Generating Random Spanning Trees

Sourav Chatterji Sumit Gulwani

EECS DepartmentUniversity of California, Berkeley

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Outline

• Applications• Determinant based algorithms (unweighted)• Random walk based algorithms (weighted)

– Broder’s algorithm (undirected)– CFTP based algorithm (strongly connected)– Wilson’s 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

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Random maze

Enter Berkeley

Get Ph.D.

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Random maze: demystified

Get Ph.D.

Enter Berkeley

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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/Broder’s 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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Wilson’s 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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Wilson’s 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 }

– Pr[T, {C1, C2, …, Cs}] =

– Pr[T] /

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Extensions

•Wilson’s algorithm removes loops

•Alternative: remove an edge which breaks the loop

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A bird’s eye view

• Determinant vs. random walk approach• Broder’s algorithm vs. CFTP based algorithm• Broder’s algorithm vs. Wilson’s algorithm

– Wilson’s 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 Wilson’s algorithm to generate a random spanning tree, and output its root

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Future Directions

• Can there be a better algorithm than Wilson’s ?– 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 0.0001

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1

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Extra Slide: Running Time of Wilson’s 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