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Mixing Times of Self-Organizing Lists Applications and Biased - - PowerPoint PPT Presentation

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Card Shuffling Mixing Times of Self-Organizing Lists Applications and Biased Permutations Previous Work New Results Amanda Pascoe Streib NIST Applied and Computational Math Division Joint work with: Prateek Bhakta, Sarah Miracle, Dana

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SLIDE 1

Mixing Times

  • f Self-Organizing Lists

and Biased Permutations

Amanda Pascoe Streib

NIST Applied and Computational Math Division Joint work with: Prateek Bhakta, Sarah Miracle, Dana Randall

Card Shuffling Applications Previous Work New Results

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SLIDE 2

Outline

  • Biased Permutations
  • New results
  • Application: Self-Organizing Lists
  • Previous work
  • Example: Card Shuffling
  • Introduction and Background on Markov chains

Applications Previous Work New Results Card Shuffling

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SLIDE 3

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 1/2

Applications Previous Work New Results Card Shuffling

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SLIDE 4

Markov Chains

i.e. a random walk on the graph G=(Ω, E): where (x,y)∈E iff can get from x to y by swapping adjacent cards

  • 1. It is a Markov chain!

Applications Previous Work New Results Card Shuffling

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SLIDE 5

Markov Chains

  • 1. It is a Markov chain!
  • 2. It is ergodic: i.e. aperiodic (not bipartite) and

irreducible (connected)

Applications Previous Work New Results Card Shuffling

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SLIDE 6

Using Markov Chains

Thm: Any finite, ergodic Markov chain converges to a unique stationary distribution π (or equivalently, for all x∈Ω, limt Pt(x,*) = π ) (i.e. for all x,y∈Ω, limt Pt(x,y) = π (y))

Applications Previous Work New Results Card Shuffling

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SLIDE 7

Using Markov Chains

Thm: Any finite, ergodic Markov chain converges to a unique stationary distribution π If a distribution π satisfies detailed balance: π(x) P(x,y) = π(y) P(y,x) for all x,y∈Ω, then it is the unique stationary distribution (or equivalently, for all x∈Ω, limt Pt(x,*) = π ) (i.e. for all x,y∈Ω, limt Pt(x,y) = π (y))

Applications Previous Work New Results Card Shuffling

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SLIDE 8

Using Markov Chains

Thm: Any finite, ergodic Markov chain converges to a unique stationary distribution π If a distribution π satisfies detailed balance: π(x) P(x,y) = π(y) P(y,x) for all x,y∈Ω, then it is the unique stationary distribution π(x) P(y,x) P(x,y) π(y) = (or equivalently, for all x∈Ω, limt Pt(x,*) = π ) (i.e. for all x,y∈Ω, limt Pt(x,y) = π (y))

Applications Previous Work New Results Card Shuffling

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SLIDE 9

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 1/2

Applications Previous Work New Results Card Shuffling

π(x) P(y,x) P(x,y) π(y) = = 1

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SLIDE 10

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 1/2

Converges to the uniform distribution

Applications Previous Work New Results Card Shuffling

π(x) P(y,x) P(x,y) π(y) = = 1

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SLIDE 11

Card Shuffling

How long until mixed?

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 1/2

Applications Previous Work New Results Card Shuffling

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SLIDE 12

= ½ ∑ | Pt(x,y) - π(y)|

Using Markov Chains

How long before for all x∈Ω, Pt(x,*) and π are close? Measure closeness using Total Variation Distance: | Pt(x,*), π |TV = ½ || Pt(x,*), π ||1

y∈Ω

The time it takes for a Markov chain to converge within ε of π is called its mixing time: τ(ε) = max min { t : | Pt(x,*), π |TV < ε }

x∈Ω

Applications Previous Work New Results Card Shuffling

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SLIDE 13

Using Markov Chains

We generally want the mixing time to be poly(n), where in this case n = # cards |Ω| = n! whereas

Applications Previous Work New Results Card Shuffling

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SLIDE 14

Previous work: Uniform sampling

How long does it take to mix? τ= O(n3 log n) - Diaconis and Shashahani (1981), Diaconis and Saloff-Coste (1993) τ= Ω(n3 log n) - Wilson (2004) 5 6 3 7 2 1 4

Applications Previous Work New Results Card Shuffling

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SLIDE 15

Previous work: Uniform sampling

How long does it take to mix? τ= O(n3 log n) - Diaconis and Shashahani (1981), Diaconis and Saloff-Coste (1993) τ= Ω(n3 log n) - Wilson (2004) 5 6 3 7 2 1 4

Applications Previous Work New Results Card Shuffling

Coupling

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SLIDE 16

Using Markov Chains

Applications Previous Work New Results Card Shuffling

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SLIDE 17

Using Markov Chains

Applications Previous Work New Results Card Shuffling

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SLIDE 18

Using Markov Chains

Applications Previous Work New Results Card Shuffling

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SLIDE 19

Using Markov Chains

Applications Previous Work New Results Card Shuffling

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SLIDE 20

Using Markov Chains

Applications Previous Work New Results Card Shuffling

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SLIDE 21

Using Markov Chains

Applications Previous Work New Results Card Shuffling

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SLIDE 22

Using Markov Chains

Applications Previous Work New Results Card Shuffling

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SLIDE 23

Using Markov Chains

Applications Previous Work New Results Card Shuffling

Coupled!

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SLIDE 24

Using Markov Chains

Applications Previous Work New Results Card Shuffling

Coupled!

