如何使用 Dijkstra 算法在地铁中找到具有换乘时间的最短路径

Question

I am facing a problem with a little trick. I need help!

The problem is to find the shortest path in a subway from the start point to the destination. Subway data provides time taken between each node and its lines. Every time you transfer(change line), it takes 5 minutes.

I tried to code this algorithm using Dijkstra's algorithm. The major difference between Dijkstra's and this problem is that this problem has the possibility of changing the shortest path among those nodes already calculated(put them in set S).

For instance, there are A, B, C, D, E noded. I want to find the shortest path from A to E.

Let's say A: line 1, line 2 B: line 1 C: line 1, line 2 D: line 2 E: line 1

A -> B: 2 minutes
B -> C: 2 minutes
A -> D: 1 minutes
D -> C: 2 minutes
C -> E: 2 minutes

In this case, the Dijkstra's algorithm will take in set S(calculated) node A first, and then node D, and then node B, and then node C, and then node E.

That is, the Dijkstra's algorithm will show A -> D -> C -> E as the shortest path with 10 minutes, since at node C, 5 minutes is required for transfer. However, the actual shortest path is A->B->C->E with 6 minutes, since it does not require transfer time!

That is, the Dijkstra's path A->D->C should be modified to A->B->C after E is taken in the set S.

However, I don't know how to implement this idea to programming code. I'm using JAVA.

Please help anyone! Give me any ideas!!

Thank you

Answer 1

Dijkstra works just fine, just modify edge weights a bit, when there's an edge from node u to v and line[u] != line[v] we just need to increase edge length by 5 (transfer time)