d aniel gerbner - 上海交通大学数学系d aniel gerbner r enyi institute joint work with viola...
TRANSCRIPT
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
Advantage in the discrete Voronoi game Daniel Gerbner
Can the second player win?
Daniel Gerbner Advantage in the discrete Voronoi game
Advantage in the discrete Voronoi game Daniel Gerbner
Can the second player win?
Daniel Gerbner Advantage in the discrete Voronoi game
Advantage in the discrete Voronoi game Daniel Gerbner
Can the second player win?
Daniel Gerbner Advantage in the discrete Voronoi game
Advantage in the discrete Voronoi game Daniel Gerbner
Can the second player win?
Daniel Gerbner Advantage in the discrete Voronoi game
Advantage in the discrete Voronoi game Daniel Gerbner
Can the second player win?
Daniel Gerbner Advantage in the discrete Voronoi game
Advantage in the discrete Voronoi game Daniel Gerbner
Can the second player win?
Daniel Gerbner Advantage in the discrete Voronoi game
Advantage in the discrete Voronoi game Daniel Gerbner
Can the second player win?
Daniel Gerbner Advantage in the discrete Voronoi game
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
Advantage in the discrete Voronoi game Daniel Gerbner
Thank you for your attention!
Daniel Gerbner Advantage in the discrete Voronoi game