Advantage in the discrete Voronoi game
Advantage in the discrete Voronoi game
Dániel Gerbner
Rényi Institute
Joint work with Viola Mészáros, Dömötör Pálvölgyi, Alexey
Pokrovskiy and Günter Rote
Dániel Gerbner
Advantage in the discrete Voronoi game
Advantage in the discrete Voronoi game
Dániel Gerbner
What is the Voronoi game?
Dániel Gerbner
Advantage in the discrete Voronoi game
Advantage in the discrete Voronoi game
Dániel Gerbner
What is the Voronoi game?
Applet by Jens Anuth.
Dániel Gerbner
Advantage in the discrete Voronoi game
Advantage in the discrete Voronoi game
Dániel Gerbner
What is the Voronoi game?
Applet by Jens Anuth.
Two players, First and Second claim points alternating for t
rounds.
Dániel Gerbner
Advantage in the discrete Voronoi game
Advantage in the discrete Voronoi game
Dániel Gerbner
What is the Voronoi game?
Applet by Jens Anuth.
Two players, First and Second claim points alternating for t
rounds.
At end area is divided, each point goes to closest claimed.
Dániel Gerbner
Advantage in the discrete Voronoi game
Advantage in the discrete Voronoi game
Dániel Gerbner
Competitive facility location problem
Dániel Gerbner
Advantage in the discrete Voronoi game
Advantage in the discrete Voronoi game
Dániel Gerbner
Competitive facility location problem
Two chains of supermarkets build shops in a city.
Dániel Gerbner
Advantage in the discrete Voronoi game
Advantage in the discrete Voronoi game
Dániel Gerbner
Competitive facility location problem
Two chains of supermarkets build shops in a city.
The customers always go to the nearest shop.
Dániel Gerbner
Advantage in the discrete Voronoi game
Advantage in the discrete Voronoi game
Dániel Gerbner
What is the discrete Voronoi game?
Dániel Gerbner
Advantage in the discrete Voronoi game
Advantage in the discrete Voronoi game
Dániel Gerbner
What is the discrete Voronoi game?
Same on a graph!
Dániel Gerbner
Advantage in the discrete Voronoi game
Advantage in the discrete Voronoi game
Dániel Gerbner
What is the discrete Voronoi game?
Same on a graph!
Players can claim only vertices and vertices are divided at end.
Dániel Gerbner
Advantage in the discrete Voronoi game
Advantage in the discrete Voronoi game
Dániel Gerbner
What is the discrete Voronoi game?
Same on a graph!
Players can claim only vertices and vertices are divided at end.
Dániel Gerbner
Advantage in the discrete Voronoi game
Advantage in the discrete Voronoi game
Dániel Gerbner
What is the discrete Voronoi game?
Same on a graph!
Players can claim only vertices and vertices are divided at end.
Dániel Gerbner
Advantage in the discrete Voronoi game
Advantage in the discrete Voronoi game
Dániel Gerbner
What is the discrete Voronoi game?
Same on a graph!
Players can claim only vertices and vertices are divided at end.
Dániel Gerbner
Advantage in the discrete Voronoi game
Advantage in the discrete Voronoi game
Dániel Gerbner
What is the discrete Voronoi game?
Same on a graph!
Players can claim only vertices and vertices are divided at end.
Dániel Gerbner
Advantage in the discrete Voronoi game
Advantage in the discrete Voronoi game
Dániel Gerbner
What is the discrete Voronoi game?
Same on a graph!
Players can claim only vertices and vertices are divided at end.
Dániel Gerbner
Advantage in the discrete Voronoi game
Advantage in the discrete Voronoi game
Dániel 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.
Dániel Gerbner
Advantage in the discrete Voronoi game
Advantage in the discrete Voronoi game
Dániel 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.
Dániel Gerbner
Advantage in the discrete Voronoi game
Advantage in the discrete Voronoi game
Dániel 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.
Dániel Gerbner
Advantage in the discrete Voronoi game
Advantage in the discrete Voronoi game
Dániel Gerbner
What percentage can each player get?
Dániel Gerbner
Advantage in the discrete Voronoi game
Advantage in the discrete Voronoi game
Dániel Gerbner
What percentage can each player get?
Definition
VR(G , t) =
# Closer to First + 12 # Tied
n
Dániel Gerbner
Advantage in the discrete Voronoi game
Advantage in the discrete Voronoi game
Dániel Gerbner
What percentage can each player get?
