find median in a fixed-size moving window along a long sequence of data

Question

Given a sequence of data (it may have duplicates), a fixed-sized moving window, move the window at each iteration from the start of the data sequence, such that (1) the oldes

Answer 1

I attach my segment tree (see my other post) which allows the frequency distribution of counts to be queried very efficiently.

This implements the following data structure:

|-------------------------------|
|---------------|---------------|
|-------|-------|-------|-------|
|---|---|---|---|---|---|---|---|
  0   1   2   3   4   5   6   7  

Each segment keeps the number of counts items in the range it covers. I use 2N segments for a range of value from 1..N. These are placed in a single rolled out vector rather than the tree format show figuratively above.

So if you are calculating rolling medians over a set of integers which vary from 1..65536, then you only need 128kb to store them, and can insert/delete/query using O(ln N) where N = the size of the range, i.e. 2**16 operations.

This is a big win if the data range is much smaller than your rolling window.

#if !defined(SEGMENT_TREE_H)
#define SEGMENT_TREE_H
#include 
#include 
#include 
#include 

#ifndef NDEBUG
#include 
#endif

template
class t_segment_tree
{
    static const unsigned                       cnt_elements    = (1 << BITS);
    static const unsigned                       cnt_storage     = cnt_elements << 1;
    std::array    counts;
    unsigned                                    count;

#ifndef NDEBUG
    std::multiset                     elements;
#endif
    public:

    //____________________________________________________________________________________

    //  constructor

    //____________________________________________________________________________________
    t_segment_tree(): count(0)
    {
        std::fill_n(counts.begin(), counts.size(),  0);
    }
    //~t_segment_tree();

    //____________________________________________________________________________________

    //  size

    //____________________________________________________________________________________
    unsigned size() const  { return count; }

    //____________________________________________________________________________________

    //  constructor

    //____________________________________________________________________________________
    void insert(unsigned x)
    {
#ifndef NDEBUG
        elements.insert(x);
        assert("...............This element is too large for the number of BITs!!..............." && cnt_elements > x);
#endif
        unsigned ii = x + cnt_elements;
        while (ii)
        {
            ++counts[ii - 1];
            ii >>= 1;
        }
        ++count;
    }

    //____________________________________________________________________________________

    //  erase 

    //      assumes erase is in the set
    //____________________________________________________________________________________
    void erase(unsigned x)
    {
#ifndef NDEBUG
        // if the assertion failed here, it means that x was never "insert"-ed in the first place
        assert("...............This element was not 'insert'-ed before it is being 'erase'-ed!!..............." && elements.count(x));
        elements.erase(elements.find(x));
#endif
        unsigned ii = x + cnt_elements;
        while (ii)
        {
            --counts[ii - 1];
            ii >>= 1;
        }
        --count;
    }

    // 
    //____________________________________________________________________________________

    //  kth element

    //____________________________________________________________________________________
    unsigned operator[](unsigned k)
    {
        assert("...............The kth element: k needs to be smaller than the number of elements!!..............." && k < size());
        unsigned ii = 1;
        while (ii < cnt_storage)
        {
            if (counts[ii - 1] <= k)
               k -= counts[ii++ - 1];
            ii <<= 1;
        }
        return (ii >> 1) - cnt_elements;
    }

};
#endif