Sampling Spanning Trees and Linear Extensions: Two Examples of Coupling

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Sampling Spanning Trees and Linear
ExtensionsTwo Examples of Coupling

• Michal Moshkovitz

Based on lectures notes by Alistair Sinclair and
Eric Vigoda
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Subjects
1. Sampling a spanning tree
2. Sampling linear extensions
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Spanning Tree

• connected
• no cycles
• ? (n-1) edges

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Sample Spanning Trees
Goal Given a connected undirected graph G(V,E),
sample a spanning tree of G. Means The Markov
Chain Monte-Carlo Method
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Convention

• rooted trees
• direct edges up associate edges with vertices

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Markov Chain
Broder Aldous
with prob. ½ DO NOTHING

• with prob. ½
• Choose a random neighbor v of the root in the
  graph

2. Delete the unique edge oriented away from v
3. Make v the new root
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Exercise
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Algorithm
start with an arbitrary spanning tree for
O(VE/?) times do a step in the Markov
chain
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Algorithm
start with an arbitrary spanning tree for
O(VE/?) times do a step in the Markov
chain
prob. 1
prob. ½
prob. ¼
prob. ¼
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?
Describe the Markov chain
The chain is ergodic
Uniform is stationary
Coupling
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The chain ergodic

• The chain is aperiodic

self loops

• The chain is irreducible

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The Chain is irreducible
Step 1 Make the two trees have the same root
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cont.
Step 2 Make an Euler Tour according to the
destination tree
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The chain is symmetric??
?
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stationary distribution
To argue p is stationary we need for every state
v, Su?v M(u,v)p(u) p(v) To argue uniform is
stationary we need for every state v, Su?v
M(u,v) 1
?
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Predecessors of a Tree
r
r
?
s

• the same root (prob. ½)
• not the same root exists r?s (d options)

Su?v M(u,v) 1?½ d?1/2d 1
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?
Describe the Markov chain
The chain is ergodic
?
?
Uniform is stationary
Coupling
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Prob how to couple two trees with different
roots??
Sol. Propp Wilson states spanning trees
with specific root
spanning tree with a root
spanning tree with THE root
? we start with the same root !
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Propp Wilson
New Markov Chain

• Simulate old MC until getting to tree with
  specific root

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The new chain ergodic
selp loop of old markov chain

• The new chain is aperiodic

• The new chain is irreducible

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The new chain is irreducible
Have the same root
return to THE root
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Stationary Distribution of New Markov Chain is
Uniform
Want ?v. Su?v M(u,v) 1
1. Initiate random walk at v on states of old MC
2. Move to each of vs predecessors in old MC w,
w.p. Mold(w,v)
3. Continue (move to random pred)
4. STOP when reached state of new MC
u1
v
u2
u3
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Stationary Distribution of New Markov Chain is
Uniform
Want ?v. Su?v M(u,v) 1
u1
v
u2
Note for all u s.t u?v, Prhitting uM(u,v)
Hence, Su?v M(u,v) 1
u3
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Coupling
The same in both copies
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Coupling Example
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Analysis

• Observation 1 if u becomes root, then both
  copies agree on edge coming out of u ever since.
• Note
• may not be such edge (whenever u is root)
• edge may change

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Observation 2 root simple random walk on G
? after walk on G visited all vertices and
returned to original root, copies coupled.
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Simple Random Walks

• Thm (Feige, 95) Let G(V,E) undirected graph.
  When performing a simple random walk on G,
• E ? O(VE)

steps until
random walk covered all vertices returned to
origin
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Concluding

• Thm (rapid mixing) For t ? ?(VE/?),
• dTV(Xt,Uniform) ? ?
• Proof
• dTV(Xt,Uniform) ? Pr By coupling
  lemma
• Pr not within t steps Observations
  1 2
• ? E / t Markovs
  inequality
• O(VE/t) Feige,
  95
• ? ?

copies didnt couple
steps until