Question

我需要创建零对角线的所有0,1 NxN矩阵。矩阵必须是对称的。在每一列和每行必须至少有一个1.任何可以帮助的想法？

Answer 1

获取该表单的所有可能的矩阵

这个想法是这种类型的大小为NxN的矩阵由其上对角线值定义。因此，迭代上对角线部分的所有可能模式，并将这些值应对到下对角线将起到作用。代码示例：

%defines N
N = 3;

%calculates degree of freedom
nValuesToRand = ((N*N) - N)/2;

%generate all possible binary patterns of size nValuesToRand 
B = dec2bin(0:2^nValuesToRand - 1);

%masks of lower and upper diagonal - will be used later on
upperTriagonalMask = logical(triu(ones(N,N)) - eye(N));
lowerTriagonalMask = logical(tril(ones(N,N)) - eye(N));

%generates a new cell to hold the matrices
allMatrices = cell(size(B,1),1);

%iterates over all possible patterns
for i=1:size(B,1)

    %generates a new matrix
    mat = zeros(N,N);

    %initializes its upper diagonal according to the binary pattern
    mat(upperTriagonalMask) = logical(B(i,:)- 48);

    %copies the upper triagonal to the lower triagonal (for symmetricality)
    upperTriagonalTransposed = triu(mat)';
    mat(lowerTriagonalMask) = upperTriagonalTransposed(lowerTriagonalMask);

    %ignores illegal Matrices 
    if sum(sum(mat,2)==0)>0
        continue;
    end

    %saves mat in the cell
    allMatrices{i} = mat;

end

%cleanes cell
allMatrices = allMatrices(~cellfun(@isempty, allMatrices));

随机矩阵生成

为大N值生成所有可能的矩阵在计算上是困难的。如果要随机生成矩阵，可以尝试以下方法：

%Dimension size
N = 6;

%Probability for appearance of 0
P = 0.5;

%A mask of the lower diagonal, to be used later on
lowerTriagonalMask = logical(tril(ones(N,N)));

%initializes the matrix
mat = zeros(N,N);

%runs the loop as long as the matrix is not valid
while (sum(mat,2)==0)>0

    %defines a random binary matrix
    mat = rand(N,N) > P;

    %zero out the diagonal values
    mat(logical(eye(N))) = 0;

    %copies the upper triagonal to the lower triagonal (for symmetricality)
    upperTriagonalTransposed = triu(mat)';
    mat(lowerTriagonalMask) = upperTriagonalTransposed(lowerTriagonalMask);

end

%testing
issymmetric(double(mat))
mat

结果：

ans = 

1

mat =

 0     1     1     1     0     0
 1     0     1     1     1     1
 1     1     0     0     0     1
 1     1     0     0     0     1
 0     1     0     0     0     1
 0     1     1     1     1     0

对称的0,1矩阵

1 个答案: