getting a submatrix of all true

Anton Vredegoor anton at
Sat Jul 5 00:24:01 CEST 2003

John Hunter <jdhunter at> wrote:

>Another way of formulating the question: for a sparse boolean matrix
>(sparse on True), what is the optimal way to remove rows and columns
>so that the total number of elements in the matrix is maximal and
>there are no True values left.

After having read Bengts code (and scraping the parts of my exploded
head from the floor) and after later getting the hint about the size
of the problem being 2**N, it finally dawned on me that the problem
would be related to getting all possible combinations of a range.

The following code has the advantage that it generates solutions by
index, for example "M[i]" returns the score and the row and column
information for index i. (note that is not the same as finding the
optimal solution, but it still could be handy to be able to generate a
specific one by index) 

However for small matrices it is possible to do exhaustive searching
with "max(M)". 

Maybe if this would be combined with some heuristic algorithm it would
be better (genetic algorithm?)

Code below (needs Python 2.3, very lightly tested), I hope this helps,


class Combinations:
    def __init__(self,n,k):
        self.n,self.k,self.count = n,k,self.noverk(n,k)
    def __getitem__(self,index):
        #combination number index
        if index > self.count - 1: raise IndexError
        res,x,rest = [],self.n,index
        for i in range(self.k,0,-1):
            while self.noverk(x,i) > rest : x -= 1
            rest -= self.noverk(x,i)
        return res
    def noverk(self,n,k):
        return reduce(lambda a,b: a*(n-b)/(b+1),range(k),1)
class AllCombinations:
    def __init__(self,n):
        self.n, self.count = n, 2**n
    def __getitem__(self,index):
        #combination number index for all k in combinations(n,k)
        if index > self.count - 1: raise IndexError        
        n,rest = self.n,index
        for k in range(n+1):
            c = Combinations(n,k)
            if rest - c.count < 0:
                return c[rest]
            rest -= c.count

class ScoreMatrix:
    def __init__(self, X):
        self.AC = AllCombinations(len(X))
        self.X = X
    def __getitem__(self,i):
        #score selected rows and columns by index
        rows = self.AC[i][::-1]    
        if rows:
            Y = [self.X[i] for i in rows]
            Z = [(i,z) for i,z in enumerate(zip(*Y)) if 1 not in z]
            cols = [i for i,z in Z]
            score = sum(map(len,[z for i,z in Z]))
            return score,rows,cols
        else: return 0,[],[]

def test():
    X = [   [1, 0, 0, 0, 1],
                [0, 0, 0, 0, 0],
                [0, 0, 0, 1, 0],
                [0, 0, 0, 0, 0],
                [0, 0, 1, 0, 0],
                [0, 0, 1, 0, 0], ]
    M = ScoreMatrix(X)
    sc,rows,cols = max(M)
    print sc
    for i,r in enumerate(X):
        for j,c in enumerate(r):
            if i in rows and j in cols: print c,
            else: print 'x',

if __name__=='__main__':


x x x x x
0 0 x 0 0
x x x x x
0 0 x 0 0
0 0 x 0 0
0 0 x 0 0

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