The document describes methods for generating random spanning trees on graphs. It introduces an algorithm that uses a random walk to generate a spanning tree in expected O(n log n) time. It also discusses the Markov chain tree theorem, which relates the stationary distribution of a Markov chain to the weights of its directed spanning trees. Finally, it proposes a "swap chain" Markov chain for generating random spanning trees by randomly adding and removing edges, and proves this chain converges rapidly to the uniform distribution.

1. Generating random spanning trees
Feliciano Colella
18th of November, 2014

2. Random Walk The Markov Chain Tree Theorem The Swap Chain
Introduction
Given a graph G with n vertices, produces a spanning trees of
G chosen uniformly at random among the spanning trees of G.
The expected running time is O(n log n) while is O(n3) in the
worst case.
A Markov chain is called rapidly mixing if it gets close to the
limit distribution in time polynomial in the log of the number
of states.
Feliciano Colella Generating random spanning trees

3. Random Walk The Markov Chain Tree Theorem The Swap Chain
Random Walk
Feliciano Colella Generating random spanning trees

4. Random Walk The Markov Chain Tree Theorem The Swap Chain
The Algorithm Generate
Input: G = (V; E), with jVj = n nodes and jEj = m edges.
1 Simulate a random walk on G starting at an arbitrary vertex s
until every vertex is visited. For each i 2 V s collect the
edge (i ; j) that corresponds to the

5. rst entrance to vertex i .
Let T be this collection of edges.
2 Output the set T.
Feliciano Colella Generating random spanning trees

6. Random Walk The Markov Chain Tree Theorem The Swap Chain
The Markov Chain Tree Theorem
Some notations ...
M is a Markov chain with V = f1; : : : ; ng and matrix P.
GM = (V; E) is a digraph s.t. E = f[i ; j ]jPi ;j 0g.
For each edge [i ; j ] 2 E, its weight, w([i ; j ]) is Pi ;j .
Directed spanning tree of GM rooted at i .
Weight of a spanning tree T is w(T) = e2Tw(e).
Family of all directed ST of GM rooted at i is Ti (GM).
Family of all directed ST of GM is T(GM).
Feliciano Colella Generating random spanning trees

7. Random Walk The Markov Chain Tree Theorem The Swap Chain
The Markov Chain Tree Theorem
Theorem 1
Let M be an irreducible Markov Chain on n states with stationary
distribution i ; : : : ; n. Let GM be the directed graph associated
with M. Then:
i =
P
T2Ti (GM) w(T)
P
T2T(GM) w(T)
Feliciano Colella Generating random spanning trees

8. Random Walk The Markov Chain Tree Theorem The Swap Chain
The Markov Chain Tree Theorem
We use the Backwards tree to prove the theorem.
Calculate the probability of being in a certain tree T after a
cover time C, namely FC .
Feliciano Colella Generating random spanning trees

9. Random Walk The Markov Chain Tree Theorem The Swap Chain
The Markov Chain Tree Theorem
We use the Backwards tree to prove the theorem.
Calculate the probability of being in a certain tree T after a
cover time C, namely FC .
Show that FC is uniformly distributed over all directed
spanning trees rooted at i .
Feliciano Colella Generating random spanning trees

10. Random Walk The Markov Chain Tree Theorem The Swap Chain
The Markov Chain Tree Theorem
We use the Backwards tree to prove the theorem.
Calculate the probability of being in a certain tree T after a
cover time C, namely FC .
Show that FC is uniformly distributed over all directed
spanning trees rooted at i .
Show that the previous property still holds when the graph is
undirected.
Feliciano Colella Generating random spanning trees

11. Random Walk The Markov Chain Tree Theorem The Swap Chain
Quasi-uniform Generation
The algorithm Generate cannot be applied to directed graphs,
because in this case the simple random walk is not reversible.
Feliciano Colella Generating random spanning trees

12. Random Walk The Markov Chain Tree Theorem The Swap Chain
Quasi-uniform Generation
The algorithm Generate cannot be applied to directed graphs,
because in this case the simple random walk is not reversible.
What can we do ?
Feliciano Colella Generating random spanning trees

13. Random Walk The Markov Chain Tree Theorem The Swap Chain
Quasi-uniform Generation
The algorithm Generate cannot be applied to directed graphs,
because in this case the simple random walk is not reversible.
What can we do ?
We need to modify all the transitions to have the same probability;
thus every tree has the same weight.
This is done by adding self-loops to the graph.
If the graph is already out-degree regular then the running time is
only O(E(C))
Feliciano Colella Generating random spanning trees

14. Random Walk The Markov Chain Tree Theorem The Swap Chain
The Swap Chain
A new approach ...
Starting from an undirected graph G.
Let S0
t be the current tree; pick and edge e from G S0
t
uniformly at random.
Add e to T.
Delete uniformly at random an edge from the new cycle to
obtain S0
t+1.
Feliciano Colella Generating random spanning trees

15. Random Walk The Markov Chain Tree Theorem The Swap Chain
The Swap Chain
A new approach ...
Starting from an undirected graph G.
Let S0
t be the current tree; pick and edge e from G S0
t
uniformly at random.
Add e to T.
Delete uniformly at random an edge from the new cycle to
obtain S0
t+1.
The authors show that this chain get close to stationary
distribution in time polynomial in n.
Feliciano Colella Generating random spanning trees

16. Random Walk The Markov Chain Tree Theorem The Swap Chain
The Swap Chain
Sketch of the proof.
Lemma 5 Fs (Pt ) tFs (P)
Lemma 6 Fs ( 1
2 (P + PT )) = Fs (P)
Lemma 7 Fs d
d0 Fs (P)
Feliciano Colella Generating random spanning trees

17. Random Walk The Markov Chain Tree Theorem The Swap Chain
The Swap Chain
Sketch of the proof.
Lemma 5 Fs (Pt ) tFs (P)
Lemma 6 Fs ( 1
2 (P + PT )) = Fs (P)
Lemma 7 Fs d
d0 Fs (P)
Theorem 8
The swap chain fStg gets close to the uniform distribution in time
polynomial in the size of the underlying graph
Use Lemma 5 for a bound for fBtg
Transform the graph for fBtg to the graph for fStg
The second eigenvalue is bounded by the matrix P of fStg
Implies that fStg is rapidly mixing
Feliciano Colella Generating random spanning trees