find indices of row subsets

Question

Michal Kvasnicka on 9 May 2018

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Edited: Michal Kvasnicka on 11 May 2018

I am trying to find vectorized matlab function ind = item2ind(item,t) to solve following problem: I have a list of row vectors

item = [2 3 1; 2 1 2; 3 1 1; 1 3 3]

and vector of all possible item elements at each item row vector

t = [1 1 2 2 2 3 3];

I need to find indexes of of separate item rows elements corresponding to the vector t in this way:

ind = [3 6 1; 3 1 4; 6 1 2; 1 6 7]

But

item = [1 1 1]

does not correspond to the vector t, because there are 3 "1" elements, and t contains only 2 "1" elements.

Note: Serial version is inefficient for large item (10000 x 100) and t (1 x 200).

function ind = item2ind(item,t)
[nlp,N] = size(item);
ind = zeros(nlp,N);
for i = 1:nlp
    auxitem = item(i,:);
    auxt = t;
    for j = 1:N
        I = find(auxitem(j) == auxt,1,'first');
        if ~isempty(I)
            auxt(I) = 0;
            ind(i,j) = I;
        else
            error('Incompatible content of item and t.');
        end
    end
end
end

Additional remarks:

Most of the time is spent on the line:

I = find(auxitem(j) == auxt,1,'first');

Is there any clever trick how to speed up this line of code? I tried this, for example, but without any speedup:

I = ipos(auxitem(j) == auxt); I = I(1);

where ipos is preallocated as:

ipos = 1:length(t);

Thanks in advance for any help ...

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Michal Kvasnicka on 9 May 2018

My concern is about "item" number of rows, which could be very large (up to 1e6).

Jan on 9 May 2018

@Michal: If you provide a relevant set of example data, we could test the speed and the correctness of the suggested code.

What is the maximum value of t?

Do you have a C compiler installed?

Michal Kvasnicka on 9 May 2018

Please read discussion under Chad's answer.

Typical length of "t" is 30-100, maxmimum value of elements at "t" is equal maximum value of elements at array "item".

Yes I have a C compiler...

Please, keep in mind, that vector "t" is not sorted.

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Answer 1

Michal Kvasnicka on 11 May 2018

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Edited: Michal Kvasnicka on 11 May 2018

So far best solution:

function ind = item2ind_new(item,t)
t = t(:);
[m,n] = size(item);
mct = max(accumarray(t,1));
G = accumarray(t,1:length(t),[],@(x) {sort(x)});
G = cellfun(@(x) padarray(x.',[0 mct-length(x)],0,'post'), G, 'UniformOutput', false);
G = vertcat(G{:});
C = cumsum(reshape(item,m,1,n)==item,3);
ia = C(sub2ind(size(C),repelem((1:m).',1,n),repelem(1:n,m,1),repelem(1:n,m,1)));
ind = G(sub2ind(size(G),item,ia));

Any idea how to improve it?

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Answer 2

Jan on 9 May 2018

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Edited: Jan on 9 May 2018

function ind = item2ind(item, t);
maxRun   = length(t) + 1;
[T , TI] = accumsort(t, maxRun);
ind      = zeros(size(item));
for k = 1:size(item, 1)
   [aItem, aItemI] = accumsort(item(k, :), maxRun);
   % [m, index]    = ismember(aItem, T);
   % Faster with undocumented function:
   [m, index]      = builtin('_ismemberhelper', aItem, T);
   if all(m)
     ind(k, aItemI) = TI(index);
   else
      error('Incompatible item.');
   end
end
end
function [T, TI] = accumsort(t, maxRun)
[sortedT, TI] = sort(t);
T = sortedT * maxRun;
c = -1;
for k = 1:numel(T)
   if T(k) ~= c
      d = 0;
      c = T(k);
   else
      d = d + 1;
   end
   T(k) = T(k) + d;
end
end

For some test data of size [10'000 x 100] I get a runtime of 0.21 sec instead of 1.3 sec of the original version.

With calling ismember the runtime is 0.41 sec. Internally ismember calls the helper function builtin('_ismemberhelper') for sorted data of type double. If it is known already, that the input is sorted, calling the internal function avoids the overhead.

If you have a C compiler, converting accumsort to a C-mex would be useful.

maxRun must a any number greater than the highest number of repetitions in t. length(t)+1 is guaranteed to be larger.

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Michal Kvasnicka on 9 May 2018

I tried to convert accumsort to C-mex (MATLAB Coder), but the mex file shows very poor performance (2x slower than MATLAB version). Actually, I am not familiar with manual C-mex file writing...

But the main problem is trade of between number of unique elements at vector "t" and overall performance. Chad's algorithm is significantly faster than yours in a case of low number unique elements of "t".

This is Chad's final code, for your info:

function ind = item2ind(item,t)
unique_t = unique(t);
ind = zeros(size(item));
% a single 'for' loop as long as the unique elements of t
for jj = 1:length(unique_t)
    O = zeros(size(item));
    O(item == unique_t(jj)) = 1;
    positions_of_t = [0 find(t == unique_t(jj))];
    % adding zero so sub_index call below will always reference a non-zero element   
    if max(sum(O,2)) > numel(positions_of_t)
        error('Incompatible content of item and t.');
    else
        sub_index = cumsum(O,2) .* O + 1;
        ind = ind + positions_of_t(sub_index);
        % this is why we needed the 0 in positions_of_t above
    end
end

Case1:

t = 1:30;
tt = repmat(t,1,1);
[~,p] = sort(rand(100000,30),2);
item = tt(p);
tic;ind1 = item2ind_Chad(item,tt);toc
tic;ind2 = item2ind_Jan(item,tt);toc
Elapsed time is 2.363570 seconds.
Elapsed time is 1.420776 seconds.

