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Adding values in vector according to labels in another vector

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I have two arrays of same size E x 1, let's call them Values and Labels. Values have real-valued data points, while Labels have integer-valued group labels for each data point, ranging from 1 to N. In my application E could be on the order of 1e6, while N is on the order of 1e5, so there are on average ~10 data points sharing the same group label.
I would like to generate the accumulated sum of values sharing the same group label. That is, I want to generate an N x 1 vector GroupSum where GroupSum(i) = sum(Values(Labels == i)). This will be done for many instances of (Values, Labels) combinations in an outer loop, so it is important to do this quickly. Therefore, I would like to find a faster alternative to
for i = 1:N
GroupSum(i) = sum(Values(Labels == i));
Any suggestions would be appreciated.
Thanks in advance.


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Accepted Answer

Raunak Gupta
Raunak Gupta on 13 Nov 2020
From the current implementation I assume that the labels run from 1 to N without missing any value in between. The best-case scenario I thought of is to at least traverse the Values array once and have an accumulating count for each label. This way you can save the overhead of doing logical indexing for all the Labels. Below code may help.
E = 1e6;
N = 1e5;
Labels = randi([1 N],E,1);
Values = randi([1 100],E,1);
GroupSum = zeros(1,N);
for i = 1:E
GroupSum(Labels(i)) = GroupSum(Labels(i)) + Values(i);
This doesn’t use (Labels == i) which can be time saving.


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Murat Azizoglu
Murat Azizoglu on 13 Nov 2020
Hi Raunak,
Thanks for the clear explanation of why the 4 orders of magnitude improvement is occurring. I'd like to understand it better so that I can improve the execution time for my application: a decoder simulation that involves running similar (but more complex, i.e. more than a simple summation) computation sequentially by about 1e7 times.
The vector Labels is known a priori due to an external design, i.e. it is not random. Therefore, the dimensions of the rectangular matrix P will not change and its row dimension d is guaranteed to be small, in the range 10-15. (By the way, I did not have this idea when I posted the original question, or else I would have posed it differently.)
What I don't understand is why the computation of GroupSum3 takes about 40-45 msec in my machine, while your solution of computing GroupSum2 in the for loop takes 10-20 msec. The P matrix in this experiment is 25 x 1e5, so the line GroupSum3 = sum(Values(P))' should be faster than the for loop over 1e6 values one by one. I suspect the memory access in forming the 25 x 1e5 matrix Values(P) is time-consuming, so that your for loop sequentially adding Values(i) to GroupSum(Label(i)) 1e6 times wins.
There is additional structure in Labels in my problem so that I can break down the sum to a small number of sums via e.g. 3 matrices P1, P2, P3 of different row dimensions, and gain an additional ~3x speedup, but that only barely catches up with your solution, which, as you say, would work for any Labels.
Can the for loop be improved if Labels were know to have such structure? E.g., say a given label can occur only 5, 10 or 15 times, i.e. unique(groupcounts(Labels)) is a very small array of only 3 elements, unlike a case with randomly generated Labels.
Thanks again.
Raunak Gupta
Raunak Gupta on 13 Nov 2020
Hi Murat,
As I suggested the time complexity here is suggested by the how many indexes from Value matrix you need to traverse. Since the sum of the values is required so the whole Value vector is needed to be traversed. So, even if we have a property which can group the number of labels based on there frequencies still in the end the Value needs to be traverse in full.
Thus as long as Sum is the final result, I don't see any further enhancement in terms of time complexity because the solution I suggested is having linear time complexity.
Murat Azizoglu
Murat Azizoglu on 14 Nov 2020
I see your point. No matter how the bins are formed via Labels, at the end there are E addition operations to be performed. Thus the complexity cannot be below O(E) which is what your for loop achieves.
This also explains why the matrix based approach is about 2-4x slower, there are 2.5e6 (=d N) elements in my P matrix many of them 0's. I had thought avoiding the for loops would be a big help in Matlab, but apparently not in this case.
Thank you very much for all the help.

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