Mixing Times of Markov Chains for Self-Organizing Lists and Biased Permutations Prateek Bhakta, Sarah Miracle, Dana Randall and Amanda Streib

Sampling Permutations 5n173...ij n pick a pair of adjacent cards uniformly at random - put j ahead of i with probability p j,i = 1- p i,j This is related to the “Move-Ahead-One Algorithm” for self-organizing lists.

Is M always rapidly mixing? Conjecture [Fill]: If {p ij } are positively biased and monotone then M is rapidly mixing. If p i,j ≥ ½ ∀ i < j we say the chain is positively biased. Q: If the {p ij } are positively biased, is M always rapidly mixing? M is not always fast...

What is already known? Uniform bias: If p i,j = ½ ∀ i, j then M mixes in θ(n 3 log n) time. [Aldous ‘83, Wilson ‘04] Constant bias: If p i,j = p > ½ ∀ i < j, then M mixes in θ(n 2 ) time. [Benjamini et al. ’04, Greenberg et al. ‘09] Linear extensions of a partial order: If p i,j = ½ or 1 ∀ i < j, then M mixes in O(n 3 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 {p ij } are positively 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 {p ij } are positively biased

Choose Your Weapon Thm 1: Let p i,j = r i ∀ i < j. Then M is rapidly mixing. Given parameters ½ ≤ r 1, …, r n-1 ≤ 1. Definition: The variation distance is Δ x (t) = ½ ∑ |P t (x,y) – π(y)|. A Markov chain is rapidly mixing if  (  ) is poly(n, log(  -1 )). Definition: Given , the mixing time is  = max min { t: Δ x (t’) < , t’ ≥ t }. A y   Ω x

The Mixing Time Definition: The variation distance is Δ x (t) = ½ ∑ |P t (x,y) – π(y)|. y   Ω A Markov chain is rapidly mixing if  (  ) is poly(n, log(  -1 )). (or polynomially mixing) Definition: Given , the mixing time is  = max min { t: Δ x (t’) < , t’ ≥ t }. A A Markov chain is slowly mixing if  (  ) is at least exp(n). x

Choose Your Weapon Thm 1: Let p i,j = r i ∀ i < j. Then M is rapidly mixing. Proof sketch: A. Define auxiliary Markov chain M inv B. Show M inv is rapidly mixing C. Compare the mixing times of M and M inv M inv can swap pairs that are not nearest neighbors -Maintains the same stationary distribution -Allowed moves are based on inversion tables Given parameters ½ ≤ r 1, …, r n-1 ≤ 1.

Inversion Tables Inversion Table I σ : Permutation σ: I σ (i) = # elements j > i appearing before i in σ The map I is a bijection from S n to T ={(x 1,x 2,...,x n ): 0 ≤ x i ≤ n-i}. I σ (1) I σ (2)

I σ (i) = # elements j > i appearing before i in σ - swap element i with the first j>i to the left w.p. r i - swap element i with the first j>i to the right w.p. 1-r i M inv on Permutations - choose a card i uniformly Inversion Tables Inversion Table I σ : Permutation σ: M inv on Inversion Tables - choose a column i uniformly - w.p. r i : subtract 1 from x i (if x i > 0) - w.p. 1- r i : add 1 to x i (if x i < n-i)

Inversion Tables Inversion Table I σ : Permutation σ: M inv on Inversion Tables - choose a column i uniformly - w.p. r i : subtract 1 from x i (if x i >0) - w.p. 1- r i : add 1 to x i (if x i <n-i) M inv is just a product of n independent biased random walks M inv is rapidly mixing. 

Choose Your Weapon Thm 1: Let p i,j = r i ∀ i < j. Then M is rapidly mixing. Proof sketch: A. Define auxiliary Markov chain M inv B. Show M inv is rapidly mixing C. Compare the mixing times of M and M inv Given r 1, …, r n-1, r i ≥ 1/2. ✔ ✔

Want stationary distribution: Permutation σ: Permutation τ: π(σ) = Π / Z i σ(j) p ji p ij P ’ (σ,τ) P ’ (τ,σ) π(τ) π(σ) = p 1,2 p 3,2 p 3,1 p 2,1 p 2,3 p 1,3 = Comparing M inv and M nn - swap element i with the first j>i to the left w.p. r i - swap element i with the first j>i to the right w.p. 1-r i M inv on Permutations - choose a card i uniformly p 3,2 p 2,3 = 1-r 2 r2r2 =

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 {p ij } are positively biased

League Hierarchy Let T be a binary tree with leaves labeled {1,…,n}. Given q v ≥ 1/2 for each internal vertex v. Thm 2: Let p i,j = q i^j for all i < j. Then M is rapidly mixing.

