What is a minimal path in a graph?

Question

In graph theory, what is the distinction between minimal distance (which Dijkstra's algorithm finds), and minimal path (which I'm not sure what it is)?

Answer 1

Minimal path is the set of edges which when traversed cover the least amount of distance between two edges. Minimal distance is the sum of the distance between the edges of a minimal path.

Answer 2

distance is scalar; a number. path is a list of vertex/edge pairs?

Answer 3

Minimal distance = least sum of edge weights. Minimal path = least edges.

Ie// It's a shorter path to fly from Vancouver to Toronto and then to Winipeg even though it's a shorter distance to fly from Vancouver to Calgary to Regina and then To Winipeg.

Edit: Flip that around I think.

Answer 4

I'm not 100% sure, but it sounds like the minimal path would be the list of vertices visited when traversing the minimal distance path from vertex A to vertex B.

Answer 5

Let me answer this question within the scope of a network with a source and a sink. I would like to distinguish between a shortest path and a minimal path, where a path is defined by a set of edges.

A shortest path is a path from source to sink that has the shortest corresponding distance. A minimal path can be any path that connects the source to the sink as long as

i) it contains no cycles; and

ii) removal of any of the edges from the path means that there is no longer a connection between source and sink.

Answer 6

Minimal distance is the same as minimal path.