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SLIDE 25

Using Markov Chains

Applications Previous Work New Results Card Shuffling

Coupled!

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SLIDE 26

Using Markov Chains

Applications Previous Work New Results Card Shuffling

Coupled!

Mixing time ≤ Coupling time

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SLIDE 27

Previous work: Uniform sampling

How long does it take to mix? Permutations:

5 6 3 7 2 1 4

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

Applications Previous Work New Results Card Shuffling

Ai,j = 1 iff j ≤ i

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SLIDE 28

Previous work: Uniform sampling

How long does it take to mix? Permutations:

5 6 3 7 2 1 4

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

Applications Previous Work New Results Card Shuffling

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SLIDE 29

Previous work: Uniform sampling

How long does it take to mix? Lattice Paths: 1 1 0 1 1 0 0 0 Permutations:

5 6 3 7 2 1 4

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

Applications Previous Work New Results Card Shuffling

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SLIDE 30

Previous work: Uniform sampling

How long does it take to mix? Lattice Paths: 1 1 0 1 1 0 0 0 Permutations:

5 6 3 7 2 1 4

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

Applications Previous Work New Results Card Shuffling

Coupling time (perm) ≤ max {Coupling time(lattice paths)}

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SLIDE 31

Previous work: Uniform sampling

How long does it take to mix? Lattice Paths: 1 1 0 1 1 0 0 0

τ = Θ(n3 log n) [Wilson]

Permutations:

5 6 3 7 2 1 4

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

Applications Previous Work New Results Card Shuffling

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SLIDE 32

Previous work: Uniform sampling

How long does it take to mix? Lattice Paths: 1 1 0 1 1 0 0 0

τ = Θ(n3 log n) [Wilson]

Permutations:

τ = Θ(n3 log n) [Wilson]

5 6 3 7 2 1 4

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

Applications Previous Work New Results Card Shuffling

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SLIDE 33

Biased Permutations

Applications Previous Work New Results Card Shuffling

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SLIDE 34

Self-Organizing Lists

Example: Pizza Delivery

  • List of clients and addresses
  • O(n) search time

Sarah Bob ... Nancy Tony Alice ...

Applications Previous Work New Results Card Shuffling

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Self-Organizing Lists

Example: Pizza Delivery

  • List of clients and addresses
  • O(n) search time
  • Each client has a different frequency of ordering

(unknown at the beginning)

  • Goal: obtain a list with most frequent clients first

Sarah Bob ... Nancy Tony Alice ...

Applications Previous Work New Results Card Shuffling

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SLIDE 36

Self-Organizing Lists

Example: Pizza Delivery

  • List of clients and addresses
  • O(n) search time
  • Each client has a different frequency of ordering

(unknown at the beginning)

  • Goal: obtain a list with most frequent clients first

Sarah Bob ... Nancy Tony Alice ...

  • Each time a person orders a pizza, move them up

in the list.

Applications Previous Work New Results Card Shuffling

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SLIDE 37

Self-Organizing Lists

Example: Pizza Delivery

  • List of clients and addresses
  • O(n) search time
  • Each client has a different frequency of ordering

(unknown at the beginning)

  • Goal: obtain a list with most frequent clients first
  • --- (Move Ahead One Algorithm)

Sarah Bob ... Nancy Tony Alice ...

  • Each time a person orders a pizza, move them up

in the list.

Applications Previous Work New Results Card Shuffling

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SLIDE 38

Self-Organizing Lists

Example: Pizza Delivery

  • List of clients and addresses
  • O(n) search time

Sarah Bob ... Nancy Tony Alice ...

Move Ahead One Algorithm Nearest Neighbor transpositions <=>

Applications Previous Work New Results Card Shuffling

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SLIDE 39

Self-Organizing Lists

Example: Pizza Delivery

  • List of clients and addresses
  • O(n) search time

Sarah Bob ... Nancy Tony Alice ...

Move Ahead One Algorithm Nearest Neighbor transpositions <=> How long does it take the list to get organized? = Mixing Time!

Applications Previous Work New Results Card Shuffling

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SLIDE 40

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

Applications Previous Work New Results Card Shuffling

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SLIDE 41

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

We make the assumption that pi,j ≥ 1/2 ∀ i < j Positively biased:

Applications Previous Work New Results Card Shuffling

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SLIDE 42

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

1 3 5

We make the assumption that pi,j ≥ 1/2 ∀ i < j Positively biased:

Applications Previous Work New Results Card Shuffling

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SLIDE 43

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

Applications Previous Work New Results Card Shuffling

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SLIDE 44

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

If pi,j ≥ 1/2 ∀ i < j, then π favors increasing permutations

Applications Previous Work New Results Card Shuffling

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SLIDE 45

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 / Z

i<j: σ(i)>σ(j)

pji pij If pi,j ≥ 1/2 ∀ i < j, then π favors increasing permutations

Applications Previous Work New Results Card Shuffling

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SLIDE 46

Previous Work

Benjamini, Berger, Hoffman, Mossel (2005):

If pi,j = p > 1/2 ∀ i < j, then θ(n2) steps

Fill (2003): Gap problem

  • proved this conjecture for n ≤ 4

Conjecture:

If the {pij} are positively biased, then M is rapidly mixing.

