How to compute distance efficiently with no loops

I have matrices for points (not the same number of rows for P1 and P2) like this

P1=[12 45
    34 7 ]

and

P2= [1 42 25
     2 37 57 
     3 55 16]

What I what is to get the Manhattan (cityblock) distance between points in P1 (X and Y coordinates are in column 1 and 2) and those in P2 (X and Y are in column 2 and 3, the first one are indices of the points.

What I have been doing so far and works but very slow is with a nested for loop. But recently I saw pdist2 and I think it can help me avoid the loop.

But in the example, I need to put the whole matrix something like this

pdist2(P1,P2, 'cityblock');

But in my case, I want to use the columns of my matrices col as follows

abs(xP1(col1) - xP2(col2)) + abs(yP1(col2) - yP2(col3))

How can I do it or is there any faster way to do it (because i saw someone stating that even pdist uses nested for) since my matrices are quite big?

Answer 1

[Guillaume]

Most likely:

pdist2(P1, P2(:, [2 3]), 'cityblock')

[etudiant_is]

Thanks a lot but can you please explain to me why you wrote it like this? I want to understand so I can do it in future by myself.

Also if I want just to have only the distances between only one point from P2 (knowing its index in the first column) and the points in P1 without computing the distances for the rest of points .

[Guillaume]

This is basic matrix indexing (see also this page in the doc). You want to use column 1 and 2 of P1 (which is the whole matrix) with column 2 and 3 of P2, so I index column 2 and 3 of P2 in the call to pdist2.

Similarly, if you just want want one point in P2, instead of using all rows (with :), you just give it index of the row:

pdist2(P1, P2(3, [2 3])) % distance between 3rd point defined by column 2 and 3 of P2

[etudiant_is]

Thank you for your help. Just one last thing, do you know if with pdist it will be faster than with nested loops or will it take similar time?

[Guillaume]

I have no idea if it'll be faster. In theory, it should be. Why don't you test it?