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I'm looking for an algorithm that can find the shortest path between two nodes in an undirected graph with a cost which is dynamic. By dynamic, I mean that the cost on edge is dependent on the next (future) step. For example, in a graph like that:
I'm looking for the shortest path from "a" to "e" but the cost of "a" to "b" depends on next step. If I go to c, it is 7, and if I go to d, it is 9.
My question is: Is there an algorithm which solves that problem?
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The Floyd Warshall Algorithm is for solving all pairs shortest path problems.
Dynamic Programming approach We fill the table in bottom up manner, we start from e=0 and fill the table till e=k . Then we have our shortest path cost stored in dp[u][v][k] where u is source, v is destination and k is number edges between path from source to destination.
Dijkstra's Algorithm finds the shortest path between a given node (which is called the "source node") and all other nodes in a graph.
The most important algorithms for solving this problem are: Dijkstra's algorithm solves the single-source shortest path problem with non-negative edge weight. Bellman–Ford algorithm solves the single-source problem if edge weights may be negative.
Reduce the problem to 'regular' shortest path problem
Create dummy vertices b1,b2 (instead of b), and the edges:
(a,b1,7), (b1,c,6), (a,b2,9), (b2,d,5)
The rest of the edges and vertices remain as they originally were.
Now, run regular shortest path algorithm (Dijkstra / Bellman Ford) from the source to the target.
(Repeat the process for any 'conditional' weight edges).
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