cake cutting is and is not a piece of cake l.
Download
Skip this Video
Loading SlideShow in 5 Seconds..
Cake Cutting is and is not a Piece of Cake PowerPoint Presentation
Download Presentation
Cake Cutting is and is not a Piece of Cake

Loading in 2 Seconds...

play fullscreen
1 / 32

Cake Cutting is and is not a Piece of Cake - PowerPoint PPT Presentation


  • 373 Views
  • Uploaded on

Cake Cutting is and is not a Piece of Cake. Jeff Edmonds, York University Kirk Pruhs , University of Pittsburgh. Informal Problem Statement. n self interested players wish to divide items of value such that each player believes that they received at least 1/n of the value

loader
I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
capcha
Download Presentation

PowerPoint Slideshow about 'Cake Cutting is and is not a Piece of Cake' - RoyLauris


An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript
cake cutting is and is not a piece of cake

Cake Cutting is and is not a Piece of Cake

Jeff Edmonds, York University

Kirk Pruhs, University of Pittsburgh

informal problem statement
Informal Problem Statement
  • n self interested players wish to divide items of value such that each player believes that they received at least 1/n of the value
  • Players may not agree on the values of items
  • Players may be deceitful, cunning, dishonest, etc.
classic problem definition
Classic Problem Definition
  • n players wish to divide a cake = [0, 1]
  • Each player p has an unknown value function Vp
    • Vp[x, y] = how much player p values piece/interval [x, y]
  • The protocol’s goal is Fairness: Each honest player p is guaranteed a piece of cake of value at least Vp[0,1]/n = 1/n
history
History
  • Originated in 1940’s school of Polish

mathematics

  • Picked up by social scientists interested in fair allocation of resources
  • Texts by Brams and Taylor, and Robertson and Webb
  • A quick Google search reveals cake cutting is used as a teaching example in many algorithms courses
classic two person discrete algorithm n 2 cut and choose
Classic Two Person Discrete Algorithm (n=2): Cut and Choose
  • Person A cuts the cake into two pieces
  • Person B selects one of the two pieces, and person A gets the other piece
two person continuous algorithm n 2 moving knife
Two Person Continuous Algorithm (n=2): Moving Knife
  • Protocol moves the knife continuously across the cake until the first player say stop
  • This player gets this piece, and the rest of the players continue
  • Moving knife algorithms are considered cheating by discrete algorithmic researchers and we do not consider them here
formalizing allowed operations
Formalizing Allowed Operations
  • Queries the protocol can make to each player:
    • Eval[p, x, y]: returns Vp[x, y]
    • Cut[p, x, v]: returns a y such Vp[x, y] = v
  • All know algorithms can be implemented with these operations
two person algorithm n 2 cut and choose
Two Person Algorithm (n=2):Cut and Choose
  • y = cut(A, 0, ½)
  • If eval(B, 0, y) ≤ ½ then
    • Player A gets [0, y] and player B gets [y, 1]
  • Else
    • Player B gets [0, y] and player A gets [y, 1]
three person algorithm n 3 steinhaus
Three Person Algorithm (n=3):Steinhaus
  • YA = cut(A, 0, 1/3)
  • YB = cut(B, 0, 1/3)
  • YC = cut(C, 0, 1/3)
  • Assume wlog YA ≤ YB ≤ YC
  • Player A gets [0, yA], and players B and C “cut and choose” on [yA, 1]
o n log n divide and conquer algorithm evan and paz
O(n log n) Divide and Conquer Algorithm: Evan and Paz
  • Yi = cut(i, 0, 1/2) for i = 1 … n
  • m = median(y1, … , yn)
  • Recurse on [0, m] with those n/2 players i for which yi < m
  • Recurse on [m, 1] with those n/2 players i for which yi > m
problem variations
Problem Variations
  • Contiguousness: Assigned pieces must be subintervals
  • Approximate fairness: A protocol is c-fair if each player is a assured a piece that he gives a value of at least c/n
  • Approximate queries (introduced by us?):
    • AEval[p, ε, x, y]: returns a value v such that Vp[x, y]/(1+ε) ≤ v ≤ (1+ ε) Vp[x, y]
    • ACut[p, ε, x, v]: returns a y such Vp[x, y]/(1+ε) ≤ v ≤ (1+ ε) Vp[x, y]
problem variations14
Problem Variations

* The proof is currently only roughly written up at this point

outline
Outline
  • Deterministic Ω(n log n) Lower Bound
    • Definition of Thin-Rich game
    • Sufficiency to lower bound Thin-Rich
    • Definition of value tree cakes
    • Lower bound for Thin-Rich
  • Hint at Randomized Ω(n log n) Lower Bound with Approximate Cuts
  • Randomized O(n) Upper Bound
thin rich game
Thin-Rich Game
  • Game Definition: Find a thin rich piece for a particular player
    • A piece is thin if it has width ≤ 2/n
    • A piece is rich if it has value ≥ 1/2n
  • Theorem: The complexity of cake cutting is at least n/2 times the complexity of thin-rich
    • Proof: In cake cutting, at least n/2 players have to end up with a thin rich piece
value tree
Value Tree

