Shortest-path weight

The shortest-path weight (distance) from $u$ to $v$ is $\delta (u,v)=\{\begin{array}{l}\text{min }w(p)\text{ over all }p\text{ from }u\text{ to }v,\text{ if there is any such path;}\\ +\infty ,\text{ if there is no path from }u\text{ to }v\end{array}$