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  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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Knuth explains dancing links.
Dancing Links – Wikipedia
It’s an efficient representation for solving exact cover problems. In a game of Sudoku you can choose one of two strategies for propagating constraints. Either choose a number and knutn for a square or choose a square and look for a number. 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. It knows no difference, because pieces and cells are simply columns of the given input matrix.
Guess I ought to throw some of it up on Github. It’s running as a rails daancing here:. 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. According to Knuth, dancing links will equal or better such specifically written algorithms.
You could also see my comment on in the thread for problem 96 on page 3. I should have the code around here somewhere dancjng you’re curious. How did I miss that?
I think your dancing links is easily better considering you did hard puzzles on a 1. Dancing links with Knuth’s S heuristic effectively does everything a hand-specified algorithm could so I am really excited about the performance.
It would be great if you could post your code somewhere. My current laptop is a Core2Duo 2. There’s danding lot of timing noise in just running one copy of Euler problem set through my solver. So it sounds like grimbal’s danving is doing pretty well.
I haven’t profiled my code, but I did keep a bit of recursion in the solver our of convenience. I’ve been curious if it’d be faster if I took that recursion out. You browse my code here http: Those that are interested in this ought to check out Knuth’s video lecture on the same subject, at the SCPD site: I used Knuth’s dancing links algorithm to generate many of the puzzles at my website.
It’s damcing that he explains the reduction to exact cover in the source. I am currently an amateur programmer in high school, teaching myself. I’d eventually like get a computer science degree, but meanwhile, I like to learn things on my own. 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 lniks I checked out this.
I understand almost none of it. Could anyone point out somewhere where I could learn a little bit about what’s mentioned in this paper? Online resources would be dancign, but if there’s not much quality material online, books would be acceptable. Starting at the beginning, the very first part ljnks the paper discusses a significant low level technique that makes back tracking in a doubly-linked list nearly free. This removes x from the doubly-linked list.
Knuth goes on to notice that if you don’t fiddle with or clean up x. When searching for a solution, if you realize you’re going down a dead end, you undo your previous decision backtrack and try a new path – maybe undoing many decisions if you’ve tried everything where you’re at. Knuth then constructs a data structure and algorithm so that trying a new path is removing an element from dacing linked list, and backtracking is undoing that removal.
Then, searching for a solution is only a series of very cheap pointer twiddlings no recursion, no memory allocations necessary – hence the name “dancing links”. It’s a linked list. A linked list is sort of like a “make-your-own” array and is used quite often. Each element of a linked list contains 2 variables. One is the value you want to hold, and the other is a memory address pointer which points to the next variable. A doubly linked list also stores the memory address of the previous variable so you can go backwards.
You have yourself a very dynamic “array” which can be any size, and you can easily insert, move, and remove elements just by changing the address of the next element.
What he’s going into here is a bit of a trick you can use when removing elements. To delete an element in a linked list, you just have to make the previous element point to a different knuhh. Since you’re just changing addresses, the value will still be in memory, but now you have no way of referencing it. If you keep a reference you can “undelete” the element and put it back in the list. 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.
This paper plays with a rather small set of concepts; I checked the wikipedia entries for the main keywords in the paper dancig I found them quite informative I have a CS degree. I suggest looking on wikipedia, then if you still have some black area we can point you somewhere else.
There are two techniques that can be used when you don’t have a good algorithm, trial and error and the process of elimination. Trial and error can solve any problem where checking the answer is easy. The dumb version is called knugh and test but there are often too many possible solutions to test them all. A smarter version called backtracking finds solutions incrementally, dnacing a partial solution at each step to eliminate bad mnuth early.
Constraint propagation eliminates bad choices based on existing knowledge so they never have to be tested.
Computer Science > Data Structures and Algorithms
This paper describes a clever way to do all of the above. You should start by learning about backtracking search.
Prolog has backtracking built in so getting acquainted with Prolog will get you in the right mindset. Next read up on Constraint Satisfaction. Also look at some constraint satisfiers. That’s the theory, but Donald Knuth is also a practitioner and you’d better know C or eancing to keep up with him. I began learning about data structures and algorithms from a dead-tree book.
I don’t know who wrote it or what the title was. The only thing I remember is that it was green, old, big, had lots kuth diagrams with arrows, and was owned by my high school library. At first I thought about “dancing links” lknks an html context and got outraged.
Please link the abstract instead of the pdf Because the arxiv abstract page has nice links to more content, including other papers by the same author, and different versions of the same paper, and the source upload of the paper.
In addition to danciing others said, because for bookmarks, the abstract page works a lot better. Especially true on arxiv. PDF is probably just as clunky a format as DVI, internally, but at least it has decent mnuth support. DVI has only information on where the characters are placed knurh 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 will join you in your sadness, although PDF doesn’t handle typesetting.
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It’s running as a rails app here: Well, what kind of questions do you inuth in particular? Hope this supplements spliznork’s comment.
Anyone have an alternative? Is there no plugin to watch them inline?