Shuffling. Shahrzad Haddadan. March 7, 2013

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1 Shufflig Shahrzad Haddada March 7, 2013 Abstract I this paper we will talk about two well-kow shufflig methods, the Top to Radom ad the Riffle Shuffle. We are iterested i the umber of shuffles that will make the deck of cards uiformly radom. 1 Itroductio Shufflig cards has bee iterestig to mathematicias for more tha fifty years. Other tha its mathematical beauty, shufflig is also studied regardig some applicatios i other fields such as Biology ([2],[3]) ad also Cryptography ([4]). I this paper, we will talk about the Top to radom shuffle ad Riffle shuffle. We will try to show their exact mixig time ad also boud the deviatio ad separatio distaces. 2 Prelimiaries Let s say we have a deck of size. Ay arragemet of cards is essetially a permutatio of the umbers 1... Durig this talk, S equals the set of all permutatios of 1... Shufflig is othig but aradomwalkos with uiform statioary distributio. The questio is, give a radom walk, how may steps will make the distributio close to uiform distributio. I order to measure beig close we eed the followig defiitios 1 : Defiitio 2.1. Give two distributios µ ad γ o some set S, we defie the stadard deviatio to be µ γ TV = 1 2 x S µ(x) γ(x). 1 The defiitios ad lemmas of this sectio ca be foud i [1] ad [5]. 1

2 Give P the trasitio matrix of a radom walk o state space S, x S the startig state, t a iteger represetig the umber of steps ad π the statioary distributio of the radom walk, we deote the deviatio distace by d(t) whichisdefiedbyd(t) := max x S P t (x,.) π TV. We say a chai is mixed at time t if d(t) 1/4. We also defie the separatio distace s x (t) by,s x (t) :=max y S [1 P t (x,y) π(y) ]. Lemma 2.1. Let P be the trasitio matrix of a Markov chai o state space S, x the startig state ad t the umber of steps. We have P t (x,.) π TV s x (t). Proof. P t (x,.) π TV = y S P t (x,y)<π(y) [π(y) P t (x, y)] = max[1 P t (x, y) ]=s x (t). y π(y) We also eed some defiitios regardig exact mixig time. y S P t (x,y)<π(y) π(y)[1 P t (x, y) ] π(y) Defiitio 2.2. Give a Markov chai M o a state space S, letw be the space of all fiite walks of M. Astoppig rule is map Γ : W [0, 1] which shows the probability of cotiuig awalkw =(w 1,...,w t ) W. We defie stoppig time to be the radom variable τ with values {0, 1,...} that is equal to the radom time that Γ stops. 2 rule. The exact mixig time of a chai is the expected stoppig time for a optimal stoppig Defiitio 2.3. For the chai M =(M t ), a strog statioary time, τ is a stoppig time which satisfies, Pr x (τ = t, M τ = y) =Pr x (τ = t)π(y). Where π is the statioary distributio. Theorem 2.2. (Lovász ad Wikler [5]) 3 A stoppig rule is optimal if ad oly if it has a haltig state. A haltig state is a state that give that we already stopped we kow it was ever exited. Lemma 2.3. (Aldous ad Diacois [6]) Let x be the startig state of a radom walk (X t ) ad τ a strog statioary time, the s x Pr x (τ>t). 2 Check out [5] for more details o stoppig rules ad stoppig time. 3 We do t brig the proof here. However, it ca be foud i [5] 2

3 The, Proof. Let y be the state for which the maximum of 1 Pr x (X t = y)/π(y) is achieved. s x (t) =1 Pr x(x t = y) π(y) 1 Pr x(x t = y, τ t) π(y) =1 Pr x(τ t)π(y) π(y) = Pr x (τ>t). 3 Top to Radom Shuffle Cosider the followig method of shufflig for a deck of size. At each step, take the first card ad isert it uiformly i ay of places that is left i the deck. Formally, we have a radom walk o S with the followig trasitio matrix: 1/ if γ =(σ2,σ Pr(σ, γ) = 3,...,σ i,σ 1,σ i+1,...,σ ), 1 i 0 otherwise. 3.1 Exact mixig Lemma 3.1. For a deck of cards, the exact mixig time for top to radom shuffle is (H 1 1)+1. Where H = i=1 1/i. Whe the exact mixig time coverges to log( 1) +1. Proof. Followig is a a optimal stoppig rule for top to radom shuffle. Let s say we start from σ =(σ 1,...,σ ). Mark the card σ 1.Shuffleutilthemarkedcardgetstothetop.Do oe more shuffle. Stop. It is easy to see that whe we stop we are i uiform distributio. Moreover, this rule is optimal sice ay state i which the deck has σ o top is a haltig state. Now, we eed to calculate the expected time that takes the rule to stop. Let T i be time that it takes util i cards get uder card σ 1. We kow that E[T 1 ]=0. Now cosider the time T i+1 T i.thisisthetimethatiseededforaothercardtogetuder the card σ 1 give that there are already i cards below σ 1. Note that T i+1 T i has geometric distributio with parameter (i+1)/. Therefore, we have E[T i+1 T i ] = /(i+1). Let τ be the stoppig time for top to radom shuffle. We have, E[τ] = E[T 1 ]+1 = E[T 1 ]+E[T 2 T 1 ]+...+E[T 1 T 2 ]+1 = 1 i=2 (1/i)+1 = (H 1 1)