  • Conjecture:

If {pij} satisfies a “monotonicity” condition, then the spectral gap is max. when pij=1/2 ∀ i,j

Applications Previous Work New Results Card Shuffling

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SLIDE 47

Previous Work

Benjamini, Berger, Hoffman, Mossel (2005):

If pi,j = p > 1/2 ∀ i < j, then θ(n2) steps

Fill (2003): Gap problem

  • proved this conjecture for n ≤ 4

Conjecture:

If the {pij} are positively biased, then M is rapidly mixing.

  • Conjecture:

If {pij} satisfies a “monotonicity” condition, then the spectral gap is max. when pij=1/2 ∀ i,j

Bubley and Dyer (1998):

If pi,j = 1/2 or 1 ∀ i<j, then O(n3 log n) steps

Applications Previous Work New Results Card Shuffling

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SLIDE 48

Previous Work

Benjamini, Berger, Hoffman, Mossel (2005):

If pi,j = p > 1/2 ∀ i < j, then θ(n2) steps

Fill (2003): Gap problem

  • proved this conjecture for n ≤ 4

Conjecture:

If the {pij} are positively biased, then M is rapidly mixing.

  • Conjecture:

If {pij} satisfies a “monotonicity” condition, then the spectral gap is max. when pij=1/2 ∀ i,j

Bubley and Dyer (1998):

If pi,j = 1/2 or 1 ∀ i<j, then O(n3 log n) steps

Applications Previous Work New Results Card Shuffling

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SLIDE 49

Previous work: Biased sampling

Permutations:

5 6 3 7 2 1 4

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

  • pick a pair of adjacent elements (i,j) u.a.r.
  • swap them with prob. p if i<j, with prob. 1-p otherwise

Simple Markov chain MNN:

Applications Previous Work New Results Card Shuffling

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SLIDE 50

Previous work: Biased sampling

Permutations:

5 6 3 7 2 1 4

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

  • pick a pair of adjacent elements (i,j) u.a.r.
  • swap them with prob. p if i<j, with prob. 1-p otherwise

Simple Markov chain MNN:

Lattice Paths:

τ = Θ(n2) Benjamini et al. if p>1/2:

1 1 0 1 1 0 0 0

Applications Previous Work New Results Card Shuffling

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SLIDE 51

Previous work: Biased sampling

Permutations:

5 6 3 7 2 1 4

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

τ = Θ(n2) Benjamini et al. if p>1/2:

  • pick a pair of adjacent elements (i,j) u.a.r.
  • swap them with prob. p if i<j, with prob. 1-p otherwise

Simple Markov chain MNN:

Lattice Paths:

τ = Θ(n2) Benjamini et al. if p>1/2:

1 1 0 1 1 0 0 0

Applications Previous Work New Results Card Shuffling

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SLIDE 52

Our Results

  • Two classes - M rapidly mixing
  • 1. pi,j = ri ≥ 1/2 ∀ i < j
  • 2. {pi,j} have tree structure
  • Thm: M is not always rapidly mixing.

We identify {pi,j} that are positively biased but where M requires exponential time to mix!

Applications Previous Work New Results Card Shuffling

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SLIDE 53

Our Results

  • Two classes - M rapidly mixing
  • 1. pi,j = ri ≥ 1/2 ∀ i < j
  • 2. {pi,j} have tree structure
  • Thm: M is not always rapidly mixing.

We identify {pi,j} that are positively biased but where M requires exponential time to mix!

Applications Previous Work New Results Card Shuffling

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SLIDE 54

Thm 1: If pi,j = ri ≥ 1/2 ∀ i < j, then M is rapidly mixing.

Proof of Thm 1

Applications Previous Work New Results Card Shuffling

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SLIDE 55

Thm 1: If pi,j = ri ≥ 1/2 ∀ i < j, then M is rapidly mixing. Proof outline:

  • A. Define auxiliary Markov chain M’
  • B. Show M’ is rapidly mixing
  • C. Compare the mixing times of M and M’

Proof of Thm 1

Applications Previous Work New Results Card Shuffling

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SLIDE 56

Thm 1: If pi,j = ri ≥ 1/2 ∀ i < j, then M is rapidly mixing. Proof outline:

  • A. Define auxiliary Markov chain M’
  • B. Show M’ is rapidly mixing
  • C. Compare the mixing times of M and M’

Proof of Thm 1

Applications Previous Work New Results Card Shuffling

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SLIDE 57

Thm 1: If pi,j = ri ≥ 1/2 ∀ i < j, then M is rapidly mixing. Proof outline:

  • A. Define auxiliary Markov chain M’
  • B. Show M’ is rapidly mixing
  • C. Compare the mixing times of M and M’

Proof of Thm 1

M’ can swap pairs that are not nearest neighbors

  • maintain same stationary distribution
  • Define the probability of swapping i and j

that are not nearest neighbors...

Applications Previous Work New Results Card Shuffling

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SLIDE 58

same stationary distribution:

Proof of Thm 1

M’ can swap pairs that are not nearest neighbors

5 6 3 1 2 7 4

Permutation σ:

5 6 2 1 3 7 4

Permutation τ: Thm 1: If pi,j = ri ≥ 1/2 ∀ i < j, then M is rapidly mixing. π(σ) = Π / Z

i<j: σ(i)>σ(j)

pji pij

Applications Previous Work New Results Card Shuffling

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SLIDE 59

same stationary distribution:

Proof of Thm 1

M’ can swap pairs that are not nearest neighbors

5 6 3 1 2 7 4

Permutation σ:

5 6 2 1 3 7 4

Permutation τ: Thm 1: If pi,j = ri ≥ 1/2 ∀ i < j, then M is rapidly mixing. π(σ) = Π / Z

i<j: σ(i)>σ(j)

pji pij

Applications Previous Work New Results Card Shuffling

P’(σ,τ) P’(τ,σ) π(τ) π(σ) = p1,2 p3,2 p3,1 p2,1 p2,3 p1,3 =

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SLIDE 60

p1,2 p3,2 p3,1 p2,1 p2,3 p1,3

Proof of Thm 1

P’(σ,τ) P’(τ,σ) π(τ) π(σ) = = Thm 1: If pi,j = ri ≥ 1/2 ∀ i < j, then M is rapidly mixing.

