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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By using our site, you acknowledge that you have read and understand our Cookie PolicyPrivacy Policyand our Terms of Service. Dancing links with Knuth’s S heuristic effectively does everything a hand-specified algorithm could so I am really excited about the performance.

When selecting a row, an entire column had to be searched for 1’s. So here’s my description of algorithm X:. Please link the abstract instead of the pdf The key point of dancing links is that in a linked list, when you remove a node which can be danding 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.

I’ve been working on a Sudoku Solver, my current solver uses the backtracking algorithm but it still takes too long.

Well then I will guess that this should help you: Hacker News new comments show ask jobs submit. Starting at the beginning, the very first part of the paper discusses a significant low level technique that makes back tracking in a doubly-linked list nearly free. Anyone have an alternative?

I haven’t profiled my code, but I did keep a bit of recursion in the solver our of convenience. What he’s going into here is a bit of a trick you can use when removing elements. The idea of DLX is based on the observation that in a circular doubly linked list of nodes.


Online resources would be preferred, but if there’s not much quality material online, books would be acceptable. Please follow proper reddiquette. If you keep a reference you can “undelete” the element and put it back in the list.

This page makes the algorithm very easy to understand: As it turns out, given the right constraints and choices, sudoku can be described as an Exact Cover problem. Now that you understand that, you can understand dancing links.

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. Trial and error can solve any problem where checking the answer is easy. So here’s my description of algorithm X: I think your dancing links is easily better considering you did hard puzzles on a 1. At all times, each node in the matrix will point to the adjacent nodes to the left and right 1’s in the same rowabove and below 1’s in the same columnand the header for its column described below.

To backtrack, the above process must be reversed using the second algorithm stated above.

Although this question is very old, I thought I’d add: It would be great if you could post your code somewhere. Can someone try to explain the Dancing Links algorithm not in terms of its derivation but its implementation?

Dancing Links

You should start by learning about backtracking search. In addition to what others said, because for bookmarks, the abstract page works a lot better. You could also see my comment on in the thread for problem 96 on page 3.

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

To remove a single column, first remove the selected column’s header. This works regardless of the number of elements in the list, even if that number is 1. I suggest looking on wikipedia, then if you still have some black area we can point you somewhere else.

This is useful when your algorithm is trying a “tree” of different possibilities to find some solution like a chess game for exapmlesince if you get stuck you can backtrack very easily.

So it sounds like grimbal’s solution is doing pretty well. Just because it has a computer in it doesn’t make it programming. Since you’re just changing addresses, the value will still be in memory, but now you have no way of referencing it.

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.


It’s running as a rails app here:. Knuth discusses optional constraints as applied to the n queens problem. You browse my code here http: When selecting a column, the entire matrix had to be searched knutj 1’s. 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.

Knuth even talks about dancing links in his new Christmas Tree Lecture [6] specifically here at 4: