Mixing Times of Self-Organizing Lists and Biased Permutations

1. Mixing Times of Self-Organizing Lists and Biased Permutations Sarah Miracle Georgia Institute of Technology

2. Joint work with . . . . PrateekBhakta Dana Randall Amanda Streib

3. pji π(σ) = Π/ Z Converges to: pij i<j:σ(i)>σ(j) Biased Card Shuffling - pick a pair of adjacent cards uniformly at random - put j ahead of i with probability pj,i= 1- pi,j 5 n 1 7 ... i j ... n-1 2 6 3 Mnn This is related to the “Move-Ahead-One Algorithm” for self-organizing lists.

4. What is already known? • Uniform bias: If pi,j =½∀ i, j then Mnn mixes in θ(n3 log n) [Wilson] • Constant bias:If pi,j = p > ½ ∀ i < j, then Mnn mixes in θ(n2) time [Benjamini, Berger, Hoffman, Mossel] If pi,j ≥ ½ ∀ i < j we say the chain is positively biased. • Q: If the {pij} are positively biased, is Mnn always rapidly mixing? • Conjecture (Fill): If {pij} satisfy a “regularity”condition, then the spectral gap is minimized when pij= 1/2 ∀ i,j • - proved this conjecture for n ≤ 4

5. What is already known? • Uniform bias: If pi,j =½∀ i, j then Mnn mixes in θ(n3 log n) [Wilson] • Constant bias:If pi,j = p > ½ ∀ i < j, then Mnn mixes in θ(n2) time [Benjamini, Berger, Hoffman, Mossel] • Linear extensions of a partial order: If pi,j = ½ or 1 ∀ i < j, then Mnn mixes in O(n3 log n) time [Bubley,Dyer] • Mnn is fast for two new classes: “Choose your weapon” and “League hierarchies” [Bhakta, M., Randall, Streib] • Mnn can be slow even when the chain is positively biased [Bhakta, M., Randall, Streib]

6. Talk Outline • Background • New Classes of Bias where Mnn is fast • Choose your Weapon • League Hierarchies • Mnn can be slow ✔

7. Definition:Given e, the mixing time is t(e) = min {t: ||Pt’,π||<e, t’ ≥ t}. A The mixing time Definition: The total variation distance is 1 ||Pt,π|| = max __ ∑ |Pt(x,y) - π(x)|. 2 xÎ Ω yÎΩ A Markov chain is rapidly mixing if t(e) is poly(n, log(e-1)). (or polynomially mixing) A Markov chain is slowly mixing if t(e) is at least exp(n).

8. M’can swap pairs that are not nearest neighbors • Maintains the same stationary distribution • Allowed moves are based on inversion tables Choose your weapon Given r1, … ,rn-1,ri≥ 1/2. Thm 1: Let pi,j = ri ∀i < j. Then MNNis rapidly mixing. • Proof sketch: • A. Define auxiliary Markov chain M’ • B. Show M’ is rapidly mixing • C. Compare the mixing times of MNNand M’

9. Inversion Table Iσ: 3 3 2 3 0 0 0 Inversion Tables Permutation σ: 5 6 3 1 2 7 4 Iσ(2) Iσ(1) Iσ(i) = # elements j > i appearing before i in σ The map I is a bijection from Sn to T ={(x1,x2,...,xn): 0 ≤ xi ≤ n-i}.

10. 3 3 2 3 0 0 0 Permutation σ: Inversion Table Iσ: 5 6 3 1 2 7 4 Inversion Tables M’on Inversion Tables - choose a column i uniformly - w.p. ri: subtract 1 from xi (if xi>0) - w.p. 1- ri: add 1 to xi (if xi<n-i) M’is just a product of n independent biased random walks M’ is rapidly mixing. ß

11. Choose your weapon Given r1, … ,rn-1,ri≥ 1/2. Thm 1: Let pi,j = ri ∀i < j. Then MNNis rapidly mixing. • Proof sketch: • A. Define auxiliary Markov chain M’ • B. Show M’ is rapidly mixing • C. Compare the mixing times of MNNand M’ ✔ ✔

12. 3 3 2 3 0 0 0 Permutation σ: Inversion Table Iσ: 5 6 3 1 2 7 4 Inversion Tables Iσ(i) = # elements j > i appearing before i in σ M’on Permutations M’on Inversion Tables - choose a card iuniformly - choose a column i uniformly - w.p. ri: subtract 1 from xi (if xi > 0) - swap element i with the first j>ito the left w.p. ri - swap element i with the first j>i to the right w.p. 1-ri - w.p. 1- ri: add 1 to xi (if xi < n-i)

13. P’(σ,τ) • p2,1 p2,3 p1,3 • p2,3 • π(τ) Permutation τ: Permutation σ: • P’(σ,τ) = r2 = p2,3 5 5 6 6 2 3 1 1 3 2 7 7 4 4 • = • = • = • p1,2 p3,2 p3,1 • p3,2 • π(σ) • P’(τ,σ) Comparing M’ and Mnn Want stationary dist.: pij π(σ) = Π/ Z pji i<j:σ(i)>σ(j) M’on Permutations - choose a card iuniformly - swap element i with the first j>ito the left w.p. ri - swap element i with the first j>i to the right w.p. 1-ri

14. League Hierarchy Let T be a binary tree with leaves labeled {1,…,n}. Given qv ≥ 1/2 for each internal vertex v. Thm 2: Let pi,j = qi^jfor all i< j. Then MNNis rapidly mixing.

