Correctness of Bellman-Ford Algorithm, can we still do better?

Question

I learned that the Bellman-Ford Algorithm has a running time of O(|E|*|V|), in which the E is the number of edges and V the number of vertices. Assume the graph does not have any negative weighted cycles.

My first question is that how do we prove that within (|V|-1) iterations (every iteration checks every edge in E), it updates the shortest path to every possible node, given a particular start node? Is it possible that we have iterated (|V|-1) times but still not ending up with shortest paths to every node?

Assume the correctness of the algorithm, can we actually do better than that? It occurs to me that not all edges are negatively weighted in a particular graph. The Bellman-Ford Algorithm seems expensive, as every iteration it goes through every edges.

Answer 1

The longest possible path from the source to any vertice would involve at most all the other vertices in the graph. In other words - you won't have a path that goes through the same vertice more than once, since that would necessarily increase the weights (this is true only thanks to the fact there are no negative cycles).
On each iteration you would update the shortest path weight on the next vertice in this path, until after |V|-1 iterations your updates would have to reach the end of that path. After that there won't be any vertices with non-tight values, since your update has covered all shortest paths up to that length.

This complexity is tight (at least for BF), think of a long line of connected vertices. Pick the leftmost as the source - your updating process would have to work its way from there to the other side once vertice at a time. Now you might argue that you don't have to check each edge that way, so let's throw in a few random edges with a very large weight (N > |V|*max-weight) - they can't help you, but your algorithm can't know that for sure, so if has to go through the process of updating the vertices with these weights (they're still better than the initial infinity).