4 3.2 Covetioal mixig time ad bouds o separatio distace ad deviatio distace Lemma 3.2. Give a deck of size, the mixig time of the top to radom shuffle is less tha log. Proof. The stoppig time τ that we gave i proof of Lemma 3.1 is i fact a strog statioary time. Let P be the trasitio matrix of the chai ad U the uiform distributio. We have P t (σ,.) U s σ (t) Pr(τ >t). Claim. Pr(τ >log +c) e c. Cosider the coupo collector problem 4. Notice that for T i siproofoflemma3.1wehavepr(t i T i 1 = j) = i (1 i )j 1 which is the same probability of how log it takes for the coupo collector to collect the i+1st coupo after collectig the ith oe. The stoppig time τ i the above proof is i fact equal to the time it takes for the coupo collector to collect the last 1st cards(2d, 3rd,... th card) plus oe extra step which is equal to the time eeded to collect all 1,..., coupos. Now, let s try to upper boud Pr(τ >log + c). Let A i be the evet that the collector does ot collect the coupo umber i till time log + c. Wehave Pr(τ >log +c) Pr(A i )= i=1 (1 1 ) log +c log + c exp( )=e c. i=1 4 Riffle Shuffle Riffle shuffle is a very commo way of shufflig. I Riffle shuffle oe divides the deck to two piles ad successively drop cards from the bottom of each pile. I 1955, Gilbert ad Shao ad idepedetly Reeds i 1981 established a good mathematical modelig of the problem. I 1992 Diacois ad Bayer aalysed the Riffle shuffle. Here, as we did for top to radom case, we will discuss the exact mixig time ad bouds o deviatio ad separatio distaces. The, we will talk about the famous result of Bayer ad Diacois that has bee famous i ews as 7 shuffles is eough. 4.1 Modelig of the problem Defiitio 4.1. The followig four modeligs of Riffle shuffle are equivalet: 4 To kow more about the coupo collector problem, read [1] Sectio 2.2 or [6]. 4

5 I. Let m be take radomly from Biomial(, 1/2). Split the card to piles of size m ad m. Let a card drop from left pile with probability a/(a+b) ad from right pile with probability b/(a+b), where a is the umber of cards left i left pile ad b is the umber of cards left i right pile. II. Let m be take radomly from Biomial(, 1/2). Split the card to piles of size m ad m. Chooseoeofthe m possible arragemets of these card uiformly at radom. III. Place poits x 1,...,x uiformly ad idepedetly i uit iterval. Assig the cards i their order to x 1,...,x. Apply the mappig x 2x to the poits. Rearrage the cards accordig to the ew orderig. Here, we do t give a proof of these defiitios beig equivalet but it ca be easily checked that all of them will yield the followig distributio. 5 (+1)/2 if γ = σ Pr(σ, γ) = 1/2 if γ = σ λ ad λ has exactly two risig sequeces(defiitio 4.2). 0 otherwise. (1) Defiitio 4.2. Arisigsequeceiapermutatioisthemaximalsetofcosecutiveumbers that occur i the correct order. For example (2, 3, 1, 4, 6, 5, 7) has three risig sequeces {2, 3, 4, 5}, {1}, {6, 7}. 4.2 Exact mixig ad bouds o separatio ad deviatio distaces I order to aalyse the Riffle shuffle it is easier to look at its reverse. Accordig to a theorem by Wikler ad Lovász ([8]) the exact mixig time of a chai ad its reverse are equal. The reverse of Riffle shuffle is famous as ushuffle. A step of ushuffle is performed by: Defiitio 4.3. (Ushuffle) To each card i the deck assig a uiformley radom bit (0 or 1). Pull the cards with label 0totopofthedeckpreservigtheirrelativeorder. Thecardswithlabel1willstayatthe bottom preservig their relative order. 5 For more details please check [1], Chapter 8, Sectio 8.3,[7] or [9] 5

