How to compute sum of binomial more efficiently?

Question

I must calculate an equation as follows:

where k1,k2 are given. I am using MATLAB to compute P. I think I have a correct implementation fo

Answer 1

You can vectorize all the process and it will make it super-fast without any need for mex.

First the nchoosek function:

function C = nCk(n,k)
% use smaller k if available
k(k>n/2) = n-k(k>n/2);
k = k(:);
kmat = ones(numel(k),1)*(1:max(n-k));
kmat = kmat.*bsxfun(@le,kmat,(n-k));
pw = bsxfun(@power,kmat,-1./(n-k));
pw(kmat==0) = 1;
kleft = ones(numel(k),1)*(min(k):n);
kleft = kleft.*bsxfun(@gt,kleft,k);
t = bsxfun(@times,kleft,prod(pw,2));
t (kleft==0) = 1;
C = prod(t,2);
end

Then beta and P computation:

function P = binomial_coefficient(k1,k2,D)
warning ('off','MATLAB:nchoosek:LargeCoefficient');
i_ind = nonzeros(triu(ones(D,1)*(1:D)))-1;
j_ind = nonzeros(tril(ones(D,1)*(1:D+1)).')-1;
valid = ~(i_ind-j_ind>=k2 | j_ind>=k1);
i_ind = i_ind(valid);
j_ind = j_ind(valid);
beta = @(ii,jj) nCk(k1,jj).*nCk(k2,ii-jj)./nCk((k1+k2),ii);
b = beta(i_ind,j_ind);
P = sum(b(:));
end

and execution time drops from 10.674 to 0.49696 seconds.

EDIT:

Taking the idea of @rahnema1, I managed to make this even faster, using a table for all unique nCk computations, so none of them will be done more than once. Using the same nCk function from above, this is how the new binomial_coefficient function will look:

function P = binomial_coefficient(k1,k2,D)
warning ('off','MATLAB:nchoosek:LargeCoefficient');
i_ind = nonzeros(triu(ones(D,1)*(1:D)))-1;
j_ind = nonzeros(tril(ones(D,1)*(1:D+1)).')-1;
valid = ~(i_ind-j_ind>=k2 | j_ind>=k1);
i_ind = i_ind(valid);
j_ind = j_ind(valid);
ni = numel(i_ind);
all_n = repelem([k1; k2; k1+k2],ni); % all n's to be used in thier order
all_k = [j_ind; i_ind-j_ind; i_ind]; % all k's to be used in thier order
% get all unique sets of 'n' and 'k':
sets_tbl = unique([all_n all_k],'rows');
uq_n = unique(sets_tbl(:,1));
nCk_tbl = zeros([max(all_n) max(all_k)+1]);
% compute all the needed values of nCk:
for s = 1:numel(uq_n)
    curret_n = uq_n(s);
    curret_k = sets_tbl(sets_tbl(:,1)==curret_n,2);
    nCk_tbl(curret_n,curret_k+1) = nCk(curret_n,curret_k).';
end
beta = @(ii,jj) nCk_tbl(k1,jj+1).*nCk_tbl(k2,ii-jj+1)./nCk_tbl((k1+k2),ii+1);
b = beta(i_ind,j_ind);
P = sum(b(:));
end

and now, when it takes only 0.01212 second to run, it's not just super-fast code, it's a flying-talking-super-fast code!