Mixing Times of Markov Chains for Self-Organizing Lists and Biased Permutations

Prateek Bhakta, Sarah Miracle,

Dana Randall and Amanda Streib

Sampling Permutations

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

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

Is M always rapidly mixing?

Conjecture [Fill]: If pij are positively biased and monotone then M is rapidly mixing.

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

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

M is not always fast . . .

What is already known?

• Uniform bias: If pi,j = ½ ∀ i, j then M mixes in θ(n3 log n) time. [Aldous ‘83, Wilson ‘04]

• Constant bias: If pi,j = p > ½ ∀ i < j, then M mixes in θ(n2) time. [Benjamini et al. ’04, Greenberg et al. ‘09]

• Linear extensions of a partial order: If pi,j = ½ or 1 ∀ i < j, then M mixes

in O(n3 log n) time. [Bubley, Dyer ‘98]

Our Results [Bhakta, M., Randall, Streib]

• M is fast for two new classes – “Choose your weapon”– “League hierarchies”– Both classes extend the uniform and

constant bias cases

•M can be slow even when the pij arepositively biased

Talk Outline

1. Background

2. New Classes of Bias where M is fast• Choose your Weapon• League Hierarchies

3. M can be slow even when the pij are positively biased

Choose Your Weapon

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

Given parameters ½ ≤ r1, … ,rn-1 ≤ 1.

Definition: The variation distance is

Δx(t) = ½ ∑ |Pt(x,y) – π(y)|.

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

Definition: Given e, the mixing time is

(t e) = max min t: Δx(t’) < e, t’ ≥ t. A

yÎ Ω

x

Choose Your Weapon

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

Proof sketch:A. Define auxiliary Markov chain Minv

B. Show Minv is rapidly mixing

C. Compare the mixing times of M and Minv

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

Given parameters ½ ≤ r1, … ,rn-1 ≤ 1.

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)

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

- 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

Minv on Permutations

- choose a card i uniformly

Inversion Tables

3 3 2 3 0 0 0

Inversion Table Iσ:

5 6 3 1 2 7 4

Permutation σ:

Minv 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)

Inversion Tables

3 3 2 3 0 0 0

Inversion Table Iσ:

5 6 3 1 2 7 4

Permutation σ:

Minv 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)

Minv is just a product of n independent biased random walks

Minv is rapidly mixing.ß

Talk Outline

1. Background

2. New Classes of Bias where M is fast• Choose your Weapon• League Hierarchies

3. M can be slow even when the pij are positively biased

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 M 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 M 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 M 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 M is rapidly mixing*.

Proof sketch:A. Define auxiliary Markov chain Mtree

B. Show Mtree is rapidly mixing

C. Compare the mixing times of M and Mtree

Mtree 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 M is rapidly mixing.

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 M is rapidly mixing.

Theorem 2: Proof sketch

Markov chain Mtree allows a transposition if it corresponds to an ASEP move on one of the internal vertices.

q2^4

1 - q2^4

Theorem 2: Proof sketch

Talk Outline

1. Background

2. New Classes of Bias where M is fast• Choose your Weapon• League Hierarchies

3. M can be slow even when the pij are positively biased

But…..M can be slow

Thm 3: There are examples of positively biased pij for

which M is slowly mixing.

1. Reduce to “biased staircase walks”

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 M is slowly mixing.

1. Reduce to biased staircase walks

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]

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

1. Reduce to biased staircase walks2. Define bias on individual cells3. Show that there is a “bad cut” in the state space

Implies that M can take exponential time to reach stationarity.

S SW

Therefore biased permutations can be slow too!

Slow Mixing Example

Open Problems

1. Is M always rapidly mixing when pi,j are positively biased and satisfy a monotonicity condition? (i.e., pi,j is decreasing in i and j)

2. When does bias speed up or slow down a chain?

Thank you!