Widest path algorithm steps I need to compute the bottleneck shortest paths from s to all vertices of a graph by modifying the Dijkstra’s algorithm. I found this explanation on Wikipedia(Link to Wikipedia) but I would appreciate if you can elaborate it a bit for me.


  If the edges are sorted by their weights, then a modified version of
  Dijkstra's algorithm can compute the bottlenecks between a designated
  start vertex and every other vertex in the graph, in linear time. The
  key idea behind the speedup over a conventional version of Dijkstra's
  algorithm is that the sequence of bottleneck distances to each vertex,
  in the order that the vertices are considered by this algorithm, is a
  monotonic subsequence of the sorted sequence of edge weights;
  therefore, the priority queue of Dijkstra's algorithm can be replaced
  by an array indexed by the numbers from 1 to m (the number of edges in
  the graph), where array cell i contains the vertices whose bottleneck
  distance is the weight of the edge with position i in the sorted
  order. This method allows the widest path problem to be solved as
  quickly as sorting; for instance, if the edge weights are represented
  as integers, then the time bounds for integer sorting a list of m
  integers would apply also to this problem


So I need to sort my vertex by weight starting from A? I would appreciate if you can explain me the steps in this algorithm. 

Thanks
The way to use Dijkstra's algorithm to solve the $s$-to-$t$ widest-path problem (in a directed graph) is described well in the following lecture notes:

http://www.cs.cmu.edu/~avrim/451f08/lectures/lect1007.pdf

Make sure to read Sections 13.1, 13.2, and 13.3.  You will see that a single-line change to Dijkstra's algorithm suffices (we change the $+$ to $\min$, and change the $\min$ to $\max$, to represent the fact that we now have a different metric for the "goodness" of a path).

The section you quoted from Wikipedia is talking about a small optimization to this variant of Dijkstra's algorithm.  However, you can use the variant of Dijkstra's algorithm in Section 13.3 of those lecture notes directly, without the optimization mentioned in Wikipedia, if you like -- and that's easier to understand.