Solution of Peter Winkler's Pizza Problemcanadam.math.ca/2009/pdf/meszaros.pdf · Solution of Peter...
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Solution of Peter Winkler’s Pizza Problem
Viola Meszaros (Charles University)
Joint work with Josef Cibulka, Jan Kyncl, Rudolf Stolar and PavelValtr
Viola Meszaros Pizza Problem
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A Problem of Peter Winkler
available slices
Figure: Bob and Alice are sharing a pizza
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A Problem of Peter Winkler
available slices
Figure: Bob and Alice are sharing a pizza
How much can Alice gain?
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Easy observations
◮ Bob can obtain half of the pizza by cutting the pizza into aneven number of slices of equal size.
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Easy observations
◮ Bob can obtain half of the pizza by cutting the pizza into aneven number of slices of equal size.
◮ If the number of slices is even, Alice has a strategy to gain atleast half of the pizza.
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Shifts and jumps
shift
jump
Figure: The two possible moves
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Shifts and jumps
shift
jump
Figure: The two possible moves
If some strategy of a player allows the player to make at most j
jumps, then it is a j-jump strategy.
Viola Meszaros Pizza Problem
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Definitions
◮ The pizza may be represented by a circular sequenceP = p0p1 . . .pn−1 and by the weights |pi | ≥ 0 for(i = 0,1, . . . ,n−1).
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Definitions
◮ The pizza may be represented by a circular sequenceP = p0p1 . . .pn−1 and by the weights |pi | ≥ 0 for(i = 0,1, . . . ,n−1).
◮ The weight of P is defined by |P | := ∑n−1
i=0|pi |.
Viola Meszaros Pizza Problem
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Definitions
◮ The pizza may be represented by a circular sequenceP = p0p1 . . .pn−1 and by the weights |pi | ≥ 0 for(i = 0,1, . . . ,n−1).
◮ The weight of P is defined by |P | := ∑n−1
i=0|pi |.
◮ A player has a strategy with gain g if that strategy guaranteesthe player a subset of slices with sum of weights at least g .
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Restricting jumps
Claim
Alice has a zero-jump strategy with gain |P |/3 and the constant
1/3 is the best possible.
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Restricting jumps
Claim
Alice has a zero-jump strategy with gain |P |/3 and the constant
1/3 is the best possible.
Theorem
Alice has a one-jump strategy with gain 7|P |/16 and the constant
7/16 is the best possible.
Viola Meszaros Pizza Problem
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Restricting jumps
Claim
Alice has a zero-jump strategy with gain |P |/3 and the constant
1/3 is the best possible.
Theorem
Alice has a one-jump strategy with gain 7|P |/16 and the constant
7/16 is the best possible.
Our main result:
Theorem
For any P, Alice has a two-jump strategy with gain 4|P |/9 and the
constant 4/9 is the best possible.
Viola Meszaros Pizza Problem
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Characteristic cycle
If the number of slices is odd, instead of the circular sequenceP = p0p1 . . .pn−1 consider the characteristic cycle defined asV = v0v1 . . .vn−1 = p0p2 . . .pn−1p1p3 . . .pn−2.
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Characteristic cycle
If the number of slices is odd, instead of the circular sequenceP = p0p1 . . .pn−1 consider the characteristic cycle defined asV = v0v1 . . .vn−1 = p0p2 . . .pn−1p1p3 . . .pn−2.
pn−1p0
p1pn−2
vn−1 = pn−2
v0 = p0
v1 = p2
vn+1
2
= p1 vn−1
2
= pn−1
.
.
.
.
.
.
. . .
Figure: A cutting of a pizza and the corresponding characteristic cycle.
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A game
Figure: Turns: A1,B2,A3, . . . , jumps: B4 and A5.
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A game
A1
Figure: Turns: A1,B2,A3, . . . , jumps: B4 and A5.
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A game
A1
B2
Figure: Turns: A1,B2,A3, . . . , jumps: B4 and A5.
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A game
A1
A3
B2
Figure: Turns: A1,B2,A3, . . . , jumps: B4 and A5.
Viola Meszaros Pizza Problem
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A game
A1
A3
B2
B4
Figure: Turns: A1,B2,A3, . . . , jumps: B4 and A5.
Viola Meszaros Pizza Problem
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A game
A1
A5
A3
B2
B4
Figure: Turns: A1,B2,A3, . . . , jumps: B4 and A5.
Viola Meszaros Pizza Problem
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A game
A1
A5
A3
B2
B4
B6
Figure: Turns: A1,B2,A3, . . . , jumps: B4 and A5.
Viola Meszaros Pizza Problem
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A game
A1
A5
A3
B2
B4
B6
A7
Figure: Turns: A1,B2,A3, . . . , jumps: B4 and A5.
Viola Meszaros Pizza Problem
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Further definitions
◮ An arc is a sequence of at most n−1 consecutive elements ofV .
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Further definitions
◮ An arc is a sequence of at most n−1 consecutive elements ofV .
◮ For an arc X = vivi+1 . . .vi+k−1, its length is l(X ) := k.
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Further definitions
◮ An arc is a sequence of at most n−1 consecutive elements ofV .
◮ For an arc X = vivi+1 . . .vi+k−1, its length is l(X ) := k.
◮ The weight of X is |X | := ∑i+k−1
j=i |vj |.
Viola Meszaros Pizza Problem
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Further definitions
◮ An arc is a sequence of at most n−1 consecutive elements ofV .
