This question refers to the Google-sponsored AI Challenge, a contest that happens every few months and in which the contenders need to submit a bot able to autonomously play a game against other robotic players. The competition that just closed was called "ants" and you can read all its specification here, if you are interested.
My question is specific to one aspect of ants: combat strategy.
The problem
Given a grid of discrete coordinates [like a chessboard] and given that each player has a number of ants that at each turn can either:
- stay still
- move east / north / west / south,
...an ant will be killed by an enemy ant if an enemy ant in range is surrounded by less (or the same) of its own enemies than the ant [equivalent to: "An ant will kill an enemy ant if an enemy in range is surrounded by more (or the same) enemies than its target"]
A visual example:
In this case the yellow ants are going to move west, and the orange ant, not being able to move away [blue tiles are blocking] will have two yellow ants "in range" and will die (if the explanation is still not clear, I invite you to visit the link above to see more examples and explained scenarios).
The question
My question is substantially about complexity. I thought to this problem extensively, but I still couldn't come up with an acceptable way to calculate the optimal set of moves in a reasonable time. It seems to me that for finding the best possible set of moves for my ants, I should evaluate the outcome for every possible scenario, but since battles can be pretty crowded with ants, this would mean that computation time would grow exponentially (5^n, with n being the number of ants involved).
Another limitation of this approach is that the solution being worked on doesn't improve its effectiveness proportionally to the time spent computing, so arbitrarily interrupting its execution might leave you with a non-acceptable solution.
I suspect that a good solution might be found via some geometrical considerations in combination with linear algebra, (maybe calculating some "centres of gravity" for groups of ants?) but I could not pass the level of "intuition" on this...
So, my question really boils down to:
How should this problem be approached to find [nearly] optimal solutions in a computation time of ~50-100 ms on a modern machine (this figure is derived by the official contest rules)?
If you are interested by the problem and need some inspiration, I highly recommend to watch some of the available game replays.
EDIT FROM THE OP:
I'm selecting this answer as accepted as the winner of the contest published a post-mortem analysis of his code, and indeed he followed the approach suggested by the author of this answer. You can read the winner's blog entry here.
For these kind of problems, MinMax algorithm with alpha beta pruning is usually used. (*) [simple explanation for minmax and alpa beta prunning is at the end, but for more details, the wikipedia page should also be read].
In order to overcome the problem you have mentioned about extremely large number of possible moves, a common improvement is doing the minmax algorithm iteratively. At first you explore all nodes until depth 1, and find the best solution. If you still have some time: explore all nodes until depth 2, and now chose a new more informed best solution, and so on...
When out of time: gives the best solution you could find, at the last level you explored.
To further improve your solution, you might want to reorder the nodes you develop: for iteration i, sort the nodes in iteration (i-1) [by their heuristical value for each vertex] and explore each possibility according the order. The idea behind it is that you are more likely to prun more vertices, if you first investigate the "best" solutions.
The problem here remains finding a good heuristical function, which evaluates "how good a state is"
(*)The MinMax algorithm is simple: You explore the game tree, and decide what will you do for each state, and what is your oponent is most likely to do for each action. This is done until depth k, where k is given to the algorithm.
The alpha beta prunning is an addition to minmax, which tells you "which nodes should not be explored anymore, since any way I am not going to chose them, because I have a better solution"
I think your problem can be solved by turning the problem around. Instead of calculating the best moves - per ant - you could caclulate the best move candidates per discrete position on your playing board.
- +1 for a save place
- +2 for a place that results in an enemy dying
- -1 point for a position of certain death
That would scale in a linear way - but have some trade off in not providing best individual movement.
Maybe worth a try :)
Tricky indeed. You may find some hints in Bee algorithms. This is a set of algorithms to use swarm cooperation and 'reasonable computation time'. Bee algorithms can for instance be used to roughly (!) solve the traveling salesman problem. I expect that these algorithms can provide you with the best solution given computing time.
Of course, the problem can be simplified using geometry: relative positions of ants in a neighbourhood are more important than absolute positions. And also light_303's solution is complementatry to the search pattern I propose.
My question is substantially about complexity. I thought to this problem extensively, but I still couldn't come up with an acceptable way to calculate the optimal set of moves in a reasonable time.
Exactly!
It's an AI competition. AI deals with problems which are too complex to be solved with optimal algorithms.
So you have to try "stuff", like your idea about centers of gravity. Even better would be some genetic algorithms where better strategies are found through natural selection (but it's hard to set up some evolving "framework" for that).
BTW: you can see the blog of the current leader and his strategy is surprisingly simple.
来源：https://stackoverflow.com/questions/8574388/combat-strategy-for-ants