connect4
I was reading "A Knowledge-based Approach of Connect-Four The Game is Solved: White Wins - Victor Allis" http://www.informatik.uni-trier.de/~fernau/DSL0607/Masterthesis-Viergewinnt.pdf
Again for some reason, and I noted in the introduction:
"For the 7 x 6 board, this means that we must be able to hold all positions of 36 and 37 men at the same time, a total of over 1,6.1013 positions. Appendix C contains a table for the number of different positions for each number of men. We can store the value of a position in 2 bits, since we have 4 possible states: win for White, win for Black, draw or not checked (we can use the address of the 2 bits as identification for the position). This way we need at least 4 Terabyte. Therefore making a database does not seem realistic yet."
And then I thought, "I have around a TB in my machine, so if I can make the board smaller to encode I could jsut bruteforce the whole thing!"
So then it spiralled out of control^W^W^W^W went like this:
- I could generate all possible boards for this game, no more upper limit, that would be cool.
- I can store this board in less that 2 bits per square (in fact, at most 63 bits) because I like cramming stuff into bits
- Having thought of a nice way to keep track of winlines on a board (with more bits) I could easily generate next board states and not have crappy for-loop code to check for wins
- Then I can generate all possible boards, but after 9 moves I was already at 800MB
- So then I thought:
I need to store all unique boards in something I'll call a "table", but it's going to be huge so it has to be stored on disk. But searching millions of records in this "table" would be slow, so I need some other thing I can search quickly, like a B+ Tree. That is also going to be rather large, so it also has to be stored on disk. I'll call this file an "index".
So now I'm implementing a rudimentary thing I call a "database".
Anyway, source code so far: encoding/decoding/writing/loading boards+winlines. generating new boards from a move sequence, visualizing boards in amazing 2D ASCII art. A quite fancy (if I say so myself) index that is a B+ Tree index that is persistent on disk, but efficiently cached in memory and also very, very fast.
And todo many things so I though it was time to throw this onto github since it has turned into a medium sized software project (You can tell easily: you've started to worry about multiple files and have implemented a binary search)
Overview
the main program takes a board, generates new boards (with associated data) and then stores the boards in the database.
The database can be queried to see if a board is alrewady there, and it's on disk and stuff so we can use a database of boards to generate the next generation of boards and eventually end up with a DAG that has all possible board states in it, which is a pretty good result.
Since there is a database of board states for every generation we can run things independantly. This makes testing and tweaking easier.
When we actually have these databases we can also do a pruning step to arrive at (yet another) database of moves that play the perfect game. If there are mutliple branches in boardspace that lead to a win for white (which we'll assume is there since the game is solved) we can prune the branch with the largest amount of moves. All cool.
Now let's discuss the database. Here we store a board. The way this happens is to create a unique key from the board state (encode as a board63) and store that in our on-disk binary+ tree. The associated data we store in another file.
The bpt is nice and balanced and makes searches easy (and fast, since we do billions of these this is important). The data is stored in a table file where we keep winlines around etc.
Finding the smallest db to play the perfect game
So I think work this backwards:
Start with the last generation (42). These are all draws or wins for Black. (since the first wins for White are at move 7, so White only wins on odd moves, Black on even moves). So what is there to do here? Nothing. It can just be thrown out. As a good correctness check I'll go over all of them and make sure we end up with only Black Wins or Draws.
Definition: a board can be Good or Bad: Good is a win for White, Bad is a draw or Win for Black.
So take gen41: take all wins for White (this is the last chance White has to win) and throw out everything else. Give an id to each of those wins and store them in a hash or something, probably on disk (the Win List). This will be important later, because it will allow me to find out that 2 different moves will lead to the same win so we can discard one of those branches. I think when there are 2 possible wins from a given board I should always go with the lowest branch id, so all code converges on those numbers and I don't end up with 2 branches that lead to the same win but I can't tell. This might not be true.
Anyway, go back to gen40: Black has made a move. Take all of those boards and do the following: for each board, generate the next possible moves and see if they are in gen41. If so: store the board in the Good list, otherwise store it in the Bad list.
now go to gen39. For all boards, take all the winning boards and put them in the Win List. Take all other boards and generate all next moves for Black, and check the Good/Bad lists to see what is up. now take all the boards that in every case lead to an entry in the Good list, and put it in the Good list. (maybe have a list per generation, or put them all together). Mark the other ones as Bad (or Bad list): these lead to Black winning.