Mixing Times of Self-Organizing Lists and Biased Permutations

Sarah Miracle

Georgia Institute of Technology

Joint work with . . . .

Prateek Bhakta Dana Randall Amanda Streib

Biased Card Shuffling

5 n 1 7 3... i j ... n-1 2 6

- pick a pair of adjacent cards uniformly at random- put j ahead of i with probability pj,i = 1- pi,j

Converges to: π(σ) = Π / Zi<j:σ(i)>σ(j) pij

pji

This is related to the “Move-Ahead-One Algorithm” for self-organizing lists.

Mnn

What is already known?

- proved this conjecture for n ≤ 4

Q: If the pij are positively biased, is Mnn always rapidly mixing?

If pi,j ≥ ½ ∀ i < j we say the chain is positively biased.

Conjecture (Fill): If pij satisfy a “regularity” condition, then the spectral gap is minimized when pij = 1/2 ∀ i,j

• 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]

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]

Talk Outline

1. Background

2. New Classes of Bias where Mnn is fast

• Choose your Weapon• League Hierarchies

3. Mnn can be slow

The mixing time

Definition: The total variation distance is

||Pt,π|| = max __ ∑ |Pt(x,y) - π(x)|.xÎ Ω yÎ Ω 2

1

A Markov chain is rapidly mixing if t(e) is poly(n, log(e-1)). (or polynomially mixing)

Definition: Given e, the mixing time is

(t e) = min t: ||Pt’,π|| < e, t’ ≥ t. A

A Markov chain is slowly mixing if t(e) is at least exp(n).

Choose your weapon

Thm 1: Let pi,j = ri ∀ i < j. Then MNN is rapidly mixing.

Proof sketch:A. Define auxiliary Markov chain M’B. Show M’ is rapidly mixingC. Compare the mixing times of MNN and M’

M’ can swap pairs that are not nearest neighbors- Maintains the same stationary distribution- Allowed moves are based on inversion tables

Given r1, … ,rn-1, ri ≥ 1/2.

Inversion Tables

3 3 2 3 0 0 0

Inversion Table Iσ:

5 6 3 1 2 7 4

Permutation σ:

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.

Iσ(1) Iσ(2)

Inversion Tables

3 3 2 3 0 0 0

Inversion Table Iσ:

5 6 3 1 2 7 4

Permutation σ:

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

Choose your weapon

Thm 1: Let pi,j = ri ∀ i < j. Then MNN is rapidly mixing.

Proof sketch:A. Define auxiliary Markov chain M’B. Show M’ is rapidly mixingC. Compare the mixing times of MNN and M’

Given r1, … ,rn-1, ri ≥ 1/2.

Inversion Tables

3 3 2 3 0 0 0

Inversion Table Iσ:

5 6 3 1 2 7 4

Permutation σ:

Iσ(i) = # elements j > i appearing before i in σ

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)

- swap element i with the first j>i to the left w.p. ri- swap element i with the first j>i to the right w.p. 1-ri

M’ on Permutations

- choose a card i uniformly

Want stationary dist.:

5 6 3 1 2 7 4

Permutation σ:

5 6 2 1 3 7 4

Permutation τ:

π(σ) = Π / Zi<j:σ(i)>σ(j) pji

pij

P’(σ,τ)

P’(τ,σ)

π(τ) π(σ)

= p1,2 p3,2 p3,1

p2,1 p2,3 p1,3=

Comparing M’ and Mnn

- swap element i with the first j>i to the left w.p. ri- swap element i with the first j>i to the right w.p. 1-ri

M’ on Permutations

- choose a card i uniformly

P’(σ,τ) = r2 = p2,3

p3,2

p2,3=

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^j for all i < j. Then MNN is rapidly mixing.

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^j for all i < j. Then MNN is rapidly mixing.

A-League B-League

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^j for all i < j. Then MNN is rapidly mixing.

1st Tier

2nd Tier

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^j for all i < j. Then MNN is rapidly mixing.

Proof sketch:A. Define auxiliary Markov chain M’B. Show M’ is rapidly mixingC. Compare the mixing times of MNN and M’

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^j for all i < j. Then MNN is rapidly mixing.

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^j for all i < j. Then MNN is rapidly mixing.

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^j for all i < j. Then MNN is rapidly mixing.

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^j for all i < j. Then MNN is rapidly mixing.

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

Want stationary distribution:

π(σ) = Π / Zi<j:σ(i)>σ(j) pji

pij

P’(σ,τ)

P’(τ,σ)

π(τ) π(σ)

= p1,5 p3,5 p6,8 p6,9 p1,6 p3,6 p5,8 p6,9 p5,6

p5,1 p5,3 p8,6 p9,6 p6,1 p6,3 p8,5 p9,6 p6,5

=

What about the Stationary Distribution?

P’(σ,τ) =1 – v5^6 = p6,5

p5,6

p6,5

=

9 3 8 5 7 4 2

Permutation τ:

169 3 8 6 7 4 2

Permutation σ:

15

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.

MNN is also rapidly mixing if p is weakly regular.

i.e., for all i, pi,j < pi,j+1 if j > i. (by comparison)

ß

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

Talk Outline

1. Background

2. New Classes of Bias where Mnn is fast

• Choose your Weapon• League Hierarchies

3. Mnn can be slow

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

1 3 n2 ... ...2 n +1 n

2

always in order

1 if i < j ≤ 2

n or < i < j 2 n

pij =

Permutation σ:

1 62 7 83 9 54 10

1’s and 0’s 2 n

2 n

1 01 0 01 0 11 0

Slow mixing example

Thm 3: There are examples of positively biased p for which MNN is slowly mixing.

1. Reduce to biased lattice paths

1 if i < j ≤ 2

n or < i < j 2 n

pij = 1/2+1/n2 if i+(n-j+1) < M

Each choice of pij where i ≤ < j2

n

determines the bias on square (i, n-j+1)

(“fluctuating bias”)

1/2 + 1/n2

M= n2/3

1- δ otherwise 1-δ

2. Define bias on individual cells (non-uniform growth proc.)

[Greenberg, Pascoe, Randall]

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

Implies that MNN can take exp. time to reach stationarity.

S SW

Therefore biased card shuffling can be slow too!

Open Problems

1. 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?

Thank you!