I have been reading the paper "Preprocessing the Steiner Problem in Graphs" by Duin (http://link.springer.com/chapter/10.1007%2F978-1-4757-3171-2_10) and I am having a bit of trouble wrapping my head around what he calls a "bottleneck distance".
Given a graph $G = (V,K,E,c)$ with $V$ the set of vertices, $K$ the subset of so-called special vertices, $E$ the set of undirected edges and c the set of weights for each edge $e \in E$.
For a path $P \subset G$, the bottleneck length $bl(P)$ is the maximum edge weight on P.
The bottleneck distance $b_{ij}$ is the minimum bottleneck length, ie.
$b_{ij} = min\{bl(P)|P \text{ is an }i-j \text{ path} \}$
it then goes on to say that "$b_{ij}$ is equal to $bl(T_{\{ij\}})$ where $T$ denotes the MST on $G$".
does this mean that in order to calculate the bottleneck distances for all pairs of vertices, I have to find $T$, the MST for $G$, and then simply find the maximum edge weight of the path on $T$ from each pair of vertices?
If that is correct, can someone please explain how that works? the main thing that i don't understand about it is that the MST will contain arbitrary paths (not necessarily shortest) between $i$ and $j$, so why are we allowed to use the bottleneck length of those paths in particular, instead of checking for the minimum bottleneck length from the set of all paths between $i$ and $j$ as is suggested by the definition of the bottleneck distance?