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## Cake Cutting is and is not a Piece of Cake

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**Cake Cutting is and is not a Piece of Cake**Jeff Edmonds, York University Kirk Pruhs, University of Pittsburgh**Informal Problem Statement**Resource allocation between n possibly deceitful players**I like**I like 0 1 Classic Problem Definition = [0, 1] • n players wish to divide a cake • Each player p has an unknown value function Vp**v**I like x y 0 1 Classic Problem Definition = [0, 1] • n players wish to divide a cake • Each player p has an unknown value function Vp • Allowed Operations : • Eval[p, x, y]: returns how much player p values piece/interval [x, y]**v**I like x y 0 1 Classic Problem Definition = [0, 1] • n players wish to divide a cake • Each player p has an unknown value function Vp • Allowed Operations : • Eval[p, x, y]: returns how much player p values piece/interval [x, y] • Cut[p, x, v]: returns a y such Eval[p,x, y] = v**1/n**I like 0 1 Classic Problem Definition = [0, 1] • n players wish to divide a cake • Each player p has an unknown value function Vp • Goal: Fair cut Each honest player p is guaranteed a piece of value at least 1/n.**1/n**I like 0 1 Classic Problem Definition = [0, 1] • n players wish to divide a cake • Each player p has an unknown value function Vp • Goal: Fair cut Each honest player p is guaranteed a piece of value at least 1/n.**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**I like**I like O(n log n) Divide and Conquer Algorithm: Evan and Paz • Yp = cut(p, 0, 1/2) for p = 1 … n My half cut is here. My half cut is here.**I like**O(n log n) Divide and Conquer Algorithm: Evan and Paz • Yp = cut(p, 0, 1/2) for p = 1 … n My half cut is here.**O(n log n) Divide and Conquer Algorithm: Evan and Paz**• Yp = cut(p, 0, 1/2) for p = 1 … n • m = median(y1, … , yn)**I like**I like so am happy with the left. so am happy with the right. O(n log n) Divide and Conquer Algorithm: Evan and Paz • Yp = cut(p, 0, 1/2) for p = 1 … n • m = median(y1, … , yn) • Recurse on [0, m] with those n/2 players p for which yp < m • Recurse on [m, 1] with those n/2 players p for which yp > m • Time O(nlogn)**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 Variations**(Approximate) * Submitted to STOC**Outline**• Deterministic Ω(n log n) Lower Bound • Randomized with Approximate Cuts Ω(n log n) Lower Bound • Randomized with Exact Cuts O(n) Upper Bound**At least n/2 players require thin rich piece**Thin-Rich Game • Game Definition: Single player must find a thin rich piece. • A piece is thin if it has width ≤ 2/n • A piece is rich if it has value ≥ 1/2n • Theorem: The deterministic complexity of Thin-Rich is Ω(log n). • Theorem: The deterministic complexity of cake cutting is Ω(nlog n).**Alg**Adv • I give sequence of Eval[x,y] & Cut[x,v]operations. • I dynamically choose how to answer**Alg**Adv • I can choose any non-continuous thin piece, • but W.L.G.I choose one of these. • I cut the cake in to n thin pieces.**Alg**Adv ... ... ... ... ... ... • I build a complete 3-ary treewith the n pieces as leaves**Alg**Adv ½ ¼ ½ ¼ ¼ ¼ ... ... ... ... ... ... • For each node, • I label edges • <½,¼,¼> or <¼,¼,½>**Alg**Adv ½ ¼ ½ ¼ ¼ ¼ ¼ ¼ ½ ... ... ¼ ¼ ... ... ... ½ ... 1/1024 = ¼×¼×½×¼×¼×½ • Value of each piece isproduct of edge labelsin path.**Alg**Adv ½ ¼ ½ ¼ ¼ ½ ¼ ¼ ½ ... ... ¼ ¼ ... ... ... ½ ... 1/256 = ½×¼×½×¼×¼×½ To get a rich pieceI need at least 40% of the edge labels in path be ½. Good luck**Alg**Adv 0.4398 0 y • I need to find a yso that V[0,y] = 0.4398. • Cut[0,0.4398]?**Alg**Adv 1/2 0 1 0.4398 0.4398 • I do binary search to find0.4398 • Cut[0,0.4398]? 1/4**Alg**Adv ¼ ½ ¼ 1/4 2/4 0.4398 • I do binary search to find0.4398 • I fix some edge labels • Cut[0,0.4398]?**Alg**Adv ¼ ½ ¼ 0.4398 0.4398 • I do binary search to find0.4398 • I fix some edge labels • Cut[0,0.4398]? 1/4 2/4 6/16 7/16 4/16 8/16**Alg**Adv ¼ ½ ½ ¼ ¼ ¼ 7/16 8/16 0.4398 • I fix some edge labels • Cut[0,0.4398]?