6 Cosiderig defiitio (II) it is ot hard to see that ushuffle is the reverse of shuffle. Lemma 4.1. Let τ to be a optimal stoppig time for Riffle shuffle ad τ to be a optimal stoppig time for ushuffle. We have E[τ] =E[ τ] 2log. Proof. The followig is a optimal stoppig rule for ushuffle. Ushuffle the cards ad at each step, keep track of the bits that are assiged to each card. After t steps ay card will be associated with a legth t biary umber. Stop whe all umbers are differet. It is easy to check that this stoppig rule geerates the uiform distributio ad the iverse of startig permutatio is the haltig state. Now, we should calculate the expectatio of stoppig time. Notice that we stop after t steps if we got differet umbers whe we are allowed to choose from {0, 1,...,2 t 1}. Therefore, we have a istace of the Birthday problem. We use the results from Birthday problem to boud the expected stoppig time. Pr(τ t) =Pr(we have distict umbers i rage {1,...2 t })=(1 1 2 t )(1 2 2 t )...(1 ) Π 2 i=1(e i/2t )=e ( 1 t 2 t ) i=1 i e (2 /2 t). Therefore, we have E(τ) = t=1 Pr(τ t) t=1 (1 /2 t) e(2 )= log 2 t=1 (1 /2 t) e(2 )+ t=log 2 +1 (1 /2 t) e(2 ) log 2 log +1 (2 /2 t ) 2log(). Usig the result from Lovász ad Wikler ([8]) 6,wekowthattheaboveexactmixig time for ushuffle is also the exact mixig time of shuffle. Corollary. For the Riffle shuffle the covetioal mixig time is bouded by 2 log. Proof. Let X t be distributio of the deck after t Riffle shuffles. Sice τ is strog statioary time, we have X t U Pr(τ > t). For t =log( 2 /c), we have Pr(τ > log( 2 /c)) 1 e c c. 4.3 Mixig time of Riffle shuffle. Is seve shuffles eough? Now, we kow the exact mixig time of the Riffle shuffle ad it also gives us a boud o mixig time. However, our discussio will ot be complete uless we discuss the famous paper of Bayer ad Diacois, Trailig the Dovetail shuffle to its lair ([7]). This result has bee i the ews 7 ad famous as I shufflig cards, 7 is the wiig umber. As you will see i the 6 The exact mixig time of a chai ad its reversal are equal. 7 Kolata, Gia (Jauary 09, 1990). I Shufflig Cards, 7 Is Wiig Number. New York Times. Retrieved

7 followig, the fact that seve shuffles is eough to make a card radom has bee questioed alot. Wewillalsodiscusshowumber seve mightbesigificatishufflig52cards although it is NOT specifically discussed i the paper. Here is the most importat theorem i [7]. Theorem 4.2. If cards are shuffled m times, the the chace that the deck is i arragemet π is 2 m + r /2 m, where r is the umber of risig sequeces i π. Sketch of Proof. First cosider a geeralizatio of Riffle shuffle to a-shuffle where the deck is cut to a piles ad the the piles will be iterleavig ito each other. Defiitio 4.1 will have the followig formulatio i geeral case. Defiitio 4.4. The followig four modeligs of a-shuffle are equivalet: I. Let m 1,...,m a be take radomly from multiomial (, 1/a,..., 1/a). Let a card drop from pile i with probability x i /(x x a )), where x i is the umber of cards left i pile i. II. Let m 1,...,m a be take radomly from multiomial (, 1/a,..., 1/a). Split the card to piles of size m 1,...,m a.chooseoeofthe m 1,...,m a possible arragemets of these card uiformly at radom. III. Place poits x 1,...,x uiformly ad idepedetly i uit iterval. Assig the cards i their order to x 1,...,x. Apply the mappig x ax to the poits. Rearrage the cards accordig to the ew orderig. Cosiderig part III of the defiitio we ca see a ab-shuffle is equivalet to performig a a-shuffle first ad the a b-shuffle. As a result, i order to fid out the distributio after k, 2-shuffles, it will be eough to calculate the distributio of the deck after oe 2 m shuffle. The proof of Theorem 4.2 will the be a result of the followig lemma 8 : Lemma 4.3. If a a-shuffle is performed o a deck of cards, the the chace that the deck is i arragemet π is a+ r /a, where r is the umber of risig sequeces i π. 8 For more details please read [7] ad [9]. 7