◮ For an arc X = vivi+1 . . .vi+k−1, its length is l(X ) := k.
◮ The weight of X is |X | := ∑i+k−1
j=i |vj |.
◮ An arc of length (n+1)/2 is called a half-circle.
Viola Meszaros Pizza Problem
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Further definitions
◮ An arc is a sequence of at most n−1 consecutive elements ofV .
◮ For an arc X = vivi+1 . . .vi+k−1, its length is l(X ) := k.
◮ The weight of X is |X | := ∑i+k−1
j=i |vj |.
◮ An arc of length (n+1)/2 is called a half-circle.
◮ For each v in V the potential of v is the minimum of theweights of half-circles covering v .
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Zero-jump strategy
◮ Lower bound: There exists v on V with potential at least 1/3.
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Zero-jump strategy
◮ Lower bound: There exists v on V with potential at least 1/3.
A
B
C
D
E
F
Figure: A covering triple of half-circles.
Viola Meszaros Pizza Problem
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Zero-jump strategy
◮ Lower bound: There exists v on V with potential at least 1/3.
A
B
C
D
E
F
Figure: A covering triple of half-circles.
◮ Upper bound: Consider the cutting V = 100100100.
Viola Meszaros Pizza Problem
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One-jump strategy
X1
X2
A13
A15
A17
A19
A21
A23
E0
B12
B14
B16
B18
B20
B22
A9
B8
A11
B10
A1A5
A3
B2
B4
B6
B
A
F D
C
A7
E
A1A5
A3
B2
B4
B6
B
A
F D
C
A7
E
Figure: One-jump strategy: Alice chooses a jump rather than a shift(left) and makes no more jumps afterwards (right).
Viola Meszaros Pizza Problem
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Two-jump strategy
B′
A′
F ′
E ′
D′
C ′
A1A3
B2B4
B6
A7
A9A11A13
A15
A17
B8
B10
B12
B14
B16
B18
A5
B
E
F ′
E ′
D′
B2B4
B6A9
B8
B
A19
B′
A′C ′
A1A3
A7
A5
E
Figure: We define two phases of the game. During the first phase Alicemakes one jump (left). She makes another jump as the first turn of thesecond phase (right).
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Analysis of Alice’s gain
For n ≥ 1, let g(n) be the maximum g ∈ [0,1] such that for anycutting of the pizza into n slices, Alice has a strategy with gaing |P |.
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Analysis of Alice’s gain
For n ≥ 1, let g(n) be the maximum g ∈ [0,1] such that for anycutting of the pizza into n slices, Alice has a strategy with gaing |P |.
Theorem
Let n ≥ 1. Then
g(n) =
1 if n = 1,4/9 if n ∈ {15,17,19,21, . . . },1/2 otherwise.
Viola Meszaros Pizza Problem
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Analysis of Alice’s gain
For n ≥ 1, let g(n) be the maximum g ∈ [0,1] such that for anycutting of the pizza into n slices, Alice has a strategy with gaing |P |.
Theorem
Let n ≥ 1. Then
g(n) =
1 if n = 1,4/9 if n ∈ {15,17,19,21, . . . },1/2 otherwise.
Alice uses a zero-jump strategy when n is even or n ≤ 7, a
one-jump strategy for n ∈ {9,11,13}, and a two-jump strategy for
n ∈ {15,17,19,21, . . . }.
Viola Meszaros Pizza Problem
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Some more results
Theorem
For any ω ∈ [0,1], Bob has a one-jump strategy with gain 5|P |/9 if
he cuts the pizza into 15 slices as follows:
Pω = 0010100(1+ω)0(2−ω)00202. These cuttings describe, up
to scaling, rotating and flipping the pizza upside-down, all the
pizza cuttings into 15 slices for which Bob has a strategy with gain
5|P |/9.
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Some more results
Theorem
For any ω ∈ [0,1], Bob has a one-jump strategy with gain 5|P |/9 if
he cuts the pizza into 15 slices as follows:
Pω = 0010100(1+ω)0(2−ω)00202. These cuttings describe, up
to scaling, rotating and flipping the pizza upside-down, all the
pizza cuttings into 15 slices for which Bob has a strategy with gain
5|P |/9.
Theorem
Up to scaling, rotating and flipping the pizza upside-down, there is
a unique pizza cutting into 21 slices of at most two different sizes
for which Bob has a strategy with gain 5|P |/9. The cutting is
001010010101001010101.
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Algorithms
Theorem
There is an algorithm that, given a cutting of the pizza with n
slices, performs a precomputation in time O(n). Then, during the
game, the algorithm decides each of Alice’s turns in time O(1) in
such a way that Alice makes at most two jumps and her gain is at
least g(n)|P |.
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Algorithms
Theorem
There is an algorithm that, given a cutting of the pizza with n
slices, performs a precomputation in time O(n). Then, during the
game, the algorithm decides each of Alice’s turns in time O(1) in
such a way that Alice makes at most two jumps and her gain is at
least g(n)|P |.
Claim
There is an algorithm that, given a cutting of the pizza with n
slices, computes an optimal strategy for each of the two players in
time O(n2). The algorithm stores an optimal turn of the player on
turn for all the n2−n+2 possible positions of the game.
Viola Meszaros Pizza Problem
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Open problem
Problem
Is there an algorithm that uses o(n2) time for some
precomputations and then computes each optimal turn in constant
time?
Viola Meszaros Pizza Problem