![]() I'm not sure exactly what the "step" is for this problem. The matrix is effectively a form of memoization.ĭynamic programming is discussed in The Algorithm Design Manual but there is also plenty of discussion of it on the web. Then, once you have filled in the matrix, you can find the best result, and then work backward to get a path through the matrix that leads to the best result. In dynamic programming, you find some sort of "step" that takes you possibly closer to your solution, and keep track of the results of each step in a big matrix. To get a faster solution than exhaustive search, I think what you want is probably dynamic programming. Note that the speed of your search is going to greatly depend on how tight your Upper Bound is and how tight your Lower Bound is. Once you have these two functions you can apply standard Bound and Branch search. ![]() You can do this several times taking the highest lower bound that you get. When your calculating U(S) for a given S it should go quicker if you choose higher K (the conditions are relaxed) so choosing the value of K will be a trade of for quickness of finding U(S) and quality (how tight an upper bound U(S) is.)įor the L(S) function calculate the score that you would get if you simply randomly kept click until you got to a state that could not be solved any further. You need two functions U(S) and L(S) that compute a lower and upper bound respectively of a given state S.įor the U(S) function I'm thinking calculate the score that you would get if you were able to freely shuffle K bubbles in the board (each move) and arrange the blocks in such a way that would result in the highest score, where K is a value you choose yourself. Given a state of the game S, you branch on S by breaking it up in m sets Si where each Si is the state after taking a legal move of all m legal moves given the state S I'm thinking you could try a branch and bound search with the following idea: Thanks to David Locke for posting the paper link which talks above a window solver which uses a constant-depth lookahead heuristic. But trending towards larger and larger bubbles groups seems to be one approach I don't seen any obvious way to divide and conquer. What other approaches could yield high scores besides the exhaustive search? Memoization table grows to 5,692,482 boards, and hits 6,713,566 times. The solver rate is ~3k-4k boards/sec and gradually decreases as the memoization search takes longer. A (3,15,5) board takes 12,384,726 boards in 50 minutes on a server. I create a prototype in python which shows a (2,15,5) board takes 8859 boards to solve in about 3 seconds. Once a board is solved we store the board and the best score in a memoization table. Some of the ideas I am using include normalized memoization. Once the bubble group is picked, we create a new board and try to solve that board, recursively descending down The first algorithm is a simple exhaustive recursive algorithm which explores going through the board row by row and column by column picking bubble groups. ![]()
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