8 Proof of lemma: Ay rearragemet cosists of a cut ad iterleave. The probability of 1 acutaditerleaveis ( ) m 1,...,m a /a =1/a. Therefore, it suffices to cout the m 1,...,ma umber of rearragemets that will geerate π. Notice that whe the cut is specified the iterleavig will be forced ad each of the risig sequeces is a uio of some piles. Therefore, r 1 cutsareforced.fortheresta r+1, the umber of possible cuts will be a+ r. Corollary. Let Q t be the distributio of the deck after t shuffles. As a result of Theorem 4.2 we have Q t U = t + r A,r /2 t 1! r=1 Where A,r is the umber of permutatios of 1... with r risig sequeces which is kow as Euleria umbers. Havig the above formulatio, for arbitrary, Bayer ad Diacois approximate the deviatio distace ad they observe that it does ot chage much till it reaches the poit t = 3 2 log the at this poit it drops i a cosiderable amout. They call this pheomeo, the cut off pheomeo which is a more powerful cocept tha mixig. You ca read more about the cut off pheomeo i Chapter 18 of [1] ad also [7]. Note. For a deck of 52 cards we ca calculate the deviatio distace after t =1,...10 Riffle shuffles. The followig table ca be foud i [7]. t dv(t) We otice that before the 7th step the deviatio distace is ot chagig much. However, at the 7th shuffle it almost halves ad it cotiues gettig half of its previous amout after each sigle shuffle. Therefore, time 7 ca be cosidered as the cut off poit. This might be the reaso that the result has bee famous as seve shuffles is eough. However, we defiitely kow that the distributio after 7 shuffles is still far from beig uiform. Also, ote that there are 11 steps eeded if you cosider exact mixig. I this regard, Peter Boyle 9 has costructed a game o which oe s chace of wiig after seve shuffles is 0.8 althoughitis 0.5 withuiformdeck. 9 You ca read about Doyle s game of New Age Solitaire i [9] 8

9 5 Coclusio ad other works There are some other papers of Diacois et al lookig at the Riffle shuffle whe we have repeated cards([10], [11]). This is specially iterestig because i most of the card games some of the cards are treated equally. They show that if the deck cosists of two differet type of cards, the mixig time will be log + c 10.Specifically,foradeckof52cards4shuffleswillbe eough 11. What came here was a very cocise talk about two well kow shuffles. There are still much more ways of shufflig that have bee studied by mathematicias. The iterested reader should also check out the radom traspositio(chapter 8 of [1]) ad Thrope shuffle([12]). Also, if you wat to kow more about Riffle shuffle, check Persi Diacois s website 12. Refereces [1] D. Levi, Y. Peres, E. Wilmer (2008), Markov Chais ad Mixig Times, America Mathematical Society Press. [2] D. Kadel, Y. Matias, R. Uger ad P. Wikler (1996), Shufflig Biological Sequeces, Disc. Appl. Math. 71. [3] R. Durret (2003), Shufflig Chromosomes, J. Theoret. Probab. 16, [4] B. Morris (2012), P. Rogaway ad V. T. Hoag A Ecipherig Scheme Based o a Card Shuffle. CRYPTO. [5] L. Lovász ad P. Wikler (1998), Mixig times, Microsurveys i Discrete Probability, D. Aldous ad J. Propp, eds., DIMACS Series i Discrete Math. ad Theoretical Computer Sciece 41, Amer. Math. Soc., Providece RI, pp [6] D. Aldous, P. Diacois (1986), Shufflig Cards ad Stoppig Times. Math l Mothly, 93(5): [10] Shufflig the cards: Math does the trick. Sciece News. Friday, November 7, Retrieved 14 November Diacois ad his colleagues are issuig a update. Whe dealig may gamblig games, like blackjack, about four shuffles are eough cgates/persi/idex.html 9

10 [7] D. Bayer, ad P. Diacois (1992), Trailig the Dovetail shuffle to its lair, A. Appl. Probab 2, [8] L. Lovász ad P. Wikler (1998), Reversal of Markov chais ad the forget time, Combiatorics, Probability ad Computig 7:1, pp [9] B. Ma (1995), How may times should you shuffle a deck of cards? I Topics i Cotemporary Probability ad its Applicatios (J.L. Sell, ed.), CRC Press, Boca Rato. [10] P. Diacois, K. Assaf, S. Soudararaja (2008), A Rule of Thumb for Riffle Shufflig. Aals of Applied Probability, 21(3): arxiv: v1. [11] P. Diacois, K. Assaf, S. Soudararaja (2009), Riffle shuffles of a deck with repeated cards. DMTCS Proceedigs, 21st Iteratioal Coferece o Formal Power Series ad Algebraic Combiatorics (FPSAC 2009),0(1): [12] B. Morris (2005), The mixig time of the Thorp shuffle. SIAM joural o computig, STOC. 10