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Knuth7: Dancing Links and SAT solvers in Java

Implementation of algorithms found in Knuth's fascicles 5C and 6, which will both become part of chapter 7 of The Art of Computer Programming.

Exact Cover

The text describes variants of the Dancing Links algorithm, which can be used to solve the Exact Cover (XC) problem. One has a set of items, together with a set of options (which are each nonempty subsets of the items), and asks whether there is a subset of the options which will exactly cover the items (in the sense that each item occurs exactly once in the chosen options).

Knuth proposes a simple text representations for XC problems which this code can read: The first line lists the items, separated by whitespace; each following line is an option, consisting of whitespace-separated items.

For example, the problem specification

a b c d e f g
c e
a d g
b c f 
a d f
b g 
d e g

is solved if we select the first, fourth and fifth options (c e, a d f, b g) and in fact this is the only solution.

In code, we can say:

ExactCoverProblem p = ExactCoverProblem.parseFrom("a b c d e f g\nc e\na d g\nb c f\na d f\nb g\nd e g");

Then p.solutions() is a Stream<List<Integer>> which will have one element: the list [3, 4, 0], indicating which options (counting from zero) form the solution.

The stream is lazy: calling solutions() only pays the cost of setting up the problem tableau. Each demand for a new element of the stream will provoke a search for the next distinct solution, which will stop as soon as one is found. The search for solutions will be exhaustive only if you consume the whole stream.

Adapters are provided for a few applications of XC that Knuth proposes.

Sudoku

You can supply a Sudoku problem in the form of a string of 81 digits or dots (indicating an empty cell), and then generate a stream of solutions:

String ex28c = ".3. .1. ... " +
                "... 4.. 1.." +
                ".5. ... .9." +
                "2.. ... 6.4" +
                "... .35 ..." +
                "1.. ... ..." +
                "4.. 6.. ..." +
                "... ... .5." +
                ".9. ... ...";
Sudoku.fromBoardString(ex28c).solutions().collect(Collectors.toList())                
// ["934 518 267 762 493 185 851 762 493 285 971 634 649 235 718 173 846 529 418 659 372 327 184 956 596 327 841 ",
//  "934 517 268 862 493 175 751 862 493 275 981 634 649 235 817 183 746 529 417 659 382 328 174 956 596 328 741 "]

The solver is fast; it can generate an answer within about 15ms.

Word Find

You can request the generation of word find puzzles. Just supply the grid width and height and a list of words to fit in the grid. Words can appear in any of the 8 orientations. The wrapper generates a stream of solutions as strings (using newlines to separate rows of the grid).

new WordFind(8, 8, Arrays.asList("gandalf", 
                                 "frodo", 
                                 "boromir", 
                                 "merry", 
                                 "pippin", 
                                 "aragorn", 
                                 "samwise", 
                                 "legolas", 
                                 "gimli", 
                                 "gollum", 
                                 "ring", 
                                 "sauron")).solutions().limit(1)

creates a stream which will produce the solution

G A N D A L F B
S N Y R R E M O
A P I F A G U R
M I L R G O L O
W P M O O L L M
I P I D R A O I
S I G O N S G R
E N S A U R O N

A dot character is used for cells that are unused after the packing is achieved; thus in this case we can see that every cell in the grid participates in the puzzle.

Secondary Items and Colored Options

Secondary items (which must occur at most once in any solution, instead of exactly once are supported. In the item line (i.e., the first line), you can have a lone semicolon which will divide the primary from the secondary items (there should be whitespace on either side of the semicolon), like: a b c ; x y.

A secondary item within an option my be given a color using a colon suffix, like b c x:Red. The XCC algorithm implemented here will ensure that all options chosen for a solution will agree on the "color" of any secondary items included.

Satisfiability

Currently we implement algorithms A, B and D, which are enough for small problems. We implement DIMACS format for SAT problem input. More to follow.