# Find minimum non-zero entry in sparse matrix

## weblog/

Using a sparse matrix to store a weighted adjacency matrix, I found myself trying to grad the minimal non-zero entry per row. This is a little tricky since taking min will just give the first zero (and obviously taking max of the inverse also returns 0). Let your sparse matrix be:
``````A = sprand(10,10,0.5)-sprand(10,10,0.1);
``````
This site proposes filling the zero entries with `inf`.
``````A(~A) = inf;
[mA,mI] = min(A);
``````
But this turns our sparse matrix into a dense one. As A gets big this becomes a disaster.
``````[sA,sI] = sort(A,'descend');
[I,J,V] = find(sA);
msI = max(sparse(I,J,I));
mA = sA(sub2ind(size(A),msI,1:size(A,1)));
mI = sI(sub2ind(size(A),msI,1:size(A,1)));
``````
Now, try this for 10000 by 10000 A with an average of 6 non-zeros per column. This replace with `inf`s method takes crashed my matlab. So now try this for a 2500 by 2500 A with an average of 6 non-zeros per row. The replace with `inf`s method takes 40.5 seconds, first converting the matrix to full format then replacing with `inf`s does a lot better at 0.08 seconds, but the truly sparse method only takes 0.035 seconds. Update: Even another way to do it, which avoids the sort:
``````  [AI,AJ,AV] = find(A);
A_i = sparse(AI,AJ,AV.^-1,size(A,1),size(A,2));
[maxA_i,maxI_i] = max(A_i);
[minA,minI] = min(A);
minA(minA==0) = inf;
mA = [maxA_i.^-1;minA];
[mA,J] = min(mA);
mI = [maxI_i;minI];
mI = mI(sub2ind(size(I),J,1:size(mI,2)));
``````