Finding a shortest path that passes through some arbitrary sequence of nodes?

Question

In this earlier question the OP asked how to find a shortest path in a graph that goes from u to v and also passes through some node w. The accepted answer, which is quite good

Answer 1

I don't see how we can do too much better, here's all I can think of. Assuming the graph is undirected, the shortest path from any node u to node v would be the same as that from v to u (in reverse of course).

Now, for your case of a shortest path in the order u1 u2 u3.. un, we could run Djikstra's algorithm on u2 (and find the shortest paths u1-u2, u2-u3 in one run), then on u4 (for u3-u4 and u4-u5), then u6.. and so on. This reduces the number of times you apply Djikstra's by roughly a half. Note that complexity wise, this is identical to the original solution.