# Intersect() with with repetition

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Yi-xiao Liu on 17 Aug 2022
Edited: Yi-xiao Liu on 14 Sep 2022
The syntax [C,ia,ib] = intersect(A,B,'rows') returns elements without repetitions. However, I need every potential combination of ia and ib that gives C. For example:
A=[1,1;1,1;1,2];
B=[0,1;1,1;1,1];
I need the output to be:
C=[1,1;1,1;1,1;1,1];
ia=[1;1;2;2];
ib=[2;3;2;3];
The answer here generates indcies into A and B that are intersecting elements: https://www.mathworks.com/matlabcentral/answers/338649-arrays-intersection-with-repetition However there are no correspondace between ia and ib generated by this method. e.g., A(ia,:)~=B(ib,:). it also doesn't give every potential combination of indices.
Any ideas?
Thanks
##### 2 CommentsShowHide 1 older comment
Yi-xiao Liu on 18 Aug 2022
Fixed the typo in ib. Good catch!

Jan on 9 Sep 2022
If you really need the redundant information in iAB_C, iAB_Cprime, idx1, idx2, this is faster than the original version tested with the data:
A = randi([0, 200], 1e6, 2);
B = randi([0, 200], 1e6, 2);
I get 1.12 s instead of 1.86 s.
function [C, iAB_C, iAB_Cprime, idx1, idx2] = repintersect_1b(A2, B2)
% Join data for faster sorting:
A = A2 * [4294967296; 1];
B = B2 * [4294967296; 1];
[uA, ~, iAx] = unique(A, "stable");
[a, idx] = sort(iAx);
aV = 1:numel(A);
aV = aV(idx).';
aI = RunLen(a);
[uB,~,iBx] = unique(B, "stable");
[b, idx] = sort(iBx);
bV = 1:numel(B);
bV = bV(idx).';
bL = cumsum([1, RunLen(b)]);
[C2, iuA, iuB] = intersect(uA, uB, "stable");
% Split data for the output:
C = [floor(C2 / 4294967296), rem(C2, 4294967296)];
iAB_C = cell(numel(iuA), 1);
a0 = 1;
for ii = 1:numel(iuA)
a1 = a0 + aI(ii); % Easier indexing for A
aa = aV(a0:a1 - 1);
a0 = a1;
na = numel(aa);
b0 = bL(iuB(ii)); % Need accumulated RunLength for B
b1 = bL(iuB(ii) + 1) - 1;
bb = bV(b0:b1);
nb = numel(bb);
qa = repmat(aa.', nb, 1); % Replace MESHGRID
qb = repmat(bb, 1, na);
iAB_C{ii} = [qa(:), qb(:)];
end
iAB_Cprime = cell2mat(iAB_C);
idx2 = cumsum(cellfun('size', iAB_C, 1)); % Faster than cellfun(@(x) size(x,1), C)
idx1 = [1; idx2(1:end-1) + 1];
end
% Helper function:
function n = RunLen(x)
d = [true; diff(x(:)) ~= 0]; % TRUE if values change
n = diff(find([d.', true])); % Indices of changes
end
I've omitted the temporarily used cell arrays.
Further time could be saved, if you do not create iAB_Cprime and the set of iAB_C and the indices, because both contain the same information.
Yi-xiao Liu on 14 Sep 2022
Edited: Yi-xiao Liu on 14 Sep 2022
That (the distribution of na/nb) was the observation when I was trying your function on the data.
However, now come to think about it, the majority of na/nb being small means that the size of repetition is consistent for most intersections. So the function can (I can) be optimized to use vectorized code to cover those na==nb==1, and then loop over only the case when na>1|nb>1. (or depending on the data, also vectorize (na==1&nb==2)|(na==2&nb==1), etc.)
I know the output is redundant, and I only need one (set) of them. I was just saying eihter is acceptable.
In short, thanks for your help. I now know how to tailor it for my data.

