Problem 42676. Histogram of histogram

Histogram of histogram (HoH) is a useful measure concerning the distribution of random data, which has diverse applications in data science, statistics, information theory, etc.

In this problem, given an n-by-m array x of integer numbers {1,2,...,S}, return the HoH along every column of x: f = HoH(x). An example for n = 5, m = 4, and S = 6 follows.

Input

                 x = [1  2  2  3
                      2  3  3  6
                      1  3  1  1
                      6  3  2  5
                      2  2  4  2]

Histogram

                 h = [2  0  1  1
                      2  2  2  1
                      0  3  1  1
                      0  0  1  0
                      0  0  0  1
                      1  0  0  1]

where the r-th (r=1,...,S) row of h is the histogram bin counts for number r along every column of x.

HoH

                 f = [1  0  3  5
                      2  1  1  0
                      0  1  0  0]

where f is a max(h(:))-by-m matrix, with the p-th row representing the histogram of number p along every column of h.

Hint : A straightforward reference scheme to obtain f could be:

    h = histc(x,1:max(x(:)),1);         
    f = histc(h,1:max(h(:)),1);

This is simple but inefficient in terms of both performance and memory (It will crash for the last test case). Note that the ultimate goal is to find f (HoH); thus, it is not necessary to go through exactly the same h as described above. Try your best to improve your code in terms of both speed and memory. Your score will be based on the speed of your code.