MHB Proving $\delta(s,u)+w(u,v)=\delta(s,v)$ in a Shortest Path

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The discussion centers on proving that for a shortest path from vertex s to vertex v through vertex u, the equation $\delta(s,u) + w(u,v) = \delta(s,v)$ holds true. The proof relies on the relaxation process, where if the distance to u is equal to the shortest path distance from s to u, then after relaxing the edge (u,v), the distance to v will equal the shortest path distance from s to v. It is established that $d[v]$ must be less than or equal to $\delta(s,v)$ and greater than or equal to it, leading to the conclusion that $d[v] = \delta(s,v)$. The key takeaway is that the relationship between the distances confirms the validity of the shortest path property.
evinda
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Hey! (Blush)

I am looking at the proof of the following sentence:

Let $s \to u \to v$ a shortest path from the vertex $s$ to the vertex $v$.
Suppose that we relax at some time the edge $(u,v)$.If,before the call of Relaxation(u,v,w),it stands that $d=\delta(s,u)$,then after the call of Relaxation(u,v,w),it stands that $d[v]=\delta(s,v)$.

The algorithm of the function Relaxation(u,v,w) is the following:

Code:
Relaxation(u,v,w)
 if d[v]>d[u]+w(u,v)
    d[v]<-d[u]+w(u,v)
    p[v]<-u

This is the proof (Wait) :

If $d$ gets at some time the value $\delta(s,u)$,$d$ remains unchanged.After the relaxation,we have:

$$d[v] \leq d+w(u,v) \\ \ \ \ = \delta(s,u)+w(u,v) \\ \ \ \ = \delta(s,v)$$

So,we have: $d[v] \leq \delta(s,v)$.

However, it is known that $d[v] \geq \delta(s,v)$.

So,we conclude that $d[v]=\delta(s,v)$.

Could you explain me why $\delta(s,u)+w(u,v)=\delta(s,v)$ ? (Thinking)
 
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Hi! (Smirk)

evinda said:
Could you explain me why $\delta(s,u)+w(u,v)=\delta(s,v)$ ? (Thinking)

Let $s \to u \to v$ a shortest path from the vertex $s$ to the vertex $v$.

Since $s \to u \to v$ is a shortest path, it follows that $\delta(s,u)+w(u,v)=\delta(s,v)$.
 
I like Serena said:
Hi! (Smirk)Since $s \to u \to v$ is a shortest path, it follows that $\delta(s,u)+w(u,v)=\delta(s,v)$.

I got it now! Thank you very much! (Smirk)
 
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