Generating Random Spanning Trees

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Generating Random Spanning Trees

• Sourav Chatterji Sumit Gulwani
• EECS Department
• University of California, Berkeley

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Outline

• Applications
• Determinant based algorithms (unweighted)
• Random walk based algorithms (weighted)
• Broders algorithm (undirected)
• CFTP based algorithm (strongly connected)
• Wilsons algorithm (directed)
• Comparison
• Relation to generating perfect random state
• Future work

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Applications

• Perfect random sampling
• Statistical physics
• generating random domino tilings
• Computer networks
• accessing network reliability
• Computational biology
• generating random permutations of genomes
• Electrical networks
• Recreation

4
Random maze
Enter Berkeley
Get Ph.D.
5
Random maze demystified
Enter Berkeley
Get Ph.D.
6
Determinant Based Algorithm

• Matrix tree theorem The number of spanning trees
  of G is N(G) Det(A),
• where Aij degree(vi), if (i j)
• -1 if (i,j)
  2 G
• 0 otherwise
• RST(G)
• Let G contract e in G
• With probability N(G)/N(G), put e in RST(G)
• If yes, then compute RST(G) else compute RST(G
  e )
• Extensions
• Can it be adapted to work for weighted graphs?
• Can we approximate the ratio faster?

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Terminology (for r.w. based algorithms)

• Arborescence directed spanning tree
• Random walk on G
• w.l.o.g. G is stochastic
• A random walk on G defines a MC on G
• Forward Tree
• Backward Tree

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Random Walk
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Forward Tree

• A random walk for t steps defines a forward tree
  Ft
• First entrances in reversed orientation

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Backward Tree

• A random walk for t steps defines a backward tree
  Bt
• Last exits

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Markov Chain on Trees

• A random walk defines a MC Bt on the set of
  trees

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Aldous/Broders Algorithm

• Undirected weighted graphs
• Works in cover time C
• Algorithm
• Perform a random walk until the cover time
• Output the forward tree FC
• Proof
• Pr(FC T) ?(T), where ? is the stationary
  distribution for the backward tree chain
• ?(T) / w(T)
• Markov Tree chain theorem

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CFTP

• Strongly connected directed weighted graphs
• Works in O(C) time
• Reduces generating random arborescence to
  generating random arborescence with root r
• Uses CFTP to generate random root r
• Generating random arborescence with root r
• Uses CFTP again!
• Markov Chain Mr
• ? arborescence with root r
• Transitions Simulate M Bt until reach a
  state in Mr
• Random Map

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Random Map
Definition excursion (path from r to
r) Application change the parent of v to be the
vertex after the last occurrence of v in the
excursion
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Random Map
Definition excursion (path from r to
r) Application change the parent of v to be the
vertex after the last occurrence of v in the
excursion Collapsing test cover time
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Loop Erasure of a Random Walk

• Erase cycles in order

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Wilsons Algorithm

• Directed weighted graphs
• Fastest r.w. based algorithm runs in O(?)
• ? mean hitting time
• Reduces generating random arborescence to
  generating arborescence with root
• Add a sink v and edges of weight ? from each
  vertex to v
• Generate a random arborescence T with root v
• If T v is a tree, output it
• Else repeat for a smaller value of ?

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Wilsons Algorithm

• Generating random arborescence with root r
• Current_Tree r
• While some vertex v not in Current_Tree
• Perform random walk from v till it reaches
  Current_Tree
• Add its loop erasure to Current_Tree
• Proof
• Algorithm finishes with a tree T and a collection
  of cycles C1, C2, , Cs
• PrT, C1, C2, , Cs
• PrT /

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Extensions

• Wilsons algorithm removes loops
• Alternative remove an edge which breaks the loop

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A birds eye view

• Determinant vs. random walk approach
• Broders algorithm vs. CFTP based algorithm
• Broders algorithm vs. Wilsons algorithm
• Wilsons algorithm starts from a vertex not in
  the Current tree and that makes all the
  difference

Clique
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Relationship with generating a perfectly random
state

• Random state to generate a random spanning tree
• via generation of random root (e.g. CFTP)
• Random spanning tree to generate a random state
• Seen few techniques for generation of random
  states
• Strong Stationary Time, CFTP
• Random spanning trees is another technique!
• Use Wilsons algorithm to generate a random
  spanning tree, and output its root

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Future Directions

• Can there be a better algorithm than Wilsons ?
• Hybrid Algorithms
• Allowing for bias
• Lower bounds
• Generating a random spanning tree with some
  property like bounded degree, diameter
• Is there an interesting pre-metric that makes the
  analysis amenable to path coupling ?
• Several natural pre-metrics do not work!

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Motivation Hybrid Algorithm
Clique
1
0.0001
1
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Extra Slide Running Time of Wilsons Algorithm

• Running Time for specified root r
• add the number of visits to each vertex u
• Standard result number of times a random walk
  starting at u visits u before reaching r is
  ?(u)Eu Tr Er Tu
• Total expected time
• When r is ? -random, then
• Total Running Time
• Remains O(?)
• When ? large, hit v twice soon restart
• When 1/? O(?), probability of failure is small