What are some good methods to finding a heuristic for the A* algorithm?

Question

You have a map of square tiles where you can move in any of the 8 directions. Given that you have function called cost(tile1, tile2) which tells you the cost of

Answer 1

It depends on the cost function.

There are a couple of common heuristics, such as Euclidean distance (the absolute distance between two tiles on a 2d plane) and Manhattan distance (the sum of the absolute x and y deltas). But these assume that the actual cost is never less than a certain amount. Manhattan distance is ruled out if your agent can efficiently move diagonally (i.e. the cost of moving to a diagonal is less than 2). Euclidean distance is ruled out if the cost of moving to a neighbouring tile is less than the absolute distance of that move (e.g. maybe if the adjacent tile was "downhill" from this one).

Edit

Regardless of your cost function, you always have an admissable and consistent heuristic in h(t1, t2) = -∞. It's just not a good one.