When is a Cut-Vertex Covered?

A block-cut tree’s edgee∈E_T is said to becoveredby an augmentation edgee_A= (u, v)∈E_A if and only if e is part of the tree path connecting u with v. In order to completely cover a cut-vertex v_c ∈ V_T, all its incident tree-edges need to be covered, but this is in general not sufficient.

If v_c and its incident edges are removed from T, the tree falls apart into l connected componentsC₁^vc, . . . , C_l^vc, wherel is the degree ofv_c inT; we call themcut-components of v_c; Figure 3.4 illustrates this. To completely coverv_c, at leastl−1 augmentation edges are needed such that all cut-componentsC₁^vc, . . . , C_l^vc are united into one connected graph.

We say that an augmentation edge e_A= (u, v)∈E_Acontributes in covering the cut-vertex

3.2. PREPROCESSING 43 v_c, if it connects two cut-componentsC_i^vc andC_j^vc. Note that two tree-edges incident tov_care covered bye_A.

For any cut-vertexv_c∈V_T, let Γ(v_c)⊆E_Abe the set of augmentation edges that contribute in covering v_c. Furthermore, for each e_A ∈ E_A, let Ψ(e_A) ⊆ V_T be the set of cut-vertices to whose covering e_A contributes, i.e., Ψ(e_A) = {v_c∈ V_T |e_A∈ Γ(v_c)}. Preprocessing explicitly computes and stores the sets Γ(v_c) for all cut-vertices and the sets Ψ(e_A) for all augmentation edges as supporting data structures. This is done by first performing a depth-first search onT and storing for each vertex its depth and a reference to its parent vertex, in order to be able to efficiently determine the tree-path between any pair of vertices. Then, the computation of all sets Γ(v_c) and Ψ(e_A) can be performed in O(|E_A| |V_T|) time. The space required for these data structures is bounded above byO(|E_A| |V_T|).

Let us consider the following special case in which the graph G is complete, and the fixed graphG₀ represents a balanced binary tree:

Definition 8. [Perfect Binary Tree]

A perfect binary tree of height h ≥ 0 is a binary tree T = {r, T_L, T_R}, with the following properties:

• If h= 0, then T_L=∅ andT_R=∅,

• If h >0, then bothT_L and T_R are perfect binary trees of height h−1.

The vertex r is called the root vertex of the tree.

It follows directly from the definition that a perfect binary tree of height h has exactly n= 2^h+1−1 vertices.

Augmenting a Perfect Binary Tree. Suppose we are given a complete graph G= (V, E) with a fixed perfect binary treeG₀ = (V, E₀) of heighth.

We want to answer the following question: What are the memory complexities of Γ and Ψ data structures in this special case? After the transformation ofG₀ into the block-cut tree T, we obtain a perfect binary block-cut tree with block-vertices representing leaves, while the rest of the vertices are cut-vertices with one-to-one correspondence to the inner vertices ofG₀⁴. Thus, the total number of vertices in T is alson= 2^h+1−1. Let us first compute the average number of edges that contribute in covering of a certain cut-vertex. The root vertex r has degree two, i.e. its removal disconnects the graph into two connected components, each having (n−1)/2 = 2^h−1 vertices. Hence, the size of Γ(r) is the number of edges representing the maximal edge-cut between two subtrees:

|Γ(r)|= (2^h−1)² .

Let the root have depth zero, and let each leaf have depthh. Consider now the removal of a cut-vertex v^k_c in depth k, where 1≤k≤h−1. Each of these cut-vertices has degree three,

4Empty block-vertices can be taken out of consideration. For an explanation, see the next section.

and the treeT will fall apart into three trees: Two of them are perfect binary subtrees T₁ and T₂ of heighth−k−1, i.e:

|V[T₁]|=|V[T₂]|= 2^h−k−1 . The total number of vertices in the third threeT₃ is:

|V[T₃]|=|V[T]| − |V[T₁]| − |V[T₂]| − |{v_c^k}| , i.e.

|V[T₃]|= 2^h+1−1−2·(2^h−k−1)−1 = 2^h+1−2^h−k+1 . The number of edges that contribute in covering of the cut-vertex v_c^k is then:

|Γ(v_c^k)|=|E[T₁ :T₂]|+|E[T₂ :T₃]|+|E[T₁ :T₃]| ,

Therefore, the memory space needed to store the whole Γ data-structure for this special case isO(n²logn).

On the other hand, it is easy to see that the average length of a path inT is:

|Ψ(e_A)|= Θ(logn) ,

which implies that the average space needed to store data-structure Ψ isO(n²logn).

How Much Space doΓ andΨ Need on Average in the General Case? If we suppose that the tree structure is not degenerated, i.e. that the tree’s diameter is O(log|V_T|), then the same as above holds, i.e. each augmentation edge e_A∈ E_A covers on average O(log|V_T|) vertices. In total, Ψ needs O(|E_A|log|V_T|) space.

The average space needed to store Γ is O(|E_A|log|V_T|) as well. One observes that the Γ and Ψ data-structures are “dual” to each other.

For each entry e∈ Γ(v_c), preprocessing also stores references to the two tree-edges being incident tov_cand covered bye; we denote them bye^v_T^c₁(e) ande^v_T^c₂(e). They directly correspond to the two cut components edge ecan connect.

3.2. PREPROCESSING 45

vc v^∗

C₂^vc C₁^vc

e⁰⁰_A vc

eA

e⁰_A

(c) (d)

(b) (a)

e⁰⁰_A

Figure 3.5: Reducing the block-cut graph: (a) Assuming c(e_A) ≤ c(e⁰_A), edge e⁰_A can be eliminated. (b)e⁰⁰_A is the only edge able to connect the cut-componentsC₁^vc and C₂^vc of v_cand is therefore fixed. This introduces a new biconnected component inT (c), which is shrunk into the single new block-vertexv^∗ (d).

Check whether a Cut-Vertex is Covered. In the memetic algorithm, it is necessary to efficiently check if a certain cut-vertex is covered by a subset of augmentation edges S ⊆ E_A. With the precomputed Γ and Ψ and the aid of a temporary union-find data structure with weight balancing and path compression [5, pp. 183–189], this check can be performed in O(|S| ·α(|V_T|,|S|)) time. In most cases the degree of the cut-vertexv_c is less than four. Then, not even a union-find data structure is needed, since it is sufficient to check whether each of the tree-edges incident tov_c is covered by some augmentation edge being not incident tov_c.