Analysis of Markov chain algorithms on spanning trees, rooted ...

6 <strong>Markov</strong> <strong>chain</strong> algorithm on spanning trees rooted forests
If Y ∉ ST(G), then we set X t+1 = X. We denote the transition matrix <strong>of</strong>
M s (G) by P s .
So the transitions <strong>of</strong> this <strong>chain</strong> are given by simple random exchanges <strong>of</strong>
two edges as long as they lead again to spanning trees.
Proposition 2.1 The <strong>Markov</strong> <strong>chain</strong> M s (G) is ergodic and the stationary distribution
<strong>of</strong> M s (G) is the uniform distribution on ST(G).
Pro<strong>of</strong>: Ergodicity is equivalent to irreducibility and aperiodicity. The aperiodicity
is obvious from construction and also for any X ∈ ST(G) holds
P s (X, X) > 1 . For any X, Y ∈ ST(G) we prove by induction on k = |X ⊕ Y |,
2
the cardinality <strong>of</strong> the symmetric difference <strong>of</strong> X and Y , that Ps(X, t Y ) > 0 for
some t ∈ IN. Note that k = |X ⊕ Y | ∈ 2IN. If k = 2, X ⊕ Y = {a, b}, a ∈ X, b ∈
Y . Then choosing (with positive probability) e = a and f = b in the definition
<strong>of</strong> the <strong>chain</strong> one gets X t+1 = Y and P s (X, Y ) > 0. If k > 2 and b ∈ Y \ X then
X ∪{b} contains a circle C with some edge a in C such that a ∈ X \Y . Choosing
(with positive probability) e = a, f = b, then X ′ = (X \{a})∪{b} ∈ ST(G)
and P s (X, X ′ ) > 0. Furthermore, |X ′ ⊕ Y | = k − 2 and so by the induction
hypothesis Ps(X t ′ , Y ) > 0 for some t ∈ IN and thus Ps t+1 (X, Y ) > 0.
For any X, Y ∈ ST(G) with P s (X, Y ) > 0, X ≠ Y holds P s (X, Y ) =
1
= P 2(n−1)m s(Y, X) where n = |V |, m = |E|, i.e. the transition matrix is symmetric
and thus reversible w.r.t. the uniform distribution π on ST(G). This
implies that the uniform distribution π is the stationary distribution <strong>of</strong> M s (G).
✷
The construction <strong>of</strong> the multicommodity flow in section three uses for its
pro<strong>of</strong> an idea <strong>of</strong> Cordovil and Moreira (1993) for graphical block-matroids.
For the ease <strong>of</strong> reference we remind that a matroid M = (S, B) is a pair <strong>of</strong>
nonempty sets, B ⊂ P(S), such that for all X, Y ∈ B holds: ∀x ∈ X \ Y exists
some y ∈ Y \ X such that (X \ {x}) ∪ {y} ∈ B. Elements <strong>of</strong> B are called bases
<strong>of</strong> M. A matroid M = (S, B) is called block-matroid if S = X ∪ Y for some
X, Y ∈ B.
In our context we consider the graphical matroid (E, ST(G)) which is a
graphical block-matroid if G can be decomposed into two disjoint spanning
trees, see Figure 1.
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Figure 1: A graphical block-matroid on the left with a decomposition into
two spanning trees.