Algorithm to find two points furthest away from each other

Question

Im looking for an algorithm to be used in a racing game Im making. The map/level/track is randomly generated so I need to find two locations, start and goal, that makes use

Answer 1

Finished a python mockup of the dijkstra solution to the problem. Code got a bit long so I posted it somewhere else: http://refactormycode.com/codes/717-dijkstra-to-find-two-points-furthest-away-from-each-other

In the size I set, it takes about 1.5 seconds to run the algorithm for one node. Running it for every node takes a few minutes.

Dont seem to work though, it always displays the topleft and bottomright corner as the longest path; 58 tiles. Which of course is true, when you dont have obstacles. But even adding a couple of randomly placed ones, the program still finds that one the longest. Maybe its still true, hard to test without more advanced shapes.

But maybe it can at least show my ambition.