Vectorizing a nested for loop which fills a dynamic programming table - matlab

Question

I was wondering if there was a way to vectorize the nested for loop in this function which is filling up the entries of the 2D dynamic programming table DP. I believe that at the very least the inner loop could be vectorized as each row only depends on the previous row. I'm not sure how to do it though. Note this function is called on large 2D arrays (images) so the nested for loop really doesn't cut it.
function [cols] = compute_seam(energy)
[r, c, ~] = size(energy);
cols = zeros(r);
DP = padarray(energy, [0, 1], Inf);
BP = zeros(r, c);
for i = 2 : r
for j = 1 : c
[x, l] = min([DP(i - 1, j), DP(i - 1, j + 1), DP(i - 1, j + 2)]);
DP(i, j + 1) = DP(i, j + 1) + x;
BP(i, j) = j + (l - 2);
end
end
[~, j] = min(DP(r, :));
j = j - 1;
for i = r : -1 : 1
cols(i) = j;
j = BP(i, j);
end
end

Answer 1

Vectorization of the innermost nested loop
You were right in postulating that at least the inner loop is vectorizable. Here's the modified code for the nested loops part -
rows_DP = size(DP,1); %// rows in DP
%// Get first row linear indices for a group of neighboring three columns,
%// which would be incremented as we move between rows with the row iterator
start_ind1 = bsxfun(#plus,[1:rows_DP:2*rows_DP+1]',[0:c-1]*rows_DP); %//'
for i = 2 : r
ind1 = start_ind1 + i-2; %// setup linear indices for the row of this iteration
[x,l] = min(DP(ind1),[],1); %// get x and l values in one go
DP(i,2:c+1) = DP(i,2:c+1) + x; %// set DP values of a row in one go
BP(i,1:c) = [1:c] + l-2; %// set BP values of a row in one go
end
Benchmarking
Benchmarking Code -
N = 3000; %// Datasize
energy = rand(N);
[r, c, ~] = size(energy);
disp('------------------------------------- With Original Code')
DP = padarray(energy, [0, 1], Inf);
BP = zeros(r, c);
tic
for i = 2 : r
for j = 1 : c
[x, l] = min([DP(i - 1, j), DP(i - 1, j + 1), DP(i - 1, j + 2)]);
DP(i, j + 1) = DP(i, j + 1) + x;
BP(i, j) = j + (l - 2);
end
end
toc,clear DP BP x l
disp('------------------------------------- With Vectorized Code')
DP = padarray(energy, [0, 1], Inf);
BP = zeros(r, c);
tic
rows_DP = size(DP,1); %// rows in DP
start_ind1 = bsxfun(#plus,[1:rows_DP:2*rows_DP+1]',[0:c-1]*rows_DP); %//'
for i = 2 : r
ind1 = start_ind1 + i-2; %// setup linear indices for the row of this iteration
[x,l] = min(DP(ind1),[],1); %// get x and l values in one go
DP(i,2:c+1) = DP(i,2:c+1) + x; %// set DP values of a row in one go
BP(i,1:c) = [1:c] + l-2; %// set BP values of a row in one go
end
toc
Results -
------------------------------------- With Original Code
Elapsed time is 44.200746 seconds.
------------------------------------- With Vectorized Code
Elapsed time is 1.694288 seconds.
Thus, you might enjoy a good 26x speedup improvement in performance with that little vectorization tweak.
More tweaks
Few more optimization tweaks could be tried into your code for performance -
cols = zeros(r) could be replaced with col(r,r) = 0.
DP = padarray(energy, [0, 1], Inf) could be replaced with
DP(1:size(energy,1),1:size(energy,2)+2)=Inf;
DP(:,2:end-1) = energy;
BP = zeros(r, c) could be replaced with BP(r, c) = 0.
The pre-allocation tweaks used here are inspired by this blog post.