Definition
VR(G , t) =
# Closer to First + 12 # Tied
n
Goal: Bound VR(G , t) for certain graph(family)
Dániel Gerbner
Advantage in the discrete Voronoi game
Advantage in the discrete Voronoi game
Dániel Gerbner
What percentage can each player get?
Definition
VR(G , t) =
# Closer to First + 12 # Tied
n
Goal: Bound VR(G , t) for certain graph(family)
VR(path, t) ≥
1
2
and →
1
2
Dániel Gerbner
Advantage in the discrete Voronoi game
Advantage in the discrete Voronoi game
Dániel Gerbner
What percentage can each player get?
Definition
VR(G , t) =
# Closer to First + 12 # Tied
n
Goal: Bound VR(G , t) for certain graph(family)
VR(path, t) ≥
1
2
and →
1
2
VR(star , t) → 1
Dániel Gerbner
Advantage in the discrete Voronoi game
Advantage in the discrete Voronoi game
Dániel Gerbner
What percentage can each player get?
Definition
VR(G , t) =
# Closer to First + 12 # Tied
n
Goal: Bound VR(G , t) for certain graph(family)
VR(path, t) ≥
1
2
and →
1
2
VR(star , t) → 1
Dániel Gerbner
Advantage in the discrete Voronoi game
Advantage in the discrete Voronoi game
Dániel Gerbner
What percentage can each player get?
Definition
VR(G , t) =
# Closer to First + 12 # Tied
n
Goal: Bound VR(G , t) for certain graph(family)
VR(path, t) ≥
1
2
and →
1
2
VR(star , t) → 1
Dániel Gerbner
Advantage in the discrete Voronoi game
Advantage in the discrete Voronoi game
Dániel Gerbner
What percentage can each player get?
Definition
VR(G , t) =
# Closer to First + 12 # Tied
n
Goal: Bound VR(G , t) for certain graph(family)
VR(path, t) ≥
1
2
and →
1
2
VR(star , t) → 1
Dániel Gerbner
Advantage in the discrete Voronoi game
Advantage in the discrete Voronoi game
Dániel Gerbner
What percentage can each player get?
Definition
VR(G , t) =
# Closer to First + 12 # Tied
n
Goal: Bound VR(G , t) for certain graph(family)
VR(path, t) ≥
1
2
and →
1
2
VR(star , t) → 1
Dániel Gerbner
Advantage in the discrete Voronoi game
Advantage in the discrete Voronoi game
Dániel Gerbner
What percentage can each player get?
Definition
VR(G , t) =
# Closer to First + 12 # Tied
n
Goal: Bound VR(G , t) for certain graph(family)
VR(path, t) ≥
1
2
and →
1
2
VR(star , t) → 1
Dániel Gerbner
Advantage in the discrete Voronoi game
Advantage in the discrete Voronoi game
Dániel Gerbner
Can the second player win?
Dániel Gerbner
Advantage in the discrete Voronoi game
Advantage in the discrete Voronoi game
Dániel Gerbner
Can the second player win?
Dániel Gerbner
Advantage in the discrete Voronoi game
Advantage in the discrete Voronoi game
Dániel Gerbner
Can the second player win?
Dániel Gerbner
Advantage in the discrete Voronoi game
Advantage in the discrete Voronoi game
Dániel Gerbner
Can the second player win?
Dániel Gerbner
Advantage in the discrete Voronoi game
Advantage in the discrete Voronoi game
Dániel Gerbner
Can the second player win?
Dániel Gerbner
Advantage in the discrete Voronoi game
Advantage in the discrete Voronoi game
Dániel Gerbner
Can the second player win?
Dániel Gerbner
Advantage in the discrete Voronoi game
Advantage in the discrete Voronoi game
Dániel Gerbner
Can the second player win?
Dániel Gerbner
Advantage in the discrete Voronoi game
Advantage in the discrete Voronoi game
Dániel Gerbner
Questions
Dániel Gerbner
Advantage in the discrete Voronoi game
Advantage in the discrete Voronoi game
Dániel Gerbner
Questions
VR(G , t) < ?
Dániel Gerbner
Advantage in the discrete Voronoi game
Advantage in the discrete Voronoi game
Dániel Gerbner
Questions
VR(G , t) < ?
What if t = 1?
Dániel Gerbner
Advantage in the discrete Voronoi game
Advantage in the discrete Voronoi game
Dániel Gerbner
Questions
VR(G , t) < ?
What if t = 1?
VR(T , t) <
1
2
for a tree?
Dániel Gerbner
Advantage in the discrete Voronoi game
Advantage in the discrete Voronoi game
Dániel Gerbner
Questions
VR(G , t) < ?