5 6 3 1 2 7 4

Permutation σ:

5 6 2 1 3 7 4

Permutation τ:

same stationary distribution:

M’ can swap pairs that are not nearest neighbors π(σ) = Π / Z

i<j: σ(i)>σ(j)

pji pij

Applications Previous Work New Results Card Shuffling

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SLIDE 61

p1,2 p3,2 p3,1 p2,1 p2,3 p1,3

Proof of Thm 1

P’(σ,τ) P’(τ,σ) π(τ) π(σ) = = Thm 1: If pi,j = ri ≥ 1/2 ∀ i < j, then M is rapidly mixing.

5 6 3 1 2 7 4

Permutation σ:

5 6 2 1 3 7 4

Permutation τ:

same stationary distribution:

M’ can swap pairs that are not nearest neighbors π(σ) = Π / Z

i<j: σ(i)>σ(j)

pji pij

Applications Previous Work New Results Card Shuffling

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SLIDE 62

p1,2 p3,2 p3,1 p2,1 p2,3 p1,3

Proof of Thm 1

p3,2 p2,3 = P’(σ,τ) P’(τ,σ) π(τ) π(σ) = = Thm 1: If pi,j = ri ≥ 1/2 ∀ i < j, then M is rapidly mixing.

5 6 3 1 2 7 4

Permutation σ:

5 6 2 1 3 7 4

Permutation τ:

same stationary distribution:

M’ can swap pairs that are not nearest neighbors π(σ) = Π / Z

i<j: σ(i)>σ(j)

pji pij

Applications Previous Work New Results Card Shuffling

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SLIDE 63

p1,2 p3,2 p3,1 p2,1 p2,3 p1,3

Proof of Thm 1

p3,2 p2,3 = P’(σ,τ) P’(τ,σ) π(τ) π(σ) = = P’(σ,τ)= p2,3 Thm 1: If pi,j = ri ≥ 1/2 ∀ i < j, then M is rapidly mixing.

5 6 3 1 2 7 4

Permutation σ:

5 6 2 1 3 7 4

Permutation τ:

same stationary distribution:

M’ can swap pairs that are not nearest neighbors π(σ) = Π / Z

i<j: σ(i)>σ(j)

pji pij

Applications Previous Work New Results Card Shuffling

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SLIDE 64

Proof of Thm 1

  • Can swap i and j across multiple smaller elements

with probability pi,j P’(σ,τ)= p2,3

5 6 3 1 2 7 4

Permutation σ:

5 6 2 1 3 7 4

Permutation τ: Thm 1: If pi,j = ri ≥ 1/2 ∀ i < j, then M is rapidly mixing. Location of element i is independent of the location of all larger elements! Idea:

Applications Previous Work New Results Card Shuffling

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SLIDE 65

Inversion Tables

5 6 3 1 2 7 4

Permutation σ:

Applications Previous Work New Results Card Shuffling

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SLIDE 66

Inversion Tables

3 3 2 3

Inversion Table Iσ:

5 6 3 1 2 7 4

Permutation σ:

Applications Previous Work New Results Card Shuffling

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SLIDE 67

Inversion Tables

3 3 2 3

Inversion Table Iσ:

5 6 3 1 2 7 4

Permutation σ:

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

Applications Previous Work New Results Card Shuffling

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SLIDE 68

3 3 2 3

Inversion Table Iσ:

5 6 3 1 2 7 4

Permutation σ:

Iσ(i) = # elements j > i appearing before i in σ I(2)

Inversion Tables

Applications Previous Work New Results Card Shuffling

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SLIDE 69

3 3 2 3

Inversion Table Iσ:

5 6 3 1 2 7 4

Permutation σ:

Iσ(i) = # elements j > i appearing before i in σ I(2)

  • 0 ≤ Iσ(i) ≤ n - i

Inversion Tables

Applications Previous Work New Results Card Shuffling

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SLIDE 70

3 3 2 3

Inversion Table Iσ:

5 6 3 1 2 7 4

Permutation σ:

Iσ(i) = # elements j > i appearing before i in σ I(2)

  • 0 ≤ Iσ(i) ≤ n - i
  • I is a bijection from Sn to T ={(x1,x2,...,xn): 0 ≤ xi ≤ n-i}

Inversion Tables

Applications Previous Work New Results Card Shuffling

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SLIDE 71

3 3 2 3

Inversion Table Iσ:

5 6 3 1 2 7 4

Permutation σ:

Iσ(i) = # elements j > i appearing before i in σ I(2)

  • 0 ≤ Iσ(i) ≤ n - i
  • I is a bijection from Sn to T ={(x1,x2,...,xn): 0 ≤ xi ≤ n-i}

Inversion Tables

We will define M’ as a MC on the Inversion Tables.