Case2:

t = 1:10;
tt = repmat(t,1,3);
[~,p] = sort(rand(100000,30),2);
item = tt(p);
tic;ind1 = item2ind_Chad(item,tt);toc
tic;ind2 = item2ind_Jan(item,tt);toc
Elapsed time is 0.804354 seconds.
Elapsed time is 1.414612 seconds.

Jan on 9 May 2018

As far as I can see, my code can be parallelized also directly:

parfor k = 1:size(item, 1)

I've made a very interesting experiment with a C-Mex function:

// file: accumsortX.c
// mex -O accumsortX.c
#include "mex.h"
void mexFunction(int nlhs, mxArray *plhs[], int nrhs, const mxArray *prhs[]) {
  double *t, *T, maxT, c, d;
  size_t n;
    t    = mxGetPr(prhs[0]);
    maxT = mxGetScalar(prhs[1]);
    n    = mxGetNumberOfElements(prhs[0]);
    plhs[0] = mxCreateUninitNumericMatrix(1, n, mxDOUBLE_CLASS, mxREAL);
    T       = mxGetPr(plhs[0]);
    c = *t - 1;
    while (n-- != 0) {
       if (c != *t) {
          d = 0.0;
          c = *t;
       } else {
          d += 1.0;
       }
       *T++ = maxT * *t++ + d;
    }
    return;
  }

It has exactly the same speed as the Matlab version of accumsort - but I've moved the sorting to the main function:

function ind = item2ind_jan2(item,t)
maxT    = max(t) + 1;
[T, TI] = sort(t);
T       = accumsort(T, maxT);
ind     = zeros(size(item));
for k = 1:size(item, 1)
   [aItem, aItemI] = sort(item(k, :));
   aItem           = accumsort(aItem, maxT);
   [m, index]      = builtin('_ismemberhelper', aItem, T);
   if all(m)
      ind(k, aItemI) = TI(index);
   else
      error('item not compatible to t.');
   end
end
end
function [T] = accumsort(t, maxT)
T = t * maxT;
c = -1;
for k = 1:numel(T)
   if T(k) ~= c
      d = 0;
      c = T(k);
   else
      d = d + 1;
   end
   T(k) = T(k) + d;
end
end

Replacing accumsort() by accumsortX() does not change the run-time, even if I do not create a new vector in the Mex but write to the original array.

But let me repeat: I still have no idea how you measure the timings, because for your given input data:

t = 1:10;
tt = repmat(t,1,3);
[~,p] = sort(rand(100000,30),2);
item = tt(p);

I get the error message for incompatible items. Please clarify this detail, because I'm eager to reproduce your timings in any way.

Michal Kvasnicka on 10 May 2018

Intel® Core™ i7-6800K Processor + X99 Asus motherboard + 64GB RAM + 240GB SSD Intel Pro disk (system) + 2TB HDD (Data) + GTX TITAN (CUDA) GPU

Ubuntu Linux 16.04 (64bit)

Wick on 10 May 2018

Interesting. Jan's code is always faster on my computer than the numbers you gave. My code is faster on my machine for the short runs but for the long runs your machine is faster. It appears your memory subsystem is a bit more efficient.

Michal Kvasnicka on 10 May 2018

@Jan any progress or new ideas on your site?

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Answer 3

Wick on 9 May 2018

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Edited: Wick on 9 May 2018

Here you go. At the sizes you suggested, this shouldn't take too long. It has a single 'for' loop that cycles through the unique values of 't'.

I'm using logical indexing to identify all the elements in 'item' that match the given unique 't' and summing across the row. If the sum exceeds the number of times that value showed up in 't' you get your error. Otherwise I'm using 'cumsum' in a creative fashion (in my ever so humble opinion) to provide the indexes back to the location of the unique value in the original vector 't'.

Good Luck!

    function ind = item2ind(item,t)
    unique_t = unique(t); 
    ind = zeros(size(item));
    try
        % a single 'for' loop as long as the unique elements of t
        for jj = 1:length(unique_t)
            O = zeros(size(item));
            O(item == unique_t(jj)) = 1;
            positions_of_t = [0 find(t == unique_t(jj))];
            % adding zero so sub_index call below will always reference a non-zero element
            sub_index = cumsum(O,2) .* O + 1;
            ind = ind + positions_of_t(sub_index);
            % this is why we needed the 0 in positions_of_t above
        end
    catch
        error('Incompatible content of item and t.');
    end

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Jan on 9 May 2018

For your test data:

t     = 1:50;
tbig  = repmat(t,1,5);
[~,p] = sort(rand(100000,length(tbig)),2);
item  = tbig(p);

Your code from the question stops with the error "Incompatible content of item and t." So how can you measure the timings?!

The same for:

t     = 1:10;
tbig  = repmat(t,1,25);
[~,p] = sort(rand(100000,length(tbig)),2);
item  = tbig(p);

I tried it with:

t    = repmat(1:50, 1, 5);
t    = t(randperm(length(t)));
item = zeros(10000, 100);
for k = 1:10000; item(k, :) = t(randperm(length(t), 100)); end

This let your (and my) item2ind() run successfully.

Do I miss a point?!

Michal Kvasnicka on 9 May 2018

This is strange, my code perfectly works with both data case examples you mentioned above … ??!!

Wick on 9 May 2018

Jan,

My code is faster for small length 't' and much, much slower for large 't'. You vectorized in a completely different way than I did (and used an undocumented function but we won't use that against you). My question is, is there some rule of thumb my snippet of code didn't follow that I should change how I code things? I've always felt I was pretty good at vectorizing my MATLAB code but I've been coming here to learn how to be better. Obviously, you know some tricks I don't.

Thanks.

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