# Thorup : What is the meaning of super distance?

While reading Thorup's Algorithm to solve SSSP problem, I have one point that I can't understand: super distance.

It says: "For each vertex we have a super distance $$D(v)\geq d(v)$$"

$$d(v)$$ must refer the shortest distance from origin to $$v$$, but what is $$D$$ ?

Is it just a distance value from origin while calculation until $$D(v)=d(v)$$ ?

It says: "For each vertex we have a super distance $$D(v)\geq d(v)$$"
"Super" refers to the "$$\ge$$" relation between $$D$$ and $$d$$. In other words, since $$D(v)\ge d(v)$$ where $$d(v)$$ is the distance from $$v$$ to $$s$$, $$D(v)$$ is called super distance.
Is it just a distance value from origin while calculation until $$D(v)=d(v)$$ ?
Correct. It is just a distance that approximates the shortest distance from the source from above, which will become that shortest distance at the time $$v$$ is moved to $$S$$, the set of vertices whose distances have been settled.