Applications Previous Work New Results Card Shuffling

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SLIDE 72

I(2) 3 3 2 3

Inversion Table Iσ:

5 6 3 1 2 7 4

Permutation σ:

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

  • 0 ≤ Iσ(i) ≤ n - i
  • I is a bijection from Sn to T ={(x1,x2,...,xn): 0 ≤ xi ≤ n-i}

Inversion Tables

What happens when you add 1 to xi?

Applications Previous Work New Results Card Shuffling

slide-73
SLIDE 73

I(2) 3 3 2 3

Inversion Table Iσ:

5 6 3 1 2 7 4

Permutation σ:

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

  • 0 ≤ Iσ(i) ≤ n - i
  • I is a bijection from Sn to T ={(x1,x2,...,xn): 0 ≤ xi ≤ n-i}

Inversion Tables

What happens when you add 1 to xi?

Applications Previous Work New Results Card Shuffling

slide-74
SLIDE 74

I(2) 3 3 2 3

Inversion Table Iσ:

5 6 3 1 2 7 4

Permutation σ:

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

  • 0 ≤ Iσ(i) ≤ n - i
  • I is a bijection from Sn to T ={(x1,x2,...,xn): 0 ≤ xi ≤ n-i}

Inversion Tables

What happens when you add 1 to xi?

Applications Previous Work New Results Card Shuffling

slide-75
SLIDE 75

I(2) 3 3 2 3

Inversion Table Iσ:

5 6 3 1 2 7 4

Permutation σ:

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

  • 0 ≤ Iσ(i) ≤ n - i
  • I is a bijection from Sn to T ={(x1,x2,...,xn): 0 ≤ xi ≤ n-i}

Inversion Tables

What happens when you add 1 to xi?

  • swap element i with the first j>i to the right

Applications Previous Work New Results Card Shuffling

slide-76
SLIDE 76

I(2) 3 3 2 3

Inversion Table Iσ:

5 6 3 1 2 7 4

Permutation σ:

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

  • 0 ≤ Iσ(i) ≤ n - i
  • I is a bijection from Sn to T ={(x1,x2,...,xn): 0 ≤ xi ≤ n-i}

Inversion Tables

What happens when you add 1 to xi?

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

Applications Previous Work New Results Card Shuffling

slide-77
SLIDE 77

I(2) 3 3 2 3

Inversion Table Iσ:

5 6 3 1 2 7 4

Permutation σ:

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

  • 0 ≤ Iσ(i) ≤ n - i
  • I is a bijection from Sn to T ={(x1,x2,...,xn): 0 ≤ xi ≤ n-i}

Inversion Tables

What happens when you subtract 1 from xi?

  • swap element i with the first j>i to the left

Applications Previous Work New Results Card Shuffling

slide-78
SLIDE 78

I(2) 3 3 2 3

Inversion Table Iσ:

5 6 3 1 2 7 4

Permutation σ:

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

  • 0 ≤ Iσ(i) ≤ n - i
  • I is a bijection from Sn to T ={(x1,x2,...,xn): 0 ≤ xi ≤ n-i}

Inversion Tables

What happens when you subtract 1 from xi?

  • swap element i with the first j>i to the left

Applications Previous Work New Results Card Shuffling

slide-79
SLIDE 79

I(2) 3 3 2 3

Inversion Table Iσ:

5 6 3 1 2 7 4

Permutation σ:

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

  • 0 ≤ Iσ(i) ≤ n - i
  • I is a bijection from Sn to T ={(x1,x2,...,xn): 0 ≤ xi ≤ n-i}

Inversion Tables

What happens when you subtract 1 from xi?

  • swap element i with the first j>i to the left
  • happens w.p. ri

Applications Previous Work New Results Card Shuffling

slide-80
SLIDE 80

Proof of Thm 1

M’ samples from {(x1,x2,...,xn): 0 ≤ xi ≤ n-i }:

  • choose a column i uniformly
  • w.p. ri: subtract 1 from xi (if possible)
  • w.p. 1- ri: add 1 to xi (if possible)

Proof outline:

  • A. Define auxiliary Markov chain M’
  • B. Show M’ is rapidly mixing
  • C. Compare the mixing times of M and M’

I(2) 3 3 2 3

Inversion Table Iσ:

5 6 3 1 2 7 4

Permutation σ:

Applications Previous Work New Results Card Shuffling

slide-81
SLIDE 81

Proof of Thm 1

M’ is just a cross-product of n independent, 1-dimensional random walks

I(2) 3 3 2 3

Inversion Table Iσ:

5 6 3 1 2 7 4

Permutation σ:

Proof outline:

  • A. Define auxiliary Markov chain M’
  • B. Show M’ is rapidly mixing
  • C. Compare the mixing times of M and M’

Applications Previous Work New Results Card Shuffling

slide-82
SLIDE 82

Proof of Thm 1

M’ is just a cross-product of n independent, 1-dimensional random walks

I(2) 3 3 2 3

Inversion Table Iσ:

5 6 3 1 2 7 4

Permutation σ:

Proof outline:

  • A. Define auxiliary Markov chain M’
  • B. Show M’ is rapidly mixing
  • C. Compare the mixing times of M and M’

Location of element i is independent of the location of all larger elements! Idea:

Applications Previous Work New Results Card Shuffling

slide-83
SLIDE 83

Proof of Thm 1

M’ is just a cross-product of n independent, 1-dimensional random walks

I(2) 3 3 2 3

Inversion Table Iσ:

5 6 3 1 2 7 4

Permutation σ:

Proof outline:

  • A. Define auxiliary Markov chain M’
  • B. Show M’ is rapidly mixing
  • C. Compare the mixing times of M and M’

Applications Previous Work New Results Card Shuffling

slide-84
SLIDE 84

Proof of Thm 1

M’ is just a cross-product of n independent, 1-dimensional random walks M’ is rapidly mixing!

