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# Dancing links C++ wrapper

`sage.combinat.matrices.dlxcpp.``AllExactCovers`(M)

  Solves the exact cover problem on the matrix M (treated as a dense binary matrix).

  EXAMPLES: No exact covers:

      sage: M = Matrix([[1,1,0],[1,0,1],[0,1,1]])
      sage: print [cover for cover in AllExactCovers(M)]
      []

  Two exact covers:

      sage: M = Matrix([[1,1,0],[1,0,1],[0,0,1],[0,1,0]])
      sage: print [cover for cover in AllExactCovers(M)]
      [[(1, 1, 0), (0, 0, 1)], [(1, 0, 1), (0, 1, 0)]]

`sage.combinat.matrices.dlxcpp.``DLXCPP`(rows)

  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:

      [
       [i_11,i_12,...,i_1r]
       ...
       [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

  If soln is a solution given by DLXCPP(rows) then

  [ rows[soln[0]], rows[soln[1]], ... rows[soln[len(soln)-1]] ]

  is an exact cover.

  Solutions are given as a list

  EXAMPLES:

      sage: rows = [[0,1,2]]
      sage: rows+= [[0,2]]
      sage: rows+= [[1]]
      sage: rows+= [[3]]
      sage: print [ x for x in DLXCPP(rows) ]
      [[3, 0], [3, 1, 2]]

`sage.combinat.matrices.dlxcpp.``OneExactCover`(M)

  Solves 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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