Exact Cover Problem via Dancing Links#
- sage.combinat.dlx.AllExactCovers(M)[source]#
Use A. Ajanki’s DLXMatrix class to solve the exact cover problem on the matrix M (treated as a dense binary matrix).
EXAMPLES:
sage: # needs sage.modules 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)]
>>> from sage.all import * >>> # needs sage.modules >>> M = Matrix([[Integer(1),Integer(1),Integer(0)],[Integer(1),Integer(0),Integer(1)],[Integer(0),Integer(1),Integer(1)]]) # no exact covers >>> for cover in AllExactCovers(M): ... print(cover) >>> M = Matrix([[Integer(1),Integer(1),Integer(0)],[Integer(1),Integer(0),Integer(1)],[Integer(0),Integer(0),Integer(1)],[Integer(0),Integer(1),Integer(0)]]) # two exact covers >>> 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)[source]#
Bases:
objectSolve 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]
>>> from sage.all import * >>> from sage.combinat.dlx import * >>> ones = [[Integer(1),[Integer(1),Integer(2),Integer(3)]]] >>> ones+= [[Integer(2),[Integer(1),Integer(3)]]] >>> ones+= [[Integer(3),[Integer(2)]]] >>> ones+= [[Integer(4),[Integer(4)]]] >>> DLXM = DLXMatrix(ones,[Integer(4)]) >>> 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()[source]#
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
- sage.combinat.dlx.OneExactCover(M)[source]#
Use 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 # needs sage.modules sage: OneExactCover(M) # needs sage.modules sage: M = Matrix([[1,1,0],[1,0,1],[0,0,1],[0,1,0]]) # two exact covers # needs sage.modules sage: OneExactCover(M) # needs sage.modules [(1, 1, 0), (0, 0, 1)]
>>> from sage.all import * >>> M = Matrix([[Integer(1),Integer(1),Integer(0)],[Integer(1),Integer(0),Integer(1)],[Integer(0),Integer(1),Integer(1)]]) # no exact covers # needs sage.modules >>> OneExactCover(M) # needs sage.modules >>> M = Matrix([[Integer(1),Integer(1),Integer(0)],[Integer(1),Integer(0),Integer(1)],[Integer(0),Integer(0),Integer(1)],[Integer(0),Integer(1),Integer(0)]]) # two exact covers # needs sage.modules >>> OneExactCover(M) # needs sage.modules [(1, 1, 0), (0, 0, 1)]