Given directed connected weighted graph, check if d(v) = delta(s,v)
If $d$ indeed represents the length of the shortest path, we must have 

$$
d(v)=
\begin{cases}
0, &amp;\text{if $v=s$,}\\
\displaystyle\min_{u:(u,v)\in E}\{d(u)+w(u,v)\}, &amp;\text{otherwise,}
\end{cases}
$$

where $w(u,v)$ is the weight of the edge $(u,v)$. 

So you can check whether $d$ satisfies this property for all $v$. It takes only linear time. If $d$ does not satisfy this property, it cannot represent the length of the shortest path. However, if $d$ does satisfy this property, does it really represent the length of the shortest path? I'll let you figure out this part.

For a further hint:


   Note what you need to prove is that, for all $v$, $d(v)$ is length of the shortest path from $s$ to $v$. Suppose the shortest path from $s$ to $v$ contains $k$ edges, you may try a mathematical induction on $k$.