如何计算具有加权顶点的图的最短路径？

Question

I'm trying to figure out, how to calculate the shortest path for a graph with weighted vertices. Classical algorithms like Dijkstra and Floyd–Warshall normally work with weighted edges and I'm not seeing a way how to apply them to my case (weighted vertices):

One of the ideas I had was to convert the graph to the more classical view with weighted edges. This is what I received:

Here we have mono and bi-directional weighted edges, but I'm still not sure which algorithm would handle this in order to find the shortest path.

Answer 1

You can certainly do this by transforming the graph. The simplest way is to transform each edge into a vertex, and connect the new vertexes together with edges that have the same cost as the vertex that used to join them.

But you don't really need to bother with any of that...

Dijkstra's algorithm is very easy to adapt to vertex costs without using any such transformation. When you traverse an edge, instead of new_vertex_cost = old_vertex_cost + edge_weight , you just do new_vertex_cost = old_vertex_cost + new_vertex_weight .

Answer 2

You can reduce the problem to the classical shortest path problem and use Dijkstra, Bellman-Ford, or Floyd-Warshal as it suits the purpose. For the sake of simplicity, in what follows, I assume all weights are non-negative. I consider such an assumption reasonable since the question mentions using Dijkstra's algorithm to solve the problem. In the end, this assumption can be removed with care.

Consider most general form of the problem: Assume G = <V, E> is a directed weighted graph with weights on both edges and vertices. Construct a graph H = <V', E'> , with weights only on edges, as follows: For any node v in G , create two nodes v_in and v_out in H; add an edge (v_in -> v_out) with weight equal to the weight of node v in G . Also, for any edge (u -> w) in G , add an edge (u_out -> w_in) in H (new edge carries the same weight as the original edge).

To summarize, for any vertex in the original graph add two vertices in H , one dedicated to ingoing edges, and the other dedicated to the outgoing edges (also, connect the new correlated nodes in H based on the weight of their corresponding vertex in G ).

Now, you have a directed weighted graph H with no weight on vertices, but only on edges. It is easy to prove that the shortest path between (s_in, t_out) in H is the same as the shortest path between (s,t) in the original graph G .

The proof is based on the fact that any such path goes through the edge (v_in, v_out) in H if and only if the corresponding path in G goes through node v .

As far as the analysis goes, we have |V'| = 2|V| |V'| = 2|V| , and |E'| = |E| + |V| |E'| = |E| + |V| . So the reduction does not affect the asymptotic behavior of the employed algorithm for finding shortest paths.