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

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18 <strong>Markov</strong> <strong>chain</strong> algorithm on spanning trees rooted forests and c ∈ X are edges to w in M, then X a := (X − {c}) ∪ {a} and X b := (X −{c})∪{b} are in ST(G) and for the canonical paths γ X a Y := (A i ) 0≤i≤l and γX b Y := (B i) 0≤i≤l holds { Bi ⊕ {a, b}, 0 ≤ i ≤ k A i = B i , k < i ≤ l where k is the place in γ X a Y where the edge b is added, i.e. b ∈ A k ⊕ A k+1 . Similarly for γ Y X a := (A ′ i) 0≤i≤l , γY X b := (B′ i) 0≤i≤l holds { B A ′ ′ i = i , 0 ≤ i ≤ k with k ∈ {0, . . . , l − 1} B i ⊕ {a, b}, k ong>of</strong> the coding is needed to prove that no transitions are used in to many <strong>of</strong> the canonical paths. This is a consequence <strong>of</strong> the following theorem which is parallel to Theorem 3.2. Theorem 4.2 For B, ¯B ∈ ST(G) and edge e ∈ B ⊕ ¯B there exists exactly one canonical path γ X,Y := (B i ) 1≤i≤l and some j ∈ {0, . . . , l − 1} such that a) B = B j , i.e. γ XY contains B b) ¯B = X ⊕ Y ⊕ B, i.e. B is coded in γ XY by ¯B c) e ∈ B j ⊕ B j+1 , i.e. in γ XY directly after B the edge e is replaced. Detailed pro<strong>of</strong>s <strong>of</strong> Lemma 4.1 and Theorem 4.2 are given in Fehrenbach (2003). As consequence we obtain a pro<strong>of</strong> <strong>of</strong> the mixing time bound in Theorem 3.3 by canonical paths: Pro<strong>of</strong> <strong>of</strong> Theorem 3.3 by canonical paths: We have for the mixing time τ = τ(ε) by (1.5) τ(ε) ≤ ϱ(Γ G )(log ˆπ + log ε −1 ) (4.1) where π is the stationary distribution, ˆπ := min x∈Ω π(x) and ϱ(Γ G ) := max (B,C)∈Ω 2 Ps(B,C)>0 1 π(B) P s (B, C) ∑ γ XY ∈P(B,C) π(X)π(Y )|γ XY |, (4.2) P(B, C) the set <strong>of</strong> canonical paths which contain the transition (B, C). The maximal length <strong>of</strong> a canonical path in Γ G is n − 1 since at most 2(n − 1) edges 1 are exchanged. Further P s (B, C) = , m = |E|. Thus we get for the 2(n−1)m congestion measure ϱ(Γ G ) ≤ 2n2 m |Ω| max |P(B, C)|. (4.3) (B,C)∈Ω 2 Ps(B,C)>0
J. Fehrenbach and L. Rüschendorf 19 The max expression can be estimated using Theorem 4.2. For a transistion (B, C) <strong>of</strong> M s (G) let e be the unique edge in B\C. In any canonical path γ XY ∈ P(B, C) the transition (B, C) is coded by some ¯B ∈ Ω. By Theorem 4.2 there exists exactly one canonical path in Γ G with these properties. Since this path not necessarily contains (B, C) we conclude that |P(B, C)| ≤ |Ω|. This implies ϱ(Γ G ) ≤ 2n2 m |Ω| With |Ω| ≤ m n we thus obtain · |Ω| = 2n 2 m. (4.4) τ s (ε) ≤ 2n 2 m(n log m + log ε −1 ). (4.5) 5 Forests with roots In this section we apply the multi-commodity flows resp. canonical paths to the analysis <strong>of</strong> the mixing time <strong>of</strong> some <strong>Markov</strong> <strong>chain</strong>s on forests. The <strong>Markov</strong> <strong>chain</strong> M s introduced in sections 2, 3 on the set <strong>of</strong> spanning trees only uses exchanges <strong>of</strong> two edges. These transitions can also be used on the class <strong>of</strong> forests i.e. circle free subgraphs <strong>of</strong> G and the corresponding <strong>Markov</strong> <strong>chain</strong> has an stationary distribution the uniform distribution. But s<strong>of</strong>ar no efficient bounds for the mixing time <strong>of</strong> this or related <strong>Markov</strong> <strong>chain</strong>s are known and also no randomized approximation schemes for the number <strong>of</strong> forests are known (see Welsh and Merino (2000)). It seems that also the canonical paths <strong>of</strong> section 4 transfered to this problem do not lead to a polynomial bound for the mixing time. In the following we consider the modified class <strong>of</strong> forests with roots F r (G) and show that for this modified space Ω = F r (G) we obtain rapid mixing results for various <strong>Markov</strong> <strong>chain</strong>s by means <strong>of</strong> the corresponding canonical paths constructed for the class <strong>of</strong> spanning trees. Definition 5.1 (Forests with roots) Let G=(V, E) be an undirected graph. A pair X := (R X , E X ) with R X ⊂ V, E X ⊂ E is called forest with roots if • the subgraph (V, E X ) <strong>of</strong> G contains no circle • any connected component Z <strong>of</strong> (V, E X ) has exactly one node in R X , which we call the root <strong>of</strong> Z. F r (G) denotes the set <strong>of</strong> all forests with roots. Counting forests with roots corresponds to counting forests X with connected components Z 1 , . . . , Z d which are weighted by the number <strong>of</strong> possibilities to choose a root system i.e. by ∏ d i=1 |Z i|(n−|X|). The class F r (G) <strong>of</strong> forests with roots can be identified with the class <strong>of</strong> spanning trees <strong>of</strong> an extended graph G ′ . Lemma 5.2 For any undirected graph G = (V, E) there exists a graph G ′ = (V ′ , E ′ ), such that there exists a bijection Sp : F r (G)) −→ ST (G ′ ).
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