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

Ok, "Hosam's algorithm" is a breadth first search with a preselection on the nodes. Dijkstra's algorithm should NOT be applied here, because your edges don't have weights.

The difference is crucial, because if the weights of the edges vary, you need to keep a lot of options (alternate routes) open and check them with every step. This makes the algorithm more complex. With the breadth first search, you simply explore all edges once in a way that garantuees that you find the shortest path to each node. i.e. by exploring the edges in the order you find them.

So basically the difference is Dijkstra's has to 'backtrack' and look at edges it has explored before to make sure it is following the shortest route, while the breadth first search always knows it is following the shortest route.

Also, in a maze the points on the outer border are not guaranteed to be part of the longest route. For instance, if you have a maze in the shape of a giant spiral, but with the outer end going back to the middle, you could have two points one at the heart of the spiral and the other in the end of the spiral, both in the middle!

So, a good way to do this is to use a breadth first search from every point, but remove the starting point after a search (you already know all the routes to and from it). Complexity of breadth first is O(n), where n = |V|+|E|. We do this once for every node in V, so it becomes O(n^2).