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