Low Stretch Spanning Trees

Keywords and Synonyms

Spanning trees with low average stretch    

Problem Definition

Consider a weighted connected multigraph ${ G = (V,E,\omega) }$, where ω is a function from the edge set E of G into the set of positive reals. For a path P in G, the weight of P is the sum of weights of edges that belong to the path P. For a pair of vertices ${ u,v \in V }$, the distance between them in G is the minimum weight of a path connecting u and v in G. For a spanning tree T of G, the stretch of an edge ${ (u,v) \in E }$ is defined by

$$ \textit{stretch}_T(u,v) = \frac{\textit{dist}_T(u,v)}{\textit{dist}_G(u,v)}\:, $$

and the average stretch over all edges of E is

$$ \textit{avestr}(G,T) = {1 \over {|E|}} \sum_{(u,v) \in E} \textit{stretch}_T(u,v)\:. $$

The average stretch of a multigraph ${ G = (V,E,\omega) }$ is defined as the smallest average stretch of a spanning tree T of G, ${ \textit{avestr}(G,T) }$. The average stretch of a positive integer n, \( { \textit{avestr}(n) }...