Mapping 2 vectors - help to vectorize

Working in Matlab I have 2 vectors of x coordinate with different length. For example:

xm = [15 20 24 25 26 35 81 84 93];
xn = [14 22 26 51 55 59 70 75 89 96];

I need to map xm to xn, or in other words to find which coordinates in xn are closest to xm. So if I have values associated with those coordinates, I can use this map as index and correlate those values.

Both vectors are sorted and there are no duplicates in each vector.

I wrote a simple function with for-loop:

function xmap = vectors_map(xm,xn)
xmap = zeros(size(xm));
for k=1:numel(xm)
    [~, ind] = min(abs(xm(k)-xn));
    xmap(k) = ind(1);
end

For the above example is returns

xmap =
    1     2     2     3     3     3     8     9    10

It works ok, but takes a while with long vectors (over 100,000 points).

Any ideas how to vectorize this code?

Oh! One other option: since you're looking for close correspondences between two sorted lists, you could go through them both simultaneously, using a merge-like algorithm. This should be O(max(length(xm), length(xn)))-ish.

match_for_xn = zeros(length(xn), 1);
last_M = 1;
for N = 1:length(xn)
  % search through M until we find a match.
  for M = last_M:length(xm)
    dist_to_curr = abs(xm(M) - xn(N));
    dist_to_next = abs(xm(M+1) - xn(N));

    if dist_to_next > dist_to_curr
      match_for_xn(N) = M;
      last_M = M;
      break
    else
      continue
    end

  end % M
end % N

EDIT: See @yuk's comment, the above code is not totally correct!

Consider this vectorized solution:

[~, xmap] = min( abs(bsxfun(@minus, xm, xn')) )

The fastest implementation I'm aware of that solves this problem is this one (C code that can be compiled as a .mex file; for me it's about 20 times faster than rescdsk's code in the accepted answer). It's surprising that such a common operation is not a MATLAB built-in function.

It looks like your input vectors are sorted. Use a binary search to find the closest match. This will give you a O(n ln n) run time.