I(2) 3 3 2 3

Inversion Table Iσ:

5 6 3 1 2 7 4

Permutation σ:

Proof outline:

  • A. Define auxiliary Markov chain M’
  • B. Show M’ is rapidly mixing
  • C. Compare the mixing times of M and M’

Applications Previous Work New Results Card Shuffling

slide-85
SLIDE 85

Proof of Thm 1

Proof outline:

  • A. Define auxiliary Markov chain M’
  • B. Show M’ is rapidly mixing
  • C. Compare the mixing times of M and M’

I(2) 3 3 2 3

Inversion Table Iσ:

5 6 3 1 2 7 4

Permutation σ:

Applications Previous Work New Results Card Shuffling

slide-86
SLIDE 86

Proof of Thm 1

M is rapidly mixing M’ rapidly mixing

Proof outline:

  • A. Define auxiliary Markov chain M’
  • B. Show M’ is rapidly mixing
  • C. Compare the mixing times of M and M’

I(2) 3 3 2 3

Inversion Table Iσ:

5 6 3 1 2 7 4

Permutation σ:

Applications Previous Work New Results Card Shuffling

slide-87
SLIDE 87

Our Results

  • Two classes - M rapidly mixing
  • 1. pi,j = ri ≥ 1/2 ∀ i < j
  • 2. {pi,j} have tree structure
  • Thm 2: M is not always rapidly mixing.

We identify {pi,j} where {pi,j} are positively biased but M requires exponential time to mix!

Applications Previous Work New Results Card Shuffling

slide-88
SLIDE 88

Slow Mixing Results

  • Thm 2: M is not always rapidly mixing.

Applications Previous Work New Results Card Shuffling

slide-89
SLIDE 89

Slow Mixing Results

Special Case: Bijection with staircase walks

  • Thm 2: M is not always rapidly mixing.

Applications Previous Work New Results Card Shuffling

slide-90
SLIDE 90

Slow Mixing Results

  • Thm 2: M is not always rapidly mixing.

Permutation σ:

1 3 n 2 ... ...

2 n +1 n 2 always in order

{

1 if i < j ≤ 2 n OR < i < j 2 n

pij =

1 6 2 7 8 3 9 5 4 10

Special Case: Bijection with staircase walks

Applications Previous Work New Results Card Shuffling

slide-91
SLIDE 91

Slow Mixing Results

  • Thm 2: M is not always rapidly mixing.

Permutation σ:

1 3 n 2 ... ...

2 n +1 n 2 always in order

{

1 if i < j ≤ 2 n OR < i < j 2 n

pij =

1 6 2 7 8 3 9 5 4 10

Special Case: Bijection with staircase walks

Applications Previous Work New Results Card Shuffling

slide-92
SLIDE 92

Slow Mixing Results

  • Thm 2: M is not always rapidly mixing.

Permutation σ:

1 3 n 2 ... ...

2 n +1 n 2 always in order

{

1 if i < j ≤ 2 n OR < i < j 2 n

pij =

1 6 2 7 8 3 9 5 4 10 1 1 1 1 1

Special Case: Bijection with staircase walks

Applications Previous Work New Results Card Shuffling

slide-93
SLIDE 93

Slow Mixing Results

  • Thm 2: M is not always rapidly mixing.

Permutation σ:

1 3 n 2 ... ...

2 n +1 n 2 always in order

{

1 if i < j ≤ 2 n OR < i < j 2 n

pij =

1 6 2 7 8 3 9 5 4 10 1’s and 0’s

2 n 2 n

1 1 1 1 1

Special Case: Bijection with staircase walks

Applications Previous Work New Results Card Shuffling

slide-94
SLIDE 94

Slow Mixing Results

  • Thm 2: M is not always rapidly mixing.

Permutation σ:

1 3 n 2 ... ...

2 n +1 n 2 always in order

{

1 if i < j ≤ 2 n OR < i < j 2 n

pij =

1 6 2 7 8 3 9 5 4 10 1’s and 0’s

2 n 2 n

1 1 1 1 1

?? else

1 2 3 4 5 7 6 8 9 10

Special Case: Bijection with staircase walks

Applications Previous Work New Results Card Shuffling

slide-95
SLIDE 95

p37

Slow Mixing Results

  • Thm 2: M is not always rapidly mixing.

Permutation σ:

1 3 n 2 ... ...

2 n +1 n 2 always in order

{

1 if i < j ≤ 2 n OR < i < j 2 n

pij =

1 6 2 7 8 3 9 5 4 10 1’s and 0’s

2 n 2 n

1 1 1 1 1

?? else

1 2 3 4 5 7 6 8 9 10

Special Case: Bijection with staircase walks

Applications Previous Work New Results Card Shuffling

slide-96
SLIDE 96

p37

Slow Mixing Results

  • Thm 2: M is not always rapidly mixing.

Permutation σ:

1 3 n 2 ... ...

