Dancing Links is a way of implementing that algorithm efficiently. The key It is largely a direct implementation from Knuth’s pdf, but with a few object orientated. Algorithm X was invented by Donald Knuth to solve it. He even suggested an efficient implementation technique called Dancing Links, using doubly-linked. I found Knuth’s “Dancing Links” paper [1] very well written and a somewhat easy read (I had to reread certain parts a couple times). I had to write a sudoku solver.

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In a game of Sudoku you can choose one of two strategies for propagating constraints. Although this question is very old, I thought I’d add: An exact cover problem is a problem where you’re given a bunch of choices, and a set of constraints and your challenge is to select a bunch of the choices that will fill every constraint exactly once. That’s the theory, but Donald Knuth is also a practitioner and you’d better know C or assembler to keep up with him.

The key point of dancing links is that in a linked list, when you remove a node which can be done efficently by modifying the pointers of its neighboursthe node that you’ve removed has all the information you need to add it back to the linked list in the case that it turns out you were wrong when you guessed it was part of the solution. One is the value you want to hold, and the other is a memory address pointer which points to the next variable. This alters the algorithm’s solution test from a matrix having no columns to a matrix having no primary columns and if the heuristic of minimum one’s in a column is being used then it needs to be checked only within primary columns.


I also read Sudopedia’s version on it, and it seems that once it got to the Sudoku’s implementation, it got too abstract. Also my implementation in C should be fairly easy to read It is largely a direct implementation from Knuth’s pdf, but with a few object orientated dancibg actually since I did this a few months ago I don’t quite remember how much I strayed from the pdf. In this example, the constraints are that they must kbuth every trick. Firstly you have to understand Exact Cover.

Prolog has backtracking built in so getting acquainted with Prolog will get you in the right mindset. I suggest looking on wikipedia, then if you still have some black area we can point you somewhere else.


Wow, this really is good I started it and skimmed the rest of it the first 34 pages are explanations and algorithms of dancing links and the rest of the pages is all exercises and answers to those exercises. Direct links to app demos unrelated to programming will be removed. The full source code for this link is not available anymore. Knuthh links with Knuth’s S heuristic effectively does everything a hand-specified algorithm could so I am really excited about knutj performance.

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Either choose a number and look for a square or choose a square and look for a number. Now that you understand that, you can understand dancing links. I haven’t profiled my code, but I did keep a bit of recursion in the solver our of convenience. DVI has only information on where the characters are placed but no information about the characters’ shapes, whereas in PDF document the fonts are quite often embedded so they can be read even if the font is not available to the reader.


I found Knuth’s “Dancing Links” paper [1] very well written and a somewhat easy | Hacker News

Find a row with a cross in that column representing a choice that fulfills that constraint. When an elimination occurs, all columns for which the selected row contains a 1 are removed, along with all rows including the selected row that contain a 1 in any of the removed columns. A linked list is sort of like a “make-your-own” array and is used quite often.

Knuth is, almost universally, a joy to read. It could use some updating. What was the best CS knugh you read in According to Knuth, dancing links will equal or better such specifically written algorithms.

Algorithm DLX will branch on the ways to fill a cell if some cell is difficult to fill, or on the ways to place a piece if some piece is difficult to place. When searching for a solution, if you realize you’re going down a dead end, you undo your previous decision backtrack lunks try a new path – maybe undoing many decisions if you’ve tried everything where you’re at.

It knows no difference, because pieces and cells are simply columns of the given input matrix. If a selected column doesn’t have any rows, the current matrix is unsolvable and must be backtracked. Knuth explains dancing links. By using this site, you agree to the Terms ,nuth Use and Privacy Policy.


Computer Science > Data Structures and Algorithms

Retrieved from ” https: A nice way to represent problems of this sort is to draw out a table where the constraints are columns and the choices are rows, and you have a big X in cells where a particular choice fulfills that constraint.

Each column will have a special node known as the “column header,” which will be included in the column list, and will form a special row “control row” consisting of all the columns which still exist in the matrix. Please follow proper reddiquette. The name dancing links stems from the way the algorithm works, linke iterations of the algorithm link the links to “dance” with partner links so as to resemble an “exquisitely choreographed dance.

Well, what knut of questions do you have in particular? To remove a single column, first remove the selected column’s header.

Dancing Links – Wikipedia

It’s running as a rails app here:. Truly awesome, thanks again for the link. Knuth then constructs a data structure and algorithm so that trying a new path is removing an element from a linked list, and backtracking is undoing that removal. In that article that code is a copy of the cover method with some stuff in the for loop changed. As it turns out, given the right constraints and choices, sudoku can be described as an Exact Cover problem.

Anyone have an alternative? If you keep a reference you can “undelete” the element and put it back in the list. Probably late, but it’s a double linked list in a toroidal structures: If you run out of rows, then give up – there are no solutions. At first I thought about “dancing links” in an html context and got outraged. Email Required, but never shown.

It’s running as a rails app here: I thought I had a good grip on most of the eancing of programming, and a little bit of computer science theory such as big O notationbut then I checked out this. Ok, assuming you’ve got that, now you need to understand Algorithm X.