Approximation Algorithms for Embedding General Metrics Into Trees

C
Approximation Algorithm for Embedding General Metrics – Omitted
Proofs
C.1 Proof of Lemma 6
Claim 6. For any A i ,A j ∈ A G , with A i ≠ A j , either T i ∩T j = ∅, or there exists v ∈ V (T), and v 1 ,... ,v l ,
for some l ≥ 0, such that v 1 ,... ,v l are children of v, and T i ∩ T j = {v,v 1 ,...,v l }.
Proof. It follows immediately from the fact that for any A i ,A j ∈ A G , the diameter of T G,A i
K
most 50c 2 .
∩ T G,A j
K
is at
Let r be the root of T . Initially, T ′ contains a single vertex r ′ . To simplify the discussion, we assume
w.l.o.g., that r is a leaf vertex of T . We also assume that for every edge {u,v} ∈ E(T), there is a tree T i
that contains {u,v}. This is because if there is no such tree, then we can simply introduce a new subtree T i ,
that contains only the vertices u, and v.
For every T i that visits r, we introduce in T ′ a copy φ i (T i ) of T i , and we connect φ i (r) to r ′ .
We proceed by visiting the vertices of T in a top-down fashion. Assume that we are visiting a vertex
v ∈ V (T), with parent p(v), and children v 1 ,... ,v t . At this step, we are going to introduce in T ′ a copy
φ i (T i ) of T i , for every T i that visits v, and we have not considered yet. We consider the following cases:
Case 1: There is no T i that visits v, and p(v).
Let T a be a subtree that visits p(v). For every T b that visits v, and we have not considered yet, we
introduce in T ′ a copy φ b (T b ) of T b , and we connect φ b (v) to φ a (p(v)).
Case 2: There exists T i that visits v, and p(p(v)), and there is no j ≠ i, such that T j visits v, and p(v).
For every T b that visits v, and we have not considered yet, we introduce in T ′ a copy φ b (T b ) of T b ,
and we connect φ b (v) to φ i (v).
Case 3: There is no T i that visits v, and p(p(v)), and there exists T j that visits v, and p(v).
Let a ∈ [k] be the minimum integer such that T a visits v, and p(v). For every T b that visits v, and we
have not considered yet, we introduce in T ′ a copy φ b (T b ) of T b , and we connect φ b (v) to φ a (v).
Case 4: There exists T i that visits v, and p(p(v)), and there exists T j , with i ≠ j, that visits v, and p(v).
Let a ∈ [k] be the minimum integer with a ≠ i, such that T a visits v, and p(v). For every T b that
visits v, and we have not considered yet, we introduce in T ′ a copy φ b (T b ) of T b . With probability
1/2, we connect φ b (v) to φ i (v), and with probability 1/2, we connect φ b (v) to φ a (v).
Claim 7. T ′ is a tree.
Proof. T ′ is a forest since each φ i (T i ) is a tree, and also each φ i (T i ) is connected to exactly one φ j (T j ),
such that T j was considered before i. Also, T ′ is connected since every vertex of T is contained in some
subtree T t .
Claim 8. For any v ∈ V (T), there exists at most one i ∈ [k], such that T i visits both v, and p(p(v)).
Proof. Assume that the assertion is not true. Let T i ,T j be subtrees that visit both v, and p(p(v)). Then, T i
and T j also visit p(v). This however contradicts the definition of the subtrees T 1 ,... ,T k .
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