2 n +1 n 2 always in order

{

1 if i < j ≤ 2 n OR < i < j 2 n

pij =

1 6 2 7 8 3 9 5 4 10 1’s and 0’s

2 n 2 n

1 1 1 1 1

?? else

1 2 3 4 5 7 6 8 9 10

Special Case: Bijection with staircase walks

Applications Previous Work New Results Card Shuffling

slide-97
SLIDE 97

p37

Slow Mixing Results

  • Thm 2: M is not always rapidly mixing.

Permutation σ:

1 3 n 2 ... ...

2 n +1 n 2 always in order

{

1 if i < j ≤ 2 n OR < i < j 2 n

pij =

1 6 2 7 8 3 9 5 4 10 1’s and 0’s

2 n 2 n

1 1 1 1 1

?? else

1 2 3 4 5 7 6 8 9 10

Special Case: Bijection with staircase walks

Applications Previous Work New Results Card Shuffling

slide-98
SLIDE 98

p37

Slow Mixing Results

  • Thm 2: M is not always rapidly mixing.

Permutation σ:

{

1 if i < j ≤ 2 n OR < i < j 2 n

pij =

1 6 2 7 8 3 9 5 4 10 1’s and 0’s

2 n 2 n

1 1 1 1 1

?? else

1 2 3 4 5 7 6 8 9 10

Special Case: Bijection with staircase walks

Applications Previous Work New Results Card Shuffling

slide-99
SLIDE 99

p37

Slow Mixing Results

  • Thm 2: M is not always rapidly mixing.

Permutation σ:

{

1 if i < j ≤ 2 n OR < i < j 2 n

pij =

1 6 2 7 8 3 9 5 4 10 1’s and 0’s

2 n 2 n

1 1 1 1 1

?? else

1 2 3 4 5 7 6 8 9 10

So each choice of pij where i ≤ < j 2 n determines the bias on square (i,n-j+1)

Special Case: Bijection with staircase walks

Applications Previous Work New Results Card Shuffling

slide-100
SLIDE 100

Staircase Walks

x

  • each box has a different bias px
  • M can add a box or remove a

box according to px

Applications Previous Work New Results Card Shuffling

slide-101
SLIDE 101

Staircase Walks

x

  • each box has a different bias px
  • M can add a box or remove a

box according to px Tile-based Self-Assembly: tiles can attach or detach at corners

  • attach w.p. px
  • detach w.p. 1- px

rapidly mixing <=> self-assembles efficiently

Applications Previous Work New Results Card Shuffling

slide-102
SLIDE 102

Staircase Walks

x

Fluctuating Bias: Thm 4: If px ≥ p ( p const. > 1/2) then rapidly mixing. Thm 5: There exists {px} s.t px >1/2 for all x but mixing time is exponential in n.

Applications Previous Work New Results Card Shuffling

slide-103
SLIDE 103

Staircase Walks

x

Fluctuating Bias: Thm 4: If px ≥ p ( p const. > 1/2) then rapidly mixing. Thm 5: There exists {px} s.t px >1/2 for all x but mixing time is exponential in n.

Applications Previous Work New Results Card Shuffling

slide-104
SLIDE 104

Staircase Walks

x

Fluctuating Bias: Thm 4: If px ≥ p ( p const. > 1/2) then rapidly mixing. Thm 5: There exists {px} s.t px >1/2 for all x but mixing time is exponential in n.

provides slow mixing example for biased permutations!

Applications Previous Work New Results Card Shuffling

slide-105
SLIDE 105

Slow Mixing

Thm 5: There exists {px} s.t px >1/2 for all x but mixing time is exponential in n.

1/2 !+ !1/n2 M= !n2/3 1-δ

Applications Previous Work New Results Card Shuffling

slide-106
SLIDE 106

Slow Mixing Results

Bijection with staircase walks:

  • Thm 2: M is not always rapidly mixing.

{

1 if i < j ≤ 2 n OR < i < j 2 n

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

So each choice of pij where i ≤ < j 2 n determines the bias on square (i,n-j+1) 1/2 !+ !1/n2 M= !n2/3 1- δ otherwise 1-δ

Applications Previous Work New Results Card Shuffling

slide-107
SLIDE 107

Thank you!

slide-108
SLIDE 108

Staircase Walks

Introduction Biased Permutations Nanoscience Colloids

x

Thm [Benjamini, Berger, Hoffman, Mossel]: If px=p for all x, then M is rapidly mixing. Thm 3 [GPR]: If px=p for all x, then M is rapidly mixing. Uniform Bias: proof by coupling with exponential distance metric new path coupling theorem *simpler, generalizes easily*

slide-109
SLIDE 109

Introduction Biased Permutations Nanoscience Colloids

Proof of Thm 1

M’ can swap pairs that are not nearest neighbors

  • maintain same stationary distribution

5 6 3 1 2 7 4

Permutation σ:

5 6 1 2 3 7 4

Permutation τ’: P’(σ,τ) P’(τ,σ) π(τ) π(σ) = Thm 1: If pi,j = ri ≥ 1/2 ∀ i < j, then M is rapidly mixing.

slide-110
SLIDE 110

Introduction Biased Permutations Nanoscience Colloids

Proof of Thm 1

M’ can swap pairs that are not nearest neighbors

  • maintain same stationary distribution

5 6 3 1 2 7 4

Permutation σ:

5 6 1 2 3 7 4

Permutation τ’: P’(σ,τ) P’(τ,σ) π(τ) π(σ) = Thm 1: If pi,j = ri ≥ 1/2 ∀ i < j, then M is rapidly mixing.

