Given a weighted undirected graph. There are N cities connected by M roads. For each road, you know the fuel needed to travel that road. A small subset of cities have gas station. The price per litre at each gas station is different. The task is to travel from city A to city B in a car with fuel capacity C minimizing the total cost.
Stuck at — Create a new graph including only the cities with gas stations along with source and destination cities. The edge weights in this graph can be found by multiple runs of Dijkstra's algorithm. Now, how to proceed with different fuel price at each node? Some DP?
What are the restrictions of N, M and C?
One possible idea is represent each vertex as a pair (v, f), where 1 <= v <= N and 0 <= f <= C. And build a graph that, if v has a gas station then it can go from (v, f) a (v, f + a), where f + a <= C, with cost equal to a * (price of fuel in city v), and can go from (v, f) to (u, f — w) with cost 0, where w is the necessary fuel to travel from v to u. Finally, perform a Dijkstra.
well the only possible options are to fill the tank completely or fill enough to travel from current to next edge i.e weight of edge u->v . so transform the give graph G into graph G` by doubling the edges .
There would be two types of edges.
Then Dijkstra's algorithm is performed from target node to source node.
cant we assure that the shortest path dijkstra finds will contain edges only of type 1?
No because the cost at filling is different at different nodes. Not filling more cheap fuel at one node may prove costly in the global solution