Speeding up the conditional filling of huge sparse matrices - matlab

Question

I was wondering if there is a way of speeding up (maybe via vectorization?) the conditional filling of huge sparse matrices (e.g. ~ 1e10 x 1e10). Here's the sample code where I have a nested loop, and I fill in a sparse matrix only if a certain condition is met:
% We are given the following cell arrays of the same size:
% all_arrays_1
% all_arrays_2
% all_mapping_arrays
N = 1e10;
% The number of nnz (non-zeros) is unknown until the loop finishes
huge_sparse_matrix = sparse([],[],[],N,N);
n_iterations = numel(all_arrays_1);
for iteration=1:n_iterations
array_1 = all_arrays_1{iteration};
array_2 = all_arrays_2{iteration};
mapping_array = all_mapping_arrays{iteration};
n_elements_in_array_1 = numel(array_1);
n_elements_in_array_2 = numel(array_2);
for element_1 = 1:n_elements_in_array_1
element_2 = mapping_array(element_1);
% Sanity check:
if element_2 <= n_elements_in_array_2
item_1 = array_1(element_1);
item_2 = array_2(element_2);
huge_sparse_matrix(item_1,item_2) = 1;
end
end
end
I am struggling to vectorize the nested loop. As far as I understand the filling a sparse matrix element by element is very slow when the number of entries to fill is large (~100M). I need to work with a sparse matrix since it has dimensions in the 10,000M x 10,000M range. However, this way of filling a sparse matrix in MATLAB is very slow.
Edits:
I have updated the names of the variables to reflect their nature better. There are no function calls.
Addendum:
This code builds the matrix adjacency for a huge graph. The variable all_mapping_arrays holds mapping arrays (~ adjacency relationship) between nodes of the graph in a local representation, which is why I need array_1 and array_2 to map the adjacency to a global representation.

Answer 1

I think it will be the incremental update of the sparse matrix, rather than the loop based conditional that will be slowing things down.
When you add a new entry to a sparse matrix via something like A(i,j) = 1 it typically requires that the whole matrix data structure is re-packed. The is an expensive operation. If you're interested, MATLAB uses a CCS data structure (compressed column storage) internally, which is described under the Data Structure section here. Note the statement:
This scheme is not effcient for manipulating matrices one element at a
time
Generally, it's far better (faster) to accumulate the non-zero entries in the matrix as a set of triplets and then make a single call to sparse. For example (warning - brain compiled code!!):
% Inputs:
% N
% prev_array and next_array
% n_labels_prev and n_labels_next
% mapping
% allocate space for matrix entries as a set of "triplets"
ii = zeros(N,1);
jj = zeros(N,1);
xx = zeros(N,1);
nn = 0;
for next_label_ix = 1:n_labels_next
prev_label = mapping(next_label_ix);
if prev_label <= n_labels_prev
prev_global_label = prev_array(prev_label);
next_global_label = next_array(next_label_ix);
% reallocate triplets on demand
if (nn + 1 > length(ii))
ii = [ii; zeros(N,1)];
jj = [jj; zeros(N,1)];
xx = [xx; zeros(N,1)];
end
% append a new triplet and increment counter
ii(nn + 1) = next_global_label; % row index
jj(nn + 1) = prev_global_label; % col index
xx(nn + 1) = 1.0; % coefficient
nn = nn + 1;
end
end
% we may have over-alloacted our triplets, so trim the arrays
% based on our final counter
ii = ii(1:nn);
jj = jj(1:nn);
xx = xx(1:nn);
% just make a single call to "sparse" to pack the triplet data
% as a sparse matrix object
sp_graph_adj_global = sparse(ii,jj,xx,N,N);
I'm allocating in chunks of N entries at a time. Assuming that you know alot about the structure of your matrix you might be able to use a better value here.
Hope this helps.