi Institute Joint work with Viola M esz aros, D om ot or P alv olgyi, Alexey Pokrovskiy and Gunter Rote D aniel Gerbner](https://reader033.vdocument.in/reader033/viewer/2022052409/60a0036b35193b63873aa4b3/html5/thumbnails/71.jpg)
Advantage in the discrete Voronoi game Daniel Gerbner
Summary
For treesif t = 1 then VR(T , 1) ≥ 1
2 sharpif t = 2 then VR(T , 2) > 1
3 sharpif t ≥ 3 then 1
4 ≤ VR(T , t) and VR(T , t) < 13 + ε possible.
For general graphsVR(G , t) < ε possible for all t.What if First makes t moves and Second makes 1?Is VR(G , t : 1) < ε?
Equivalent problem: Is there a finite function family, F , such thatfor any f1, . . . , ft ∈ F there is a g ∈ F such thatg > max(f1, . . . , ft) on (1− ε) fraction of inputs?
Daniel Gerbner Advantage in the discrete Voronoi game
![Page 72: D aniel Gerbner - 上海交通大学数学系D aniel Gerbner R enyi Institute Joint work with Viola M esz aros, D om ot or P alv olgyi, Alexey Pokrovskiy and Gunter Rote D aniel Gerbner](https://reader033.vdocument.in/reader033/viewer/2022052409/60a0036b35193b63873aa4b3/html5/thumbnails/72.jpg)
Advantage in the discrete Voronoi game Daniel Gerbner
SummaryFor treesif t = 1 then VR(T , 1) ≥ 1
2 sharpif t = 2 then VR(T , 2) > 1
3 sharpif t ≥ 3 then 1
4 ≤ VR(T , t) and VR(T , t) < 13 + ε possible.
For general graphsVR(G , t) < ε possible for all t.What if First makes t moves and Second makes 1?Is VR(G , t : 1) < ε?
Equivalent problem: Is there a finite function family, F , such thatfor any f1, . . . , ft ∈ F there is a g ∈ F such thatg > max(f1, . . . , ft) on (1− ε) fraction of inputs?
Daniel Gerbner Advantage in the discrete Voronoi game
![Page 73: D aniel Gerbner - 上海交通大学数学系D aniel Gerbner R enyi Institute Joint work with Viola M esz aros, D om ot or P alv olgyi, Alexey Pokrovskiy and Gunter Rote D aniel Gerbner](https://reader033.vdocument.in/reader033/viewer/2022052409/60a0036b35193b63873aa4b3/html5/thumbnails/73.jpg)
Advantage in the discrete Voronoi game Daniel Gerbner
SummaryFor treesif t = 1 then VR(T , 1) ≥ 1
2 sharpif t = 2 then VR(T , 2) > 1
3 sharpif t ≥ 3 then 1
4 ≤ VR(T , t) and VR(T , t) < 13 + ε possible.
For general graphsVR(G , t) < ε possible for all t.
What if First makes t moves and Second makes 1?Is VR(G , t : 1) < ε?
Equivalent problem: Is there a finite function family, F , such thatfor any f1, . . . , ft ∈ F there is a g ∈ F such thatg > max(f1, . . . , ft) on (1− ε) fraction of inputs?
Daniel Gerbner Advantage in the discrete Voronoi game
![Page 74: D aniel Gerbner - 上海交通大学数学系D aniel Gerbner R enyi Institute Joint work with Viola M esz aros, D om ot or P alv olgyi, Alexey Pokrovskiy and Gunter Rote D aniel Gerbner](https://reader033.vdocument.in/reader033/viewer/2022052409/60a0036b35193b63873aa4b3/html5/thumbnails/74.jpg)
Advantage in the discrete Voronoi game Daniel Gerbner
SummaryFor treesif t = 1 then VR(T , 1) ≥ 1
2 sharpif t = 2 then VR(T , 2) > 1
3 sharpif t ≥ 3 then 1
4 ≤ VR(T , t) and VR(T , t) < 13 + ε possible.
For general graphsVR(G , t) < ε possible for all t.What if First makes t moves and Second makes 1?
Is VR(G , t : 1) < ε?
