Understanding Correctness of Bidirectional Dijkstra I'm trying to understand the correctness of the bidirectional version of Dijkstras algorithm as mentioned here on slide 10: 

https://www.cs.princeton.edu/courses/archive/spr06/cos423/Handouts/EPP%20shortest%20path%20algorithms.pdf

For a contradiction, they consider an $s-t$ path $p$ that is shorter than the minimum sum $\mu$ of tentative distances from the forward and backward search.
The next step I don't understand: there is an edge $(v,w)$ on this path, such that:


$dist(s,v) &lt; top_f$
$dist(w,t) &lt; top_b$


Where do these relationships come from? 

From what I see, after stopping, it holds that $\mu \leq top_f + top_b$. 

So if we consider any edge $(v,w)$ of $p$, we have that 

$dist(s,v) + length(v,w) + dist(w,t) &lt; top_f + top_b$ 

We can leave out that edge to get the inequality 

$dist(s,v) + dist(w,t) &lt; top_f + top_b$ 

But from this, I can't derive these two relationships.

I appreciate any help! Thanks!
I came across these slides where on slide 23 another proof is given, however the idea of this proof is the same as in the slides of my original post.



But this still doesn't show, why $dist(s,v)$ and $dist(w,t)$ need to be less than $top_f$ and $top_b$ respectively.

EDIT: Note that with this proof $dist(s,v) \leq top_f$ and $dist(w,t) \leq top_b$ hold, since both vertices have already been settled. I'm in doubt, whether the original statement $dist(s,v) &lt; top_f$ and $dist(w,t) &lt; top_b$ is actually correct and therefore I'm marking this question as 'answered'. However if anyone finds an explanation, feel free to answer!