1/4

1/2

1/4

1/4

1/4

1/4

1/4

1/2

1/4

1/4

1/2

1/2

0 1/9 2/9 3/9 4/9 5/9 6/9 7/9 8/9 1

Value = Product of edge labels

deterministic log n lower bound for thin rich
Deterministic Ω(log n) Lower Bound for Thin-Rich
  • Theorem: To win at Thin-Rich, when the input is derived from a value tree, the protocol has to find a leaf where at least 40% of the edge labels on root to leaf path are ½
  • Theorem: From each query, the protocol learns the edge labels on at most two root to leaf paths
  • Theorem: The deterministic complexity of Thin-Rich is Ω(log n)
    • Proof: Reveal edges with label ¼ on the two paths learned by the protocol
randomized lower bound
Randomized Lower Bound
  • Theorem: From each approximate query, the protocol learns the edge labels on at most two root to leaf paths, and at most one constant depth triangle
  • Theorem: The randomized complexity of thin-rich with approximate queries is Ω(log n)
    • Proof:Use Yao’s technique. For each vertex in the value tree, uniformly at random pick the edge to label ½.The expected number of labels of ½ on all known labeled paths after k queries is O( (log3 n)/3 + k)
outline20
Outline
  • Deterministic Ω(n log n) Lower Bound
  • Hint at Randomized Ω(n log n) Lower Bound with Approximate Cuts
  • Randomized O(n) Upper Bound
    • O(1) complexity randomized protocol for Thin-Rich
    • Cake cutting algorithm
    • Generalized offline power of two choices lemma
    • Non-independent random graph model
o 1 complexity randomized protocol for for thin rich
O(1) Complexity Randomized Protocol for for Thin-Rich
  • Pick an i uniformly at random from 0 … n-1
  • x = Cut[0, i/n]
  • y = Cut[ 0, (i+1)/n]
  • If (y-x) ≤ 2/n then return piece [x, y]
  • Else goto step 1
randomized protocol for cake cutting
Randomized Protocol for Cake Cutting
  • Protocol Description:
    • Each player repeatedly applies randomized thin-rich protocol to get 2d pieces
    • For each player, pick one of the two tentative pieces in such a way that every point of cake is covered by at most O(1) pieces. If this is not possible, then start over again.
  • Theorem: This protocol is approximately fair
  • We need to show that the second step of the protocol is successful with probability Ω(1)
digression 1
Digression(1)
  • Power of Two Choices Setting: n balls, each of which can be put into two of n bins that are selected independently uniformly at random
  • Online Theorem: The online greedy assignment guarantees maximum load of O(log log n) whp
  • Offline Theorem: There is an assignment with maximum load O(1) whp
digression 2 proof of offline power of two choices theorem
Digression(2): Proof of Offline Power of Two Choices Theorem
  • Consider a graph G
    • Vertices = bins
    • One edge for each ball connecting the corresponding vertices
    • Important: Edges are independent
  • Lemma: If G is acyclic then the maximum load is 1
  • Classic Theorem: If a graph G has n/3 independent edges, then G is acyclic whp
    • Proof: Union Bound.
    • Prob[G contains a cycle C] ≤ ΣC Prob[C is in the graph] ~ Σi (n choose i) * (1/3n)i
key theorem for o n bound generalized offline balls and bins
Key Theorem for O(n) Bound: Generalized Offline Balls and Bins
  • Each of n players arbitrarily partition [0, 1] into n pieces
  • Each player picks uniformly at random 2*d pieces
  • Then with probability Ω(1), we can assign to each player one of its 2*d pieces so that every point is covered by at most O(1) pieces
  • This is equivalent to offline balls and bins if the partition is into equal sized pieces, except that:
    • We may need d > 1, and
    • We don’t get high probability bound
why a high probability result is not achievable
Why a High Probability result is Not Achievable

.

.

.

Probability of overlap of k ~ (n choose k) / nk

problem case forks
Problem Case: Forks
  • Theorem: With probability Ω(1) there is no fork of depth ω(1)
  • Therefore we throw out forked paths, and proceed

Fork of depth 3

sufficiency condition
Sufficiency Condition
  • Theorem: The maximum load is at most 1 if there is not directed path between the two pieces of the same person
dealing with dependent edges
Dealing with Dependent Edges
  • Lemma: There are not many dependent edges
  • Lemma: Each possible path, between two pieces of the same player, can have at most two dependent edges
  • Lemma: With probability Ω(1) there is no path between two pieces of the same player
conclusions
Conclusions
  • Generalized offline balls and bins theorem may be useful elsewhere
  • The model of random graphs, where there are some dependencies on the edges, and our analysis may be useful elsewhere
    • Is dependent random graph model novel ?