Optimizing three nested loops with multiple calculation in MATLAB

Question

For the following code I want to optimize it using the pattern introduced in this solution. However, the problem is how to deal with referring to three nested loops in a single statement. Moreover, the condition is far different from that post.

hint: W and S are NxN sparse double matrices.

    for i=1:N
    for j=1:N
        for k=1:N
            if W(j,k)~=0       
                temp(k)=S(i,j)-S(i,k); 
            end
        end
              sum_temp=max(temp)+sum_temp;
              temp=0;
    end
    B(i,i)=sum_temp;
    sum_temp=0;
end

The inner two loops are very similar to the code piece I address in the last section of this answer on Code Review. You should be able to follow that recipe to turn those two loops into a single statement. — Cris Luengo
– Cris Luengo, Commented Feb 20, 2020 at 15:57
I obviously have no clue what your code does, but I am surprised that temp is never reset to 0. Before we start trying to optimize this, could you confirm that this is correct? — Daniel
– Daniel, Commented Feb 20, 2020 at 19:11

Daniel · Accepted Answer · 2020-02-22 10:13:53Z

In this situation I would opt against fully vectorizing your solution. Calculating S(i,j)-S(i,k) for each combination would mean an intermediate result of size [N,N,N]. Instead I went through your code and eliminated as much iteration as possible without increasing the memory consumption. Step by step so you can understand how I ended up there.

N=30;
S=rand(N,N);
W=rand(N,N)<.1;
sum_temp=0;
temp=0;
%Your original code for reference
for i=1:N
    for j=1:N
        for k=1:N
            if W(j,k)~=0
                temp(k)=S(i,j)-S(i,k);
            end
        end
        sum_temp=max(temp)+sum_temp;
        temp=0;
    end
    B(i,i)=sum_temp;n
    sum_temp=0;
end
B_orig=B;
%1) you only want the max, no need to make temp a vector
for i=1:N
    sum_temp=0;
    for j=1:N
        temp=0;
        for k=1:N
            if W(j,k)~=0
                temp=max(temp,S(i,j)-S(i,k));
            end
        end
        sum_temp=temp+sum_temp;
    end
    B(i,i)=sum_temp;
end
assert(all(all(B==B_orig)))
%2) eliminate the outer loop
sum_temp=zeros(N,1);
for j=1:N
    temp=zeros(N,1);
    for k=1:N
        if W(j,k)~=0
            temp=max(temp,S(:,j)-S(:,k));
        end
    end
    sum_temp=temp+sum_temp;
end
B=diag(sum_temp);
assert(all(all(B==B_orig)))

%3) combine the inner loop with the condition
sum_temp=zeros(N,1);
for j=1:N
    temp=zeros(N,1);
    for k=find(W(j,:))
        temp=max(temp,S(:,j)-S(:,k));
    end
    sum_temp=temp+sum_temp;
end
B=diag(sum_temp);
assert(all(all(B==B_orig)))

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