slide-111
SLIDE 111

Introduction Biased Permutations Nanoscience Colloids

Proof of Thm 1

M’ can swap pairs that are not nearest neighbors

  • maintain same stationary distribution

5 6 3 1 2 7 4

Permutation σ:

5 6 1 3 2 7 4

Permutation τ’’: π(σ)P’(σ,τ) = π(τ)P’(τ,σ) detailed balance: P’(σ,τ) P’(τ,σ) π(τ) π(σ) = Thm 1: If pi,j = ri ≥ 1/2 ∀ i < j, then M is rapidly mixing.

slide-112
SLIDE 112

Introduction Biased Permutations Nanoscience Colloids

Proof of Thm 1

M’ can swap pairs that are not nearest neighbors

  • maintain same stationary distribution

5 6 3 1 2 7 4

Permutation σ:

5 6 3 1 2 7 4

Permutation σ: π(σ)P’(σ,τ) = π(τ)P’(τ,σ) detailed balance: P’(σ,τ) P’(τ,σ) π(τ) π(σ) = Thm 1: If pi,j = ri ≥ 1/2 ∀ i < j, then M is rapidly mixing. = p1,2 p3,2 p3,1 p2,1 p2,3 p1,3

slide-113
SLIDE 113

)( ( ) )

p1,2 p3,2 p3,1 p2,1 p2,3 p1,3 =(

Introduction Biased Permutations Nanoscience Colloids

Proof of Thm 1

M’ can swap pairs that are not nearest neighbors

  • maintain same stationary distribution

5 6 3 1 2 7 4

Permutation σ:

5 6 3 1 2 7 4

Permutation σ: π(σ)P’(σ,τ) = π(τ)P’(τ,σ) detailed balance: P’(σ,τ) P’(τ,σ) π(τ) π(σ) = Thm 1: If pi,j = ri ≥ 1/2 ∀ i < j, then M is rapidly mixing. = p1,2 p3,2 p3,1 p2,1 p2,3 p1,3

slide-114
SLIDE 114

)( ( ) )

p1,2 p3,2 p3,1 p2,1 p2,3 p1,3 =(

Introduction Biased Permutations Nanoscience Colloids

Proof of Thm 1

M’ can swap pairs that are not nearest neighbors

  • maintain same stationary distribution

5 6 3 1 2 7 4

Permutation σ:

5 6 3 1 2 7 4

Permutation σ: π(σ)P’(σ,τ) = π(τ)P’(τ,σ) detailed balance: P’(σ,τ) P’(τ,σ) π(τ) π(σ) = Thm 1: If pi,j = ri ≥ 1/2 ∀ i < j, then M is rapidly mixing. = p1,2 p3,2 p3,1 p2,1 p2,3 p1,3

slide-115
SLIDE 115

)( ( ) )

p1,2 p3,2 p3,1 p2,1 p2,3 p1,3 =(

Introduction Biased Permutations Nanoscience Colloids

Proof of Thm 1

M’ can swap pairs that are not nearest neighbors

  • maintain same stationary distribution

5 6 3 1 2 7 4

Permutation σ:

5 6 3 1 2 7 4

Permutation σ: π(σ)P’(σ,τ) = π(τ)P’(τ,σ) detailed balance: P’(σ,τ) P’(τ,σ) π(τ) π(σ) = Thm 1: If pi,j = ri ≥ 1/2 ∀ i < j, then M is rapidly mixing. = p1,2 p3,2 p3,1 p2,1 p2,3 p1,3

slide-116
SLIDE 116

)( ( ) )

p1,2 p3,2 p3,1 p2,1 p2,3 p1,3 =(

Introduction Biased Permutations Nanoscience Colloids

Proof of Thm 1

M’ can swap pairs that are not nearest neighbors

  • maintain same stationary distribution

p3,2 p2,3 =

5 6 3 1 2 7 4

Permutation σ:

5 6 3 1 2 7 4

Permutation σ: π(σ)P’(σ,τ) = π(τ)P’(τ,σ) detailed balance: P’(σ,τ) P’(τ,σ) π(τ) π(σ) = Thm 1: If pi,j = ri ≥ 1/2 ∀ i < j, then M is rapidly mixing. = p1,2 p3,2 p3,1 p2,1 p2,3 p1,3

slide-117
SLIDE 117

)( ( ) )

p1,2 p3,2 p3,1 p2,1 p2,3 p1,3 =(

Introduction Biased Permutations Nanoscience Colloids

Proof of Thm 1

M’ can swap pairs that are not nearest neighbors

  • maintain same stationary distribution

p3,2 p2,3 =

5 6 3 1 2 7 4

Permutation σ:

5 6 3 1 2 7 4

Permutation σ: P’(σ,τ) P’(τ,σ) π(τ) π(σ) = P’(σ,τ)= p2,3 = π(τ) π(τ’) π(τ’’) π(τ’) π(τ’’) π(σ)

( ( ( ) ) )

Thm 1: If pi,j = ri ≥ 1/2 ∀ i < j, then M is rapidly mixing.

slide-118
SLIDE 118

Previous work: Uniform sampling

How long does it take to mix? Permutations:

5 6 3 7 2 1 4

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

Applications Previous Work New Results Card Shuffling

1 4 5 3 6 2 7

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

Coupling time (perm) ≤ max {Coupling time(lattice paths)}