I'm not sure what the following algorithm does but it seems that it calculates the shortest paths from a node $t$.
Initially we're given a graph $G=(V,E)$ with non-negative weights $c(e) \ge0$ for each edge $e\in E$ and we're given a node $t\in V$.
The algorithm is as follows:
Initialize array $A$ such that $A[v] = 0$ if $v=t$ else $A[v]=\infty$.
Outer loop which is repeated continually
$\quad$ Inner loop: we scan all edges in lexicographical order and for every $e=(u,v)\in E$ we $\quad$ do the following:
$\qquad$ if $A[v] >A[u]+c(e)$ then $A[v] =A[u]+c(e)$
$\quad$ if no updates were performed in the inner loop the algorithm exits.
I need to prove by induction what the algorithm does. For example for the following graph below and if $s=b$ the initialization of $A$ will be as follows:
Then the first run of the inner loop would be:
1) $A$ is not bigger than $A$ so we do nothing
2) $A$ is bigger than $A$ so $A=2$
3) $A$ is bigger than $A$ so $A=5$
The next time the inner loop will not make any updates because all the reachable nodes were already updated.
I'm not really sure if I understood correctly what the algorithm does and how to prove this inductively.
How does the fact that the algorithm checks nodes in lexicographical order come into play?