What if t = 1?
VR(T , t) <
1
2
for a tree?
1
2
for trees.
Claim
VR(T , 1) ≥
Dániel Gerbner
Advantage in the discrete Voronoi game
Advantage in the discrete Voronoi game
Dániel Gerbner
Questions
VR(G , t) < ?
What if t = 1?
VR(T , t) <
1
2
for a tree?
1
2
for trees.
Claim
VR(T , 1) ≥
Proof.
First takes center of tree. (A vertex which cuts the tree into
connected components of order at most n/2.)
Dániel Gerbner
Advantage in the discrete Voronoi game
Advantage in the discrete Voronoi game
Dániel Gerbner
Trees
Dániel Gerbner
Advantage in the discrete Voronoi game
Advantage in the discrete Voronoi game
Dániel Gerbner
Trees
For any t ≥ 2 there exists a tree T with VG (T , t) < 0.34:
Dániel Gerbner
Advantage in the discrete Voronoi game
Advantage in the discrete Voronoi game
Dániel Gerbner
Trees
For any t ≥ 2 there exists a tree T with VG (T , t) < 0.34:
N
N
N
head
kN
N
legs
x
x
h c
=
2
N
4
N
8
N
Dániel Gerbner
N
Advantage in the discrete Voronoi game
Advantage in the discrete Voronoi game
Dániel Gerbner
Trees
For any t ≥ 2 there exists a tree T with VG (T , t) < 0.34:
N
N
N
head
kN
N
legs
x
x
h c
=
2
N
4
N
8
N
N
Basic idea: most of the legs are controlled by the player who
claims the last vertex there; the head is controlled by the player
who claims h.
Optimal for t = 2: VG (T , 2) > 1/3 for any tree T .
Dániel Gerbner
Advantage in the discrete Voronoi game
Advantage in the discrete Voronoi game
Dániel Gerbner
Strategy stealing
Dániel Gerbner
Advantage in the discrete Voronoi game
Advantage in the discrete Voronoi game
Dániel Gerbner
Strategy stealing
Theorem
For every graph and t we have
1
1
VR(G , 1) ≤ VR(G , t) ≤ (VR(G , 1) + 1).
2
2
Dániel Gerbner
Advantage in the discrete Voronoi game
Advantage in the discrete Voronoi game
Dániel Gerbner
Strategy stealing
Theorem
For every graph and t we have
1
1
VR(G , 1) ≤ VR(G , t) ≤ (VR(G , 1) + 1).
2
2
Proof of Lower bound.
Dániel Gerbner
Advantage in the discrete Voronoi game
Advantage in the discrete Voronoi game
Dániel Gerbner
Strategy stealing
Theorem
For every graph and t we have
1
1
VR(G , 1) ≤ VR(G , t) ≤ (VR(G , 1) + 1).
2
2
Proof of Lower bound.
First starts with best pick in one round game v
Dániel Gerbner
Advantage in the discrete Voronoi game
Advantage in the discrete Voronoi game
Dániel Gerbner
Strategy stealing
Theorem
For every graph and t we have
1
1
VR(G , 1) ≤ VR(G , t) ≤ (VR(G , 1) + 1).
2
2
Proof of Lower bound.
First starts with best pick in one round game v
then plays Second’s strategy
Dániel Gerbner
Advantage in the discrete Voronoi game
Advantage in the discrete Voronoi game
Dániel Gerbner
Strategy stealing
Theorem
For every graph and t we have
1
1
VR(G , 1) ≤ VR(G , t) ≤ (VR(G , 1) + 1).
2
2
Proof of Lower bound.
First starts with best pick in one round game v
then plays Second’s strategy
(ignoring own first pick).
Dániel Gerbner
Advantage in the discrete Voronoi game
Advantage in the discrete Voronoi game
Dániel Gerbner
Strategy stealing
Theorem
For every graph and t we have
1
1
VR(G , 1) ≤ VR(G , t) ≤ (VR(G , 1) + 1).
2
2
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 0 of Second’s strategy
Dániel Gerbner
Advantage in the discrete Voronoi game
Advantage in the discrete Voronoi game
Dániel Gerbner
Strategy stealing
Theorem
For every graph and t we have
1
1
VR(G , 1) ≤ VR(G , t) ≤ (VR(G , 1) + 1).
2
2
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 0 of Second’s strategy
but v is there.
Dániel Gerbner
Advantage in the discrete Voronoi game
Advantage in the discrete Voronoi game
Dániel Gerbner
Strategy stealing
Theorem
For every graph and t we have
1
1
VR(G , 1) ≤ VR(G , t) ≤ (VR(G , 1) + 1).
