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# Exact Cover Problem via Dancing Links

`sage.combinat.dlx.``AllExactCovers`(M)

  Utilizes A. Ajanki’s DLXMatrix class to solve the exact cover problem on the matrix M (treated as a dense binary matrix).

  EXAMPLES:

      sage: M = Matrix([[1,1,0],[1,0,1],[0,1,1]])  #no exact covers
      sage: for cover in AllExactCovers(M):
      ...       print cover
      sage: M = Matrix([[1,1,0],[1,0,1],[0,0,1],[0,1,0]]) #two exact covers
      sage: for cover in AllExactCovers(M):
      ...       print cover
      [(1, 1, 0), (0, 0, 1)]
      [(1, 0, 1), (0, 1, 0)]

class `sage.combinat.dlx.``DLXMatrix`(ones, initialsolution=None)

  Solves the Exact Cover problem by using the Dancing Links algorithm described by Knuth.

  Consider a matrix M with entries of 0 and 1, and compute a subset of the rows of this matrix which sum to the vector of all 1’s.

  The dancing links algorithm works particularly well for sparse matrices, so the input is a list of lists of the form: (note the 1-index!):

      [
       [1, [i_11,i_12,...,i_1r]]
       ...
       [m,[i_m1,i_m2,...,i_ms]]
      ]

  where M[j][i_jk] = 1.

  The first example below corresponds to the matrix:

      1110
      1010
      0100
      0001

  which is exactly covered by:

      1110
      0001

  and

      1010
      0100
      0001

  EXAMPLES:

      sage: from sage.combinat.dlx import *
      sage: ones = [[1,[1,2,3]]]
      sage: ones+= [[2,[1,3]]]
      sage: ones+= [[3,[2]]]
      sage: ones+= [[4,[4]]]
      sage: DLXM = DLXMatrix(ones,[4])
      sage: for C in DLXM:
      ...      print C
      [4, 1]
      [4, 2, 3]

  Note

  The 0 entry is reserved internally for headers in the sparse representation, so rows and columns begin their indexing with 1. Apologies for any heartache this causes. Blame the original author, or fix it yourself.

  `next`()

    Search for the first solution we can find, and return it.

    Knuth describes the Dancing Links algorithm recursively, though actually implementing it as a recursive algorithm is permissible only for highly restricted problems. (for example, the original author implemented this for Sudoku, and it works beautifully there)

    What follows is an iterative description of DLX:

        stack <- [(NULL)]
        level <- 0
        while level >= 0:
            cur <- stack[level]
            if cur = NULL:
                if R[h] = h:
                    level <- level - 1
                    yield solution
                else:
                    cover(best_column)
                    stack[level] = best_column
            else if D[cur] != C[cur]:
                if cur != C[cur]:
                    delete solution[level]
                    for j in L[cur], L[L[cur]], ... , while j != cur:
                        uncover(C[j])
                cur <- D[cur]
                solution[level] <- cur
                stack[level] <- cur
                for j in R[cur], R[R[cur]], ... , while j != cur:
                    cover(C[j])
                level <- level + 1
                stack[level] <- (NULL)
            else:
                if C[cur] != cur:
                    delete solution[level]
                    for j in L[cur], L[L[cur]], ... , while j != cur:
                        uncover(C[j])
                uncover(cur)
                level <- level - 1

    TESTS:

        sage: from sage.combinat.dlx import *
        sage: M = DLXMatrix([[1,[1,2]],[2,[2,3]],[3,[1,3]]])
        sage: while 1:
        ...     try:
        ...         C = M.next()
        ...     except StopIteration:
        ...         print "StopIteration"
        ...         break
        ...     print C
        StopIteration
        sage: M = DLXMatrix([[1,[1,2]],[2,[2,3]],[3,[3]]])
        sage: for C in M:
        ...       print C
        [1, 3]
        sage: M = DLXMatrix([[1,[1]],[2,[2,3]],[3,[2]],[4,[3]]])
        sage: for C in M:
        ...       print C
        [1, 2]
        [1, 3, 4]

`sage.combinat.dlx.``OneExactCover`(M)

  Utilizes A. Ajanki’s DLXMatrix class to solve the exact cover problem on the matrix M (treated as a dense binary matrix).

  EXAMPLES:

      sage: M = Matrix([[1,1,0],[1,0,1],[0,1,1]])  #no exact covers
      sage: print OneExactCover(M)
      None
      sage: M = Matrix([[1,1,0],[1,0,1],[0,0,1],[0,1,0]]) #two exact covers
      sage: print OneExactCover(M)
      [(1, 1, 0), (0, 0, 1)]

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