Design

In this section, we extend the idea of vertex balls and vicinities to inverse-balls and vicinities. We then give efficient algorithms to construct inverse-ball and inverse-vicinities of vertices in weighted undirected graphs.

Definition 16 (Inverse-ball of a vertex). Let G = (V, E) be a connected weighted undirected graph and let L⊂ V be a subset of vertices. The inverse-ball of a vertex v ∈ V , denoted by B(v), is the set of vertices w ∈ V that contain v in their ball, that is, the set of vertices w∈ V for which d(w, v) < d(w, ℓ(w)).

Definition 17 (Inverse-vicinity of a vertex). Let G = (V, E) be a connected weighted undirected graph and let L ⊂ V be a subset of vertices. The inverse-vicinity of a vertex v ∈ V , denoted by B^⋆(v), is the set of vertices w ∈ V that contain v in their vicinity, that is, the set of vertices w∈ V for which v ∈ B^⋆(w).

Constructing inverse-balls and inverse-vicinities of bounded size. The re-sult of Lemma 3 bounds the size of vertex balls and vicinities; while this leads to bounds on average size of inverse-balls and inverse-vicinities, we would like a bound on the worst-case size. We now discuss how to efficiently construct inverse-balls and inverse-vicinities of bounded worst-case size. We will need the following result:

Lemma 4 ( [58]). Let G = (V, E) be a weighted undirected graph with n vertices, m edges and maximum degreeµ = 2m/n. For any fixed 1≤ α ≤ n, there exists a subset of vertices L of expected size 8n log n/α such that for each vertex v∈ V , we have that|B(v)| = α. Moreover, such a set L and the distance from each vertex v to each vertex w∈ B(v) can be computed in time eO(mα).

For sake of completeness, we informally describe the algorithm for con-structing such a set L. Fix some 1≤ α ≤ n. The algorithm maintains two set of vertices — a set L that constitutes the final output of the algorithm and an-other set W that contains all vertices that have inverse-ball of size more than α. The set L is initialized to an empty set and W is initialized to the vertex set V . The algorithm runs in multiple iterations; in each iteration, it uniform randomly samples 4n/α vertices from W , inserts them to set L; re-computes the inverse-ball of each vertex and updates W to all vertices that still contains more than α vertices in their inverse-ball. The algorithm terminates when W contains 4n/α or fewer vertices; in this case, all vertices in W are inserted in set L. The main idea behind the proof of correctness is as follows. Clearly, by construction, each vertex has inverse-ball of size at most α. The main chal-lenge is to bound the size of set L. It is shown in [58] that the expected number of iterations performed by the algorithm before termination is at most 2 log n; since 4n/α vertices are added to L in each iteration, the size of the set

Loutput by the algorithm is at most 8n log n/α.

It is easy to verify that the set of vertices in the inverse-vicinity of any vertex vis given by B^⋆(v) =S

w∈N(v)B(w). Hence, once the inverse-ball for each vertex has been computed, the inverse-vicinity of any vertex v can be computed easily by iterating through each vertex w∈ N(v), and letting each vertex in B(w) to be in the inverse-vicinity of v. Hence, we get:

Lemma 5. Let G = (V, E) be a weighted undirected graph with n vertices, m edges and maximum degree µ = 2m/n. For any fixed 1≤ α ≤ n, there exists a subset of vertices L of expected size 8n log n/α such that for each vertex v ∈ V , we have that |B(v)| = α and |B^⋆(v)| ≤ µ · α. It is possible to compute, in time O(mα), such a set L, the distance from each vertex v to each vertex we ∈ B(v) and the candidate distance from each vertex v to each vertex w ∈ B^⋆(v).

Lemma 5 gives an efficient way to sample a set of vertices of size eO(n/α) such that the size of the inverse-ball of each vertex is bounded by O(α); com-pare this with the sampling technique of Lemma 3 that gives an efficient way to sample a set of vertices of the same size such that the ball of each vertex is bounded by O(α). We emphasize that the above lemma bounds the size of set L in expectation, while the size of inverse-ball and inverse-vicinity for any vertex is bounded deterministically.

It is, in fact, possible to combine the sampling technique of Lemma 3 and Lemma 5 to construct a set L of size eO(n/α) such that the ball, the vicinity, the inverse-ball and inverse-vicinity of each vertex is of bounded size. Specifically, fix some 1≤ α ≤ n. Then, first the algorithm samples a set of vertices L1 of size eO(n/α) using the algorithm of Lemma 3. The set L₁ is used as a seed set for the algorithm of Lemma 5. Then, another set of vertices L₂ of size eO(n/α) using the algorithm of Lemma 5. This gives us the final set of sampled vertices

L = L₁∪ L2 with the following property:

Lemma 6. Let G = (V, E) be a weighted undirected graph with n vertices, m edges and maximum degreeµ = 2m/n. For any fixed 1≤ α ≤ n, there exists a subset of vertices L of expected size eO(n/α) such that for each vertex v ∈ V, we have that |B(v)| = O(α), |B(v)| = O(α), |B^⋆(v)| = O(αµ) and |B^⋆(v)| = O(αµ). It is possible to compute, in time eO(mα), such a set L, the distance from each vertex v to each vertex w ∈ B(v) and to each vertex w ∈ B(v) and the candidate distance from each vertex v to each vertex w ∈ B^⋆(v) and to each vertex w∈ B^⋆(v).

The notation used in the last two sections in summarized in Table 2.2.

Table 2.2: Notation on balls and vicinities used throughout the dissertation.

ℓ(v) Landmark of vertex v B(v) Ball of vertex v

r_v Ball radius of vertex v B^⋆(v) Vicinity of vertex v B(v) Inverse-ball of vertex v B^⋆(v) Inverse-vicinity of vertex v