League Hierarchy Let T be a binary tree with leaves labeled {1,…,n}. Given q v ≥ 1/2 for each internal vertex v. Thm 2: Let p i,j = q i^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 q v ≥ 1/2 for each internal vertex v. Thm 2: Let p i,j = q i^j for all i < j. Then M is rapidly mixing. 1 st Tier 2 nd Tier

League Hierarchy Let T be a binary tree with leaves labeled {1,…,n}. Given q v ≥ 1/2 for each internal vertex v. Thm 2: Let p i,j = q i^j for all i < j. Then M is rapidly mixing*. Proof sketch: A. Define auxiliary Markov chain M tree B. Show M tree is rapidly mixing C. Compare the mixing times of M and M tree M tree 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 q v ≥ 1/2 for each internal vertex v. Thm 2: Let p i,j = q i^j for all i < j. Then M is rapidly mixing.

Theorem 2: Proof sketch Let T be a binary tree with leaves labeled {1,…,n}. Given q v ≥ 1/2 for each internal vertex v. Thm 2: Let p i,j = q i^j for all i < j. Then M nn is rapidly mixing.

Theorem 2: Proof sketch Let T be a binary tree with leaves labeled {1,…,n}. Given q v ≥ 1/2 for each internal vertex v. Thm 2: Let p i,j = q i^j for all i < j. Then M nn is rapidly mixing.

Let T be a binary tree with leaves labeled {1,…,n}. Given q v ≥ 1/2 for each internal vertex v. Thm 2: Let p i,j = q i^j for all i < j. Then M is rapidly mixing. Theorem 2: Proof sketch

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

Want stationary distribution: π(σ) = Π / Z i σ(j) p ji p ij P ’ (σ,τ) P ’ (τ,σ) π(τ) π(σ) = p 1,5 p 3,5 p 6,8 p 6,9 p 1,6 p 3,6 p 5,8 p 6,9 p 5,6 p 5,1 p 5,3 p 8,6 p 9,6 p 6,1 p 6,3 p 8,5 p 9,6 p 6,5 = What about the Stationary Distribution? P ’ (σ,τ) =1 – v 5^6 = p 6,5 p 5,6 p 6,5 = Permutation τ: Permutation σ: 15

Markov chain M tree allows a transposition if it corresponds to an ASEP move on one of the internal vertices. Each ASEP is rapidly mixing M tree is rapidly mixing. M nn is also rapidly mixing if {p} is weakly regular. i.e., for all i, p i,j i. (by comparison)  Theorem 2: Proof sketch

Relating the Two Classes Choose your weapon: p i,j = r i ≥ 1/2 ∀ i < j Tree hierarchies: p i,j = q i^j > 1/2 ∀ i < j

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 {p ij } are positively biased

But….. M can be slow Thm 3: There are examples of positively biased {p ij } for which M is slowly mixing. 1. Reduce to “biased staircase walks” 13n n +1 n 2 always in order { 1 if i < j ≤ 2 n or < i < j 2 n p ij = Permutation σ: ’ s and 0 ’ s 2 n 2 n

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 p ij = 1/2+1/n 2 if i+(n-j+1) < M Each choice of p ij where i ≤ < j 2 n determines the bias on square (i, n-j+1) (“fluctuating bias”) 1/2 + 1/n 2 M= n 2/3 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 walks 2. Define bias on individual cells 3. Show that there is a “bad cut” in the state space Implies that M can take exponential time to reach stationarity. S S  Therefore biased permutations can be slow too! Slow Mixing Example

Open Problems 1.Is M always rapidly mixing when {p i,j } are positively biased and satisfy a monotonicity condition? (i.e., p i,j is decreasing in i and j) 2. When does bias speed up or slow down a chain?

Thank you!