function [ X, y, beta, ind ] = gen_sync1( p, N, d, nns, ratio )

%% generate synthetic data with zero pair-wise correlation

nl = p./nns; % the lenght of nl is d+1; n[end] = 1 as we only has 

                      % one root node; nl[1] is the number of nodes at d

                      % layer; nl[2] is the number of nodes at d-1 layer

% construct sparse matrix to vectorize the computation

gind = zeros(d,p);

for l = 1:d

    gind(l,nns(l):nns(l):p) = 1;

    gind(l,:) = nl(l)+1-cumsum(gind(l,:),'reverse');

end

Gind = cell(1,d);

if nl(1)==p

    j=2;

else

    j=1;

end

for l=j:d

    Gind{l}=sparse(gind(l,:),1:p,ones(1,p),nl(l),p);

end

% generate the data matrix

mu = zeros(1,p);

SIGMA = speye(p);

X = mvnrnd(mu,SIGMA,N);

% generate the response

T = true(p,1);

for l = 1:d

    if l==1&&nl(1)==p

        Tf = unifrnd(0,1,[p,1])<=ratio(l);

    else

        Tl = unifrnd(0,1,[nl(l),1])<=ratio(l);

        Tf = logical((Gind{l})'*Tl);

    end

    T = T&Tf;

end

beta = zeros(p,1);

beta(T) = normrnd(0,1,[nnz(T),1]);

y = X*beta + 0.01*normrnd(0,1,[N,1]);  

% construct ind

if nl(1) == p % if the nodes at the first layer only has one feature

    cumnl = [0,1,cumsum(nl(2:end))+1];

    ind = zeros(3,cumnl(end));

    ind(:,1)=[-1, -1, 1]';

    ind(:,end)=[1,p,0]';

    for i = 2:d

        ind(1,cumnl(i)+1:cumnl(i+1)) = 1:nns(i):p;

        ind(2,cumnl(i)+1:cumnl(i+1)) = ind(1,cumnl(i)+1:cumnl(i+1))+nns(i)-1;

        ind(3,cumnl(i)+1:cumnl(i+1)) = sqrt(nns(i));

    end

else

    cumnl = [0,cumsum(nl)];

    ind = zeros(3,cumnl(end));

    ind(:,end)=[1,p,0]';

    for i = 1:d

        ind(1,cumnl(i)+1:cumnl(i+1)) = 1:nns(i):p;

        ind(2,cumnl(i)+1:cumnl(i+1)) = ind(1,cumnl(i)+1:cumnl(i+1)) + nns(i);

        ind(3,cumnl(i)+1:cumnl(i+1)) = sqrt(nns(i));

    end

end

end