2
2
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 0 of Second’s strategy
but v is there.
If he could now change v to v 0 , he could not
control more than 1 − VR(G , 1) new vertices,
as a one round game with v and v 0 shows.
Dániel Gerbner
Advantage in the discrete Voronoi game
Advantage in the discrete Voronoi game
Dániel Gerbner
Strategy stealing
Theorem
For every graph and t we have
1
1
VR(G , 1) ≤ VR(G , t) ≤ (VR(G , 1) + 1).
2
2
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 0 of Second’s strategy
but v is there.
If he could now change v to v 0 , he could not
control more than 1 − VR(G , 1) new vertices,
as a one round game with v and v 0 shows.
Dániel Gerbner
VR(G , 1)
Advantage in the discrete Voronoi game
Advantage in the discrete Voronoi game
Dániel Gerbner
Strategy stealing
Theorem
For every graph and t we have
1
1
VR(G , 1) ≤ VR(G , t) ≤ (VR(G , 1) + 1).
2
2
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 0 of Second’s strategy
but v is there.
If he could now change v to v 0 , he could not
control more than 1 − VR(G , 1) new vertices,
as a one round game with v and v 0 shows.
Dániel Gerbner
< VR(G , 1)/2
Advantage in the discrete Voronoi game
Advantage in the discrete Voronoi game
Dániel Gerbner
Strategy stealing
Theorem
For every graph and t we have
1
1
VR(G , 1) ≤ VR(G , t) ≤ (VR(G , 1) + 1).
2
2
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 0 of Second’s strategy
but v is there.
If he could now change v to v 0 , he could not
control more than 1 − VR(G , 1) new vertices,
as a one round game with v and v 0 shows.
Dániel Gerbner
> 1 − VR(G , 1)/2
< VR(G , 1)/2
Advantage in the discrete Voronoi game
Advantage in the discrete Voronoi game
Dániel Gerbner
Strategy stealing
Theorem
For every graph and t we have
1
1
VR(G , 1) ≤ VR(G , t) ≤ (VR(G , 1) + 1).
2
2
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 0 of Second’s strategy
but v is there.
If he could now change v to v 0 , he could not
control more than 1 − VR(G , 1) new vertices,
as a one round game with v and v 0 shows.
Dániel Gerbner
> 1 − VR(G , 1)/2
> 1 − VR(G , 1) < VR(G , 1)/2
Advantage in the discrete Voronoi game
Advantage in the discrete Voronoi game
Dániel Gerbner
Trees
Theorem
For every graph and t we have 12 VR(G , 1) ≤ VR(G , t).
Dániel Gerbner
Advantage in the discrete Voronoi game
Advantage in the discrete Voronoi game
Dániel Gerbner
Trees
Theorem
For every graph and t we have 12 VR(G , 1) ≤ VR(G , t).
Corollary
For every tree and t we have VR(T , t) ≥ 14 .
Dániel Gerbner
Advantage in the discrete Voronoi game
Advantage in the discrete Voronoi game
Dániel Gerbner
Trees
Theorem
For every graph and t we have 12 VR(G , 1) ≤ VR(G , t).
Corollary
For every tree and t we have VR(T , t) ≥ 14 .
Theorem
For every tree we have VR(T , 2) > 13 .
Dániel Gerbner
Advantage in the discrete Voronoi game
Advantage in the discrete Voronoi game
Dániel Gerbner
Trees
Theorem
For every graph and t we have 12 VR(G , 1) ≤ VR(G , t).
Corollary
For every tree and t we have VR(T , t) ≥ 14 .
Theorem
For every tree we have VR(T , 2) > 13 .
Where is truth for t > 2 between
Dániel Gerbner
1
4
and 13 ?
Advantage in the discrete Voronoi game
Advantage in the discrete Voronoi game
Dániel Gerbner
VR(G , t) < Dániel Gerbner
Advantage in the discrete Voronoi game
Advantage in the discrete Voronoi game
Dániel Gerbner
VR(G , t) < Theorem
For all t and there is G with VR(G , t) < .
Dániel Gerbner
Advantage in the discrete Voronoi game
Advantage in the discrete Voronoi game
Dániel Gerbner
VR(G , t) < Theorem
For all t and there is G with VR(G , t) < .
Proof for t = 1 and semicontinuous case.
Dániel Gerbner
Advantage in the discrete Voronoi game
Advantage in the discrete Voronoi game
Dániel Gerbner
VR(G , t) < Theorem
For all t and there is G with VR(G , t) < .
Proof for t = 1 and semicontinuous case.
Play on d-dimensional simplex with weights on vertices.
Dániel Gerbner
Advantage in the discrete Voronoi game
Advantage in the discrete Voronoi game
Dániel Gerbner
VR(G , t) < Theorem
For all t and there is G with VR(G , t) < .
Proof for t = 1 and semicontinuous case.
Play on d-dimensional simplex with weights on vertices.
Dániel Gerbner
Advantage in the discrete Voronoi game
Advantage in the discrete Voronoi game
Dániel Gerbner
VR(G , t) < Theorem
For all t and there is G with VR(G , t) < .
Proof for t = 1 and semicontinuous case.
Play on d-dimensional simplex with weights on vertices.
Dániel Gerbner
Advantage in the discrete Voronoi game
Advantage in the discrete Voronoi game
Dániel Gerbner
VR(G , t) < Theorem
For all t and there is G with VR(G , t) < .
Proof for t = 1 and semicontinuous case.
Play on d-dimensional simplex with weights on vertices.
Dániel Gerbner
Advantage in the discrete Voronoi game
Advantage in the discrete Voronoi game
Dániel Gerbner
Summary
Dániel Gerbner
Advantage in the discrete Voronoi game
Advantage in the discrete Voronoi game
Dániel Gerbner
Summary
For trees
if t = 1 then VR(T , 1) ≥ 12 sharp
if t = 2 then VR(T , 2) > 13 sharp
if t ≥ 3 then 14 ≤ VR(T , t) and VR(T , t) <
Dániel Gerbner
1
3
+ possible.
Advantage in the discrete Voronoi game
Advantage in the discrete Voronoi game
Dániel Gerbner
Summary
For trees
if t = 1 then VR(T , 1) ≥ 12 sharp
if t = 2 then VR(T , 2) > 13 sharp
if t ≥ 3 then 14 ≤ VR(T , t) and VR(T , t) <
1
3
+ possible.
For general graphs
VR(G , t) < possible for all t.
Dániel Gerbner
Advantage in the discrete Voronoi game
Advantage in the discrete Voronoi game
Dániel Gerbner
Summary
For trees
if t = 1 then VR(T , 1) ≥ 12 sharp
if t = 2 then VR(T , 2) > 13 sharp
if t ≥ 3 then 14 ≤ VR(T , t) and VR(T , t) <
1
3
+ possible.
For general graphs
VR(G , t) < possible for all t.
What if First makes t moves and Second makes 1?
Dániel Gerbner
Advantage in the discrete Voronoi game
Advantage in the discrete Voronoi game
Dániel Gerbner
Summary
For trees
if t = 1 then VR(T , 1) ≥ 12 sharp
if t = 2 then VR(T , 2) > 13 sharp
if t ≥ 3 then 14 ≤ VR(T , t) and VR(T , t) <
1
3
+ possible.
For general graphs
VR(G , t) < possible for all t.
What if First makes t moves and Second makes 1?
Is VR(G , t : 1) < ?
Dániel Gerbner
Advantage in the discrete Voronoi game
Advantage in the discrete Voronoi game
Dániel Gerbner
Summary
For trees
if t = 1 then VR(T , 1) ≥ 12 sharp
if t = 2 then VR(T , 2) > 13 sharp
if t ≥ 3 then 14 ≤ VR(T , t) and VR(T , t) <
1
3
+ possible.
For general graphs
VR(G , t) < possible for all t.
What if First makes t moves and Second makes 1?
Is VR(G , t : 1) < ?
Equivalent problem:
Dániel Gerbner
Advantage in the discrete Voronoi game
Advantage in the discrete Voronoi game
Dániel Gerbner
Summary
For trees
if t = 1 then VR(T , 1) ≥ 12 sharp
if t = 2 then VR(T , 2) > 13 sharp
if t ≥ 3 then 14 ≤ VR(T , t) and VR(T , t) <
1
3
+ possible.
For general graphs
VR(G , t) < possible for all t.
What if First makes t moves and Second makes 1?
Is VR(G , t : 1) < ?
Equivalent problem: Is there a finite function family, F, such that
for any f1 , . . . , ft ∈ F there is a g ∈ F such that
g > max(f1 , . . . , ft ) on (1 − ) fraction of inputs?
Dániel Gerbner
Advantage in the discrete Voronoi game
Advantage in the discrete Voronoi game
Dániel Gerbner
Thank you for your attention!
Dániel Gerbner
Advantage in the discrete Voronoi game