7/2/2023 0 Comments Wikipedia sudoku strategy![]() ![]() In step 1 you can choose an item to cover however you want. Recursing will choose the other items, and you will reach all possible solutions. Note that even though you iterate through all the options that cover an item in step 2, you don’t need to iterate through the items in step 1. If you are only looking for one solution instead of all possible solutions, you can exit out after finding a solution. If any item runs out of valid options, you’ve hit a dead end.ĭepending on the problem you are trying to solve, you may find 0 solutions, 1 solution, or multiple solutions. If the item list becomes empty (or only has optional items left), you have a valid solution. Undo making this option part of the solution and try the next option.Recurse, having Algorithm X solve the problem with reduced items.They are no longer valid choices since they conflict. Remove all options that have items that were removed.All those items can be considered covered. For each option that covers that item, try it as part of the solution:.Choose an item to cover from the list of items.At the highest level, it works like this. Knuth’s Algorithm X is a method for solving exact cover problems. There are other extensions that I’ll briefly explain at the bottom of this post. We’ll make use of this in the N-Queens example. These items don’t have to be covered, but if they are covered, they can only be covered at most once. The goal of an exact cover problem is to find a set of options which cover all items exactly once.Īn extension to this idea is to have some items be optional. That option (matrix row) would have a 1 at each of those locations and a 0 at the rest of the locations. For example, a tetromino at a specific position on a chess board would cover multiple chess board locations. Rows are called options and they contain one or more items that they cover. For example, for placing objects on a chess board, there would be an item (matrix column) per chess board location. Using Knuth’s terminology, the columns are called items and are the things that are looking to be covered. Exact Cover ProblemsĪn exact cover problem is described by a binary matrix of 0s and 1s. Let’s talk about the details of the algorithms a bit and then look at some examples. I wanted to learn this algorithm to have something in my tool belt to approach these sorts of problems more intelligently, but I also think this will lead to new algorithms for generating noise textures (aka sampling matrices), and/or new types of situationally optimal noise. For instance, the problem of tiling a board with pentominoes, and solving Sudoku can both be viewed as exact cover problems. However, for some well known problems, the reduction is particularly direct. Interestingly it says that you can turn any NP problem into an exact cover problem!ĭue to its NP-completeness, any problem in NP can be reduced to exact cover problems, which then can be solved with techniques such as Dancing Links. Solving sudoku is also an exact cover problem.Ī more formal definition is on Wikipedia at. If you want to put N queens on an NxN chess board such that none can attack each other, that is an exact cover problem. So what is an exact cover problem? If you have some tetrominoes (or Tetris pieces) and you want to put them together to form a rectangle, that is an exact cover problem. Giving it a google, I found this 2018 video by Knuth talking about his “Algorithm X” method to solve exact cover problems, specifically using “Dancing Links” to implement it efficiently: In that video, algorithm X was mentioned and I was intrigued. Right angles give you a lot of information regarding the empty columns and rows in the cell they’re in, which can help you cancel out incorrect candidates in the adjacent cells.The C++ code that goes with this post can be found at Ī fun youtube video came out recently showing how people optimized an algorithm from running for an entire month, to eventually running for less than a millisecond: Right angles (any 3 given numbers in an L-shape inside of a cell).This pattern can help you isolate rows and columns to solve entire rows or columns of the puzzle. Skyscrapers (two rows or columns of a given candidate that are unequal in length).Revisit these regularly to make sure you don’t provide a false solution. Corner patterns help eliminate a ton of potential candidates in the rows and columns connected to it. Corners (a collection of 4 solved squares in any of the 4 corners). ![]() X Research source A few common patterns include: There are a bunch of different patterns out there, but if you can spot one, they’ll typically help you solve some element of the puzzle that you’re struggling with. Patterns refer to configurations of solved squares that help players regularly solve a sequence of candidates. There are a handful of patterns most players look for at this point. ![]()
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