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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Wow, this really is good I started it and skimmed the dacning of it the first 34 pages are explanations and algorithms of dancing links and the rest of the pages is all exercises lihks answers to those exercises. Next, for each row where the selected column contains a 1, traverse the row and remove it from other columns this makes those rows inaccessible and is how conflicts are prevented.

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 knugh a particular choice fulfills that constraint. To remove a single column, first remove the selected column’s header.

The inner loop’s iterator needs to go left not right, and the links need to be restored to the original, not repeat the cover operation. PDF is probably just as clunky a format as DVI, internally, but at least it has decent application support. DVI has only information on where the characters are placed but no information about the characters’ shapes, whereas linkd PDF document the fonts are quite often embedded so they can be read even if the font is not available to the reader.

It is lijks a direct implementation from Knuth’s pdf, but with a few object orientated optimizations actually since I did this a few months ago I don’t quite remember how much I strayed from the pdf.

After selecting a row, that row and a number of columns had to be searched for 1’s.

If you can’t, remove the row you tried from the potential solution structure, restore all the rows and columns you removed in steps 3 and 4 and try a different row. A very fast solution for problem 96 “sudoku” at project Euler look for grimbal in the forum does a propagate and search using these two methods explicitly.


From Wikipedia, the free encyclopedia. I thought I had a good grip on most of the basics of programming, and a little bit of computer science theory such as big O notationbut then I checked out this. I haven’t profiled my code, but I did keep a bit of recursion in the solver our of convenience. What was the best ,inks paper you read in Also my implementation in C should be fairly easy to read I should have the code around here somewhere if you’re curious.

Dancing Links – Wikipedia

If you’ve got a partial solution stored, then it’s actually a real solution, return it. Nkuth up using Facebook. Guess I ought to throw some of it up on Github. If you read the Wikipedia entry on exact cover problems there is a section on sudoku as an exact cover problem [5] this is one thing necessary to understand in order danciny implement a sudoku solver with algorithm x and dancing links.

Add it to some kind of structure where you’re storing potential solutions. I’ve been curious if it’d be faster if I took that recursion out. The Dancing Links Algorithm – An explanation that is less explanatory but more on implementation?

Dancing Links

Are you interested in promoting your own content? There are two techniques that can be used when you don’t have a good algorithm, trial and error and the process of elimination.

This page makes the algorithm very easy to understand: Prolog has backtracking built in so getting acquainted with Prolog will get you in the right mindset. According to Knuth, dancing links will equal or better such specifically written algorithms. I also read Sudopedia’s version on it, and it seems that once it got to the Sudoku’s implementation, it got too abstract.

If a selected column doesn’t have any rows, the current matrix is unsolvable and must be backtracked. The columns are removed because they have been filled, and the rows are removed because they conflict with the selected row.



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. In addition to what others said, because for bookmarks, the abstract page works a lot better.

In a game of Sudoku you can choose one of two strategies for propagating constraints. Do you have something funny to share with fellow programmers? Knuth said of it “Algorithm X is simply a statement of the obvious trial-and-error approach.

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

Now recursively try to solve the reduced table. Assume that the row you found in 3 is in the solution, so remove all columns that it have an X in that row. By using this site, you agree to the Terms of Use and Privacy Policy. For example, consider the case of someone creating their ice dance routine.

As the remainder of this article discusses the details of an implementation technique for Algorithm X, the reader is strongly encouraged to read the Algorithm X article first. Trial and error can solve any problem where checking the answer is easy.

It’s an efficient representation for solving exact cover problems. Dancing Links is a way of implementing that algorithm efficiently. 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 knuht being used then it needs to be checked only within primary columns.

By clicking “Post Your Answer”, kunth acknowledge that you have read our updated terms of serviceprivacy policy and cookie policyand that your continued use of the website is subject to these policies. Each row and column in the matrix will consist of a circular doubly-linked list of nodes.