Sparse Graph Spanners

Keywords and Synonyms

(1 + ϵ, β)-spanners; Almost additive spanners    

Problem Definition

For a pair of numbers ${ \alpha,\beta }$, ${ \alpha \ge 1 }$, ${ \beta \ge 0 }$, a subgraph ${ G^{\prime} = (V,H) }$ of an unweighted undirected graph ${ G = (V,E) }$, ${ H \subseteq E }$, is an ${ (\alpha,\beta) }$-spanner of G if for every pair of vertices ${ u,w \in V }$, $\text{dist}_{G^{\prime}}(u,w) \le \alpha \cdot \text{dist}_G(u,w) + \beta$, where ${ \text{dist}_G(u,w) }$ stands for the distance between u and w in G. It is desirable to show that for every n-vertex graph there exists a sparse ${ (\alpha,\beta) }$-spanner with as small values of α and β as possible. The problem is to determine asymptotic tradeoffs between α and β on one hand, and the sparsity of the spanner on the other.

Key Results

The main result of Elkin and Peleg [6] establishes the existence and efficient constructibility of ${ (1+\epsilon,\beta) }$-spanners of size \( {...