Shortest path in matrix with obstacles with cheat paths

Question

First of all this is an assingment and I am not looking for direct answers but instead the complexity of the best solution as you might be thinking it .

This is the know

Answer 1

Your matrix is a representation of a graph. Without the cheat paths it is quite easy to implement a nice BFS. Implementing the cheat paths is not a big deal. Just add the same matrix as another 'layer' on top of the first one. bottom layer is 'carry', top layer is 'no carry'. You can move to the other layer only at B-points for the given cost. This is the same BFS with a third dimension.

You have n^2 nodes and (n-1)^2 edges per layer and additionally a maximum of n^2 eges connecting the layers. That's O(n^2).