15. League Hierarchy Let T be a binary tree with leaves labeled {1,…,n}. Given qv ≥ 1/2 for each internal vertex v. Thm 2: Let pi,j = qi^jfor all i< j. Then MNNis rapidly mixing. A-League B-League

16. League Hierarchy Let T be a binary tree with leaves labeled {1,…,n}. Given qv ≥ 1/2 for each internal vertex v. Thm 2: Let pi,j = qi^jfor all i< j. Then MNNis rapidly mixing. 1st Tier 2ndTier

17. M’can swap pairs that are not nearest neighbors • Maintains the same stationary distribution • Allowed moves are based on the binary tree T League Hierarchy Let T be a binary tree with leaves labeled {1,…,n}. Given qv ≥ 1/2 for each internal vertex v. Thm 2: Let pi,j = qi^jfor all i< j. Then MNNis rapidly mixing. • Proof sketch: • A. Define auxiliary Markov chain M’ • B. Show M’ is rapidly mixing • C. Compare the mixing times of MNNand M’

18. League Hierarchy Let T be a binary tree with leaves labeled {1,…,n}. Given qv ≥ 1/2 for each internal vertex v. Thm 2: Let pi,j = qi^jfor all i< j. Then MNNis rapidly mixing.

19. Thm 2: Proof sketch Let T be a binary tree with leaves labeled {1,…,n}. Given qv ≥ 1/2 for each internal vertex v. Thm 2: Let pi,j = qi^jfor all i< j. Then MNNis rapidly mixing.

20. Thm 2: Proof sketch Let T be a binary tree with leaves labeled {1,…,n}. Given qv ≥ 1/2 for each internal vertex v. Thm 2: Let pi,j = qi^jfor all i< j. Then MNNis rapidly mixing.

21. Thm 2: Proof sketch Let T be a binary tree with leaves labeled {1,…,n}. Given qv ≥ 1/2 for each internal vertex v. Thm 2: Let pi,j = qi^jfor all i< j. Then MNNis rapidly mixing.

22. Thm 2: Proof sketch Markov chain M’ allows a transposition if it corresponds to an ASEP move on one of the internal vertices. q2^4 1 - q2^4

23. p6,5 • p5,1 p5,3 p8,6 p9,6 p6,1 p6,3 p8,5 p9,6 p6,5 • P’(σ,τ) • π(τ) • P’(σ,τ) =1 – v5^6 = p6,5 9 9 3 3 8 8 5 6 7 7 4 4 2 2 • = • = • p1,5 p3,5 p6,8 p6,9 p1,6 p3,6 p5,8 p6,9 p5,6 • p5,6 • π(σ) • P’(τ,σ) What about the Stationary Distribution? Want stationary distribution: pij π(σ) = Π/ Z pji i<j:σ(i)>σ(j) Permutation τ: Permutation σ: 1 5 6 1 • =

24. Thm 2: Proof sketch Markov chain M’ allows a transposition if it corresponds to an ASEP move on one of the internal vertices. Each ASEP is rapidly mixing M’ is rapidly mixing. ß MNNis also rapidly mixing if {p} is weakly regular. i.e., for all i, pi,j < pi,j+1 if j > i. (by comparison)

25. So MNN is rapidly mixing when Choose your weapon: pi,j = ri ≥ 1/2 ∀i < j Tree hierarchies: pi,j = qi^j> 1/2 ∀ i< j

26. Talk Outline • Background • New Classes of Bias where Mnn is fast • Choose your Weapon • League Hierarchies • Mnn can be slow

27. 1 n n n n n 1 n 1 2 6 0 7 0 1 3 0 8 9 0 4 1 5 1 0 10 1’s and 0’s or < i < j 1 if i < j ≤ +1 2 2 2 2 2 2 But…..MNN can be slow Thm 3: There are examples of positively biased {pij} for which MNN is slowly mixing. 1. Reduce to biased lattice paths { always in order pij = 1 2 3 ... ... n Permutation σ:

28. n Each choice of pij where i ≤ < j n n 2 2 2 Slow mixing example Thm 3: There are examples of positively biased {p} for which MNN is slowly mixing. 1. Reduce to biased lattice paths 2. Define bias on individual cells (non-uniform growth proc.) { 1 if i < j ≤ or < i < j pij = M= n2/3 1/2+1/n2 if i+(n-j+1) < M 1- δ otherwise 1-δ 1/2 + 1/n2 determines the bias on square (i, n-j+1) (“fluctuating bias”) [Greenberg, Pascoe, Randall]

29. Slow mixing example Thm 3: There are examples of positively biased {p} for which MNN is slowly mixing. 1. Reduce to biased lattice paths 2. Define bias on individual cells 3. Show that there is a “bad cut” in the state space W S S Implies that MNNcan take exp. time to reach stationarity. Therefore biased card shuffling can be slow too!

30. Open Problems Is MNN always rapidly mixing when {pi,j} are positively biased and satisfy a regularity condition? (i.e., pi,j is decreasing in i and j) 2. When does bias speed up or slow down a chain?

31. Thank you!