I need an advice about the algorithm I have to use in order to find and return the shortest path in a graph like the one I uploaded. I'm lloking for the shortest path between two points, the lower left and the upper right. Thanks

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You might consider some algorithms such as

Breadth first search
and
Depth first search.

and tweak them a bit :)

From an algorithmic point of view, the A* Algorithm is probably the most commonly used path-finding algorithm. Google fishing will come up with results aplenty.

I would agree with Ket or firstPerson. A* and Dijkstra's algorithms are going to be your best when you need to do a heuristic search involving cost for moving between nodes. I prefer A* most of the time for most purposes, but most all algorithms have their merits.

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