the cyclist on 18 Aug 2022
I frankly have not quite figured out the logic of what you want as output, but I'm pretty sure that you can construct what you want by using the ismember command instead of intersect. You'll probably need both "directions", and possibly all outputs:
A=[1,1;1,1;1,2];
B=[0,1;1,1;1,1];
[tf_ab, loc_ab] = ismember(A,B,"rows")
tf_ab = 3×1 logical array
1 1 0
loc_ab = 3×1
2 2 0
[tf_ba, loc_ba] = ismember(B,A,"rows")
tf_ba = 3×1 logical array
0 1 1
loc_ba = 3×1
0 1 1
##### 2 CommentsShowHide 1 older comment
the cyclist on 18 Aug 2022
Ah, I get it now. Also, in my mind I was thinking that ismember had that third output like unique does, that gives the mapping back to all elements of the original vector.

Yi-xiao Liu on 18 Aug 2022
Here is my current solution.
A=[1,1;1,1;1,2];
B=[0,1;1,1;1,1];
[uA,~,iA]=unique(A,"rows","stable");
iA=sortrows([iA,(1:size(A,1))'],1);
iA=mat2cell(iA(:,2),accumarray(iA(:,1),1));
[uB,~,iB]=unique(B,"rows","stable");
iB=sortrows([iB,(1:size(B,1))'],1);
iB=mat2cell(iB(:,2),accumarray(iB(:,1),1));
[C,iuA,iuB]=intersect(uA,uB,"rows","stable");
iA_C=cell(size(iuA));iB_C=cell(size(iuA));iAB_C=cell(size(iuA));
for ii=1:numel(iAB_C)
iA_C{ii}=iA{iuA(ii)};
iB_C{ii}=iB{iuB(ii)};
[iAB_C_iiA,iAB_C_iiB]=meshgrid(iA_C{ii},iB_C{ii});
iAB_C{ii}=[iAB_C_iiA(:),iAB_C_iiB(:)];
end
disp(iAB_C)
{4×2 double}
disp(iAB_C{1})
1 2 1 3 2 2 2 3
s
Yi-xiao Liu on 9 Sep 2022
Sorry. I was posting late last night just before sleep.
For typical inputs, (elements of) A and B are integers in range of 1 to 2^24. They are expected to have ~10^10 rows each, in the same scale of output C. If it helps, A usually is a subset of B (not always).
And about our enivorment, the typical node we can request has 48 CPU cores and 177GB memory. We can request large memory (~1TB) nodes if necessary.

Jan on 18 Aug 2022
Edited: Jan on 18 Aug 2022
A simple loop approach:
A = [1,1; ...
1,1; ...
1,2];
B = [0,1; ...
1,1; ...
1,1];
[C, iA, iB] = RepIntersectRows(A, B)
C = 4×2
1 1 1 1 1 1 1 1
iA = 4×1
1 1 2 2
iB = 4×1
2 3 2 3
function [C, iA, iB] = RepIntersectRows(A, B)
% Sizes of inputs:
nA = size(A, 1);
nB = size(B, 1);
w = size(A, 2);
% Pre-allocate output:
C = zeros(nA * nB, w);
iA = zeros(nA * nB, 1);
iB = zeros(nA * nB, 1);
% Find matchs:
iC = 0;
for kA = 1:nA
a = A(kA, :);
for kB = 1:nB
if isequal(a, B(kB, :))
iC = iC + 1;
C(iC, :) = a;
iA(iC) = kA;
iB(iC) = kB;
end
end
end
% Crop unused elements:
C = C(1:iC, :);
iA = iA(1:iC);
iB = iB(1:iC);
end
If A and B have 1e4 rows, the runtime is 4 seconds already. But it is thought to clarify,what you exactly need. The output of your approach does not match the original question exactly. Before I optimize my code, I want to know, if it matchs your needs at all.
Jan on 18 Aug 2022
@Yi-xiao Liu: Matlab's unique and intersect are based on binary searchs, while the loops (e.g. in my prove of concept code) perform a linear search.
Binary search means, that the data are sorted at first, such that you do not have to compare all elements, but log2 of the elements only by dividing the inteval of interest by 2 in each step.
I'm try to find a shorter and faster solution, when I'm at home.

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