Implementing Bellman-Ford algorithm on finite state machine
Snarky half-answer: Of course you can implement this in a FSM. Imagine an ordinary x86_64 processor sitting in a desktop computer. If I call each of the 2^32 bytes in memory part of my "state", wouldn't you agree that my desktop computer is a FSM?

The Wikipedia page on this algorithm gives a Python implementation, which can run on my computer, so yes. My FSM can implement the Bellman-Ford algorithm.

Now that we've proven it's possible to do with a non-infinite number of states, I'm sure it's possible to do with fewer states.