Equivalent problem: Is there a finite function family, F , such thatfor any f1, . . . , ft ∈ F there is a g ∈ F such thatg > max(f1, . . . , ft) on (1− ε) fraction of inputs?
Daniel Gerbner Advantage in the discrete Voronoi game
![Page 75: D aniel Gerbner - 上海交通大学数学系D aniel Gerbner R enyi Institute Joint work with Viola M esz aros, D om ot or P alv olgyi, Alexey Pokrovskiy and Gunter Rote D aniel Gerbner](https://reader033.vdocument.in/reader033/viewer/2022052409/60a0036b35193b63873aa4b3/html5/thumbnails/75.jpg)
Advantage in the discrete Voronoi game Daniel Gerbner
SummaryFor treesif t = 1 then VR(T , 1) ≥ 1
2 sharpif t = 2 then VR(T , 2) > 1
3 sharpif t ≥ 3 then 1
4 ≤ VR(T , t) and VR(T , t) < 13 + ε possible.
For general graphsVR(G , t) < ε possible for all t.What if First makes t moves and Second makes 1?Is VR(G , t : 1) < ε?
Equivalent problem: Is there a finite function family, F , such thatfor any f1, . . . , ft ∈ F there is a g ∈ F such thatg > max(f1, . . . , ft) on (1− ε) fraction of inputs?
Daniel Gerbner Advantage in the discrete Voronoi game
![Page 76: D aniel Gerbner - 上海交通大学数学系D aniel Gerbner R enyi Institute Joint work with Viola M esz aros, D om ot or P alv olgyi, Alexey Pokrovskiy and Gunter Rote D aniel Gerbner](https://reader033.vdocument.in/reader033/viewer/2022052409/60a0036b35193b63873aa4b3/html5/thumbnails/76.jpg)
Advantage in the discrete Voronoi game Daniel Gerbner
SummaryFor treesif t = 1 then VR(T , 1) ≥ 1
2 sharpif t = 2 then VR(T , 2) > 1
3 sharpif t ≥ 3 then 1
4 ≤ VR(T , t) and VR(T , t) < 13 + ε possible.
For general graphsVR(G , t) < ε possible for all t.What if First makes t moves and Second makes 1?Is VR(G , t : 1) < ε?
Equivalent problem:
Is there a finite function family, F , such thatfor any f1, . . . , ft ∈ F there is a g ∈ F such thatg > max(f1, . . . , ft) on (1− ε) fraction of inputs?
Daniel Gerbner Advantage in the discrete Voronoi game
![Page 77: D aniel Gerbner - 上海交通大学数学系D aniel Gerbner R enyi Institute Joint work with Viola M esz aros, D om ot or P alv olgyi, Alexey Pokrovskiy and Gunter Rote D aniel Gerbner](https://reader033.vdocument.in/reader033/viewer/2022052409/60a0036b35193b63873aa4b3/html5/thumbnails/77.jpg)
Advantage in the discrete Voronoi game Daniel Gerbner
SummaryFor treesif t = 1 then VR(T , 1) ≥ 1
2 sharpif t = 2 then VR(T , 2) > 1
3 sharpif t ≥ 3 then 1
4 ≤ VR(T , t) and VR(T , t) < 13 + ε possible.
For general graphsVR(G , t) < ε possible for all t.What if First makes t moves and Second makes 1?Is VR(G , t : 1) < ε?
Equivalent problem: Is there a finite function family, F , such thatfor any f1, . . . , ft ∈ F there is a g ∈ F such thatg > max(f1, . . . , ft) on (1− ε) fraction of inputs?
Daniel Gerbner Advantage in the discrete Voronoi game
![Page 78: D aniel Gerbner - 上海交通大学数学系D aniel Gerbner R enyi Institute Joint work with Viola M esz aros, D om ot or P alv olgyi, Alexey Pokrovskiy and Gunter Rote D aniel Gerbner](https://reader033.vdocument.in/reader033/viewer/2022052409/60a0036b35193b63873aa4b3/html5/thumbnails/78.jpg)
Advantage in the discrete Voronoi game Daniel Gerbner
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Daniel Gerbner Advantage in the discrete Voronoi game