Given a directed graph $ G = (V,E)$  with non-negative(zero and positive) weights on the edges, and a vertex $ s \in V $Problem: Find the lightest path from $s $ to each and every vertex $v \in V$ and that's from the shortest paths from $ s$ to all $v \in V$Length of a path is the number of edges in the path.Weight of a path is the sum of all weights of the path's edges-------- Edit: Elaborating the question: --------Suppose there are $x$ different paths between $s$ and some $v \in V$.Each path has its own weight.Among those $x$ paths, there are $y$ shortest paths(pay attention $ y \subseteq x$ and all the paths in $y$ have the same length).What I'm trying to find: The lightest path among the above $y$ paths. Also, i want to calculate that weight for every vertex in $V$.Initially, i thought about this algorithm:run BFS on $G$ from source $s$run Dijkestra on the BFS tree from step 1However, i ran into this problem:With source vertex $A$, the red path and the black path, both lead to the same vertex $D$.Pay attention how $ A\rightarrow B\rightarrow D$ and  $A \rightarrow C \rightarrow D$ are both of length 2Although they are both the shortest paths from $A$ to $D$, the red path is lighter than the black one.I thought about 'tweaking' BFS:Each time i arrive at a vertex $v$ which i already discovered, i check whether the new path i already walked to $v$ is lighter than the one already set to $v$.However, i don't want to change in the algorithm, because i need to prove that it actually work from the start.How can i modify my algorithm so that the problem i pointed won't pop-out?
Algorithm to find shortest lightest path in a graph from source
algorithms;graphs;shortest path