**Alg**Adv ¼ ¼ ¼ ½ ½ ½ ½ ½ ¼ ¼ ¼ ¼ ¼ ¼ ¼ 0.4398 y ¼ ¼ ½ • I find a yso that V[0,y] = 0.4398. • Cut[0,0.4398]?**Alg**Adv ¼ ¼ ¼ ½ ½ ½ ½ ½ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ½ • I learned a path,but all its labels are ¼ • Hee Hee**Alg**Adv ¼ ¼ ¼ ½ ½ ½ ½ ½ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ½ • YesAfter t operationsevery path has only t known ½ labels. • Every path has oneknown ½ label.**Alg**Adv ¼ ¼ ¼ ½ ½ ½ ½ ½ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ½ y x • I fix labels in path to x & y to ¼. • and give • Eval[x,y] = 0.00928 • Eval[x,y] 0.00928**Alg**Adv ¼ ¼ ¼ ½ ½ ½ ½ ½ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ½ ¼ ¼ ¼ ¼ ½ • YesAfter t operationsevery path has only t½ known labels.**Deterministic Ω(log n) Lower Bound**• Theorem: To win at Thin-Rich, the alg has to get a rich piece with at least 40% of the edge labels in path be ½. • Theorem: After t operations, every path has only t ½ labels. • Theorem: The deterministic complexity of Thin-Rich is Ω(depth) =Ω(log n)**At least n/2 players require thin rich piece**Deterministic Ω(nlog n) Lower Bound • Theorem: The deterministic complexity of Thin-Rich is Ω(log n). • Theorem: The deterministic complexity of cake cutting is Ω(n log n).**Outline**Done • Deterministic Ω(n log n) Lower Bound • Randomized with Approximate Cuts Ω(n log n) Lower Bound • Randomized with Exact Cuts O(n) Upper Bound Randomized Approximate Cuts**Adv**• I must choose the value functions • without knowing • coin flips • I define a randomized algorithm Rand Alg**Adv**Yau Alg RandAdv • I must choose the value functions • without knowing • coin flips • I define a randomized algorithm Rand Alg • I flip coins to choose value function. • I dynamically choose error in answers • I deterministically give sequence of Eval[x,y] & Cut[x,v]operations.**Alg**AdvRand ½ ¼ ¼ ¼ ½ ¼ ¼ ½ ¼ ... ... ... ... ... ... • For each node, • I randomly label edges • <½,¼,¼>, <¼,½,¼>, or <¼,¼,½>**Alg**Adv ¼ ½ ¼ ¼ ¼ ½ ¼ ½ ¼ ¼ ½ ½ ¼ ¼ ¼ ½ ¼ ¼ y x • Consider path to x and y. • Flip coins for labels. • 33% of labels will be ½. • Eval[x,y] • But I need 40%!**Alg**Adv ¼ ¼ ½ ¼ ¼ ½ ¼ ½ ¼ ¼ ½ ½ ¼ ¼ ½ ¼ ½ ¼ ¼ ¼ ¼ ½ ¼ ½ ¼ ¼ y x • I flip coins for path to x’and get 33% ½. • Cut[x’,0.4398]? x’**Alg**Adv ¼ ¼ ½ ¼ ¼ ½ ¼ ½ ¼ ¼ ½ ¼ ½ ¼ ¼ ½ ¼ ½ ¼ ¼ ¼ ¼ ¼ ½ ¼ ½ ¼ ¼ y x • I do binary search for 0.4398, • but for some odd reasonit finds 40%½ labels. • Cut[x’,0.4398]? ½ ½ ½ x’**Alg**Adv ¼ ¼ ½ ¼ ¼ ½ ½ ¼ ½ ¼ ½ ¼ ½ ¼ ½ ¼ ¼ ½ ½ ¼ ½ ¼ ¼ ¼ ¼ ¼ ½ ¼ ½ ¼ ¼ y x • Luckily I can give error and this hides most of the labels. • Cut[x’,0.4398]? x’**Outline**Done • Deterministic Ω(n log n) Lower Bound • Randomized with Approximate Cuts Ω(n log n) Lower Bound • Randomized with Exact Cuts O(n) Upper Bound Done Randomized Exact Cuts O(n) Upper**O(1) Complexity Randomized Protocol for Thin-Rich**Protocol Description: • “Cuts” cake into n “candidate” pieces of value 1/n • Randomly chooses one. • It is likely thin and rich.**Randomized Protocol for Cake Cutting**Protocol Description: • Each player randomly selects 2d candidate pieces. • For each player, we carefully pick one of these**Randomized Protocol for Cake Cutting**Protocol Description: • Each player randomly selects O(1) candidate pieces. • For each player, we carefully pick one of these • so that every point of cake is covered by at most O(1) pieces. • Where there is overlap, recurs. Works with O(1) probability.**Balls and Bins**Two Random Choices: n balls, n bins each ball randomly chooses two bins. Choice: Select one of two bins for each ball. Whp no bin has more than O(1) balls.**Balls and Bins**Two Random Choices: n balls, n bins each ball randomly chooses two bins. Choice: Select one of two bins for each ball. Whp no bin has more than O(1) balls.**A piece of cake.**Outline Done • Deterministic Ω(n log n) Lower Bound • Randomized with Approximate Cuts Ω(n log n) Lower Bound • Randomized with Exact Cuts O(n) Upper Bound Done More at STOC