windowed_median.cpp File Reference

An implementation of a median calculation of a sliding window along a data stream. More...

#include <cassert>
#include <cstdlib>
#include <ctime>
#include <list>
#include <set>
#include <vector>

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Classes
class	probability::windowed_median::WindowedMedian
	A class to calculate the median of a leading sliding window at the back of a stream of integer values. More...

Namespaces
namespace	probability
	for std::vector - needed in testing
namespace	windowed_median
	Functions for the Windowed Median algorithm implementation.

Typedefs
using	probability::windowed_median::Window = std::list< int >
using	probability::windowed_median::size_type = Window::size_type

Functions
static void	test (const std::vector< int > &vals, int windowSize)
	namespace probability More...
int	main (int argc, const char *argv[])
	Main function. More...

Detailed Description

An implementation of a median calculation of a sliding window along a data stream.

Given a stream of integers, the algorithm calculates the median of a fixed size window at the back of the stream. The leading time complexity of this algorithm is O(log(N), and it is inspired by the known algorithm to [find median from (infinite) data stream](https://www.tutorialcup.com/interview/algorithm/find-median-from-data-stream.htm), with the proper modifications to account for the finite window size for which the median is requested

Algorithm

The sliding window is managed by a list, which guarantees O(1) for both pushing and popping. Each new value is pushed to the window back, while a value from the front of the window is popped. In addition, the algorithm manages a multi-value binary search tree (BST), implemented by std::multiset. For each new value that is inserted into the window, it is also inserted to the BST. When a value is popped from the window, it is also erased from the BST. Both insertion and erasion to/from the BST are O(logN) in time, with N the size of the window. Finally, the algorithm keeps a pointer to the root of the BST, and updates its position whenever values are inserted or erased to/from BST. The root of the tree is the median! Hence, median retrieval is always O(1)

Time complexity: O(logN). Space complexity: O(N). N - size of window

Author

Yaniv Hollander

Function Documentation

main()

int main	(	int	argc,
		const char *	argv[]
	)

Main function.

Parameters

argc	command line argument count (ignored)
argv	command line array of arguments (ignored)

Returns

0 on exit

A few fixed test cases

Array of sorted values; odd window size

Array of sorted values - decreasing; odd window size

Even window size

Array with repeating values

Array with same values except one

Array that includes repeating values including negatives

Array with large values - sum of few pairs exceeds MAX_INT. Window size is even - testing calculation of average median between two middle values

Random test cases

Array size in the range [5, 20]

Window size in the range [3, 10]

Random array values (positive/negative)

Testing randomized test

                                       {
    
    /// A few fixed test cases
    test({1, 2, 3, 4, 5, 6, 7, 8, 9}, 3);   /// Array of sorted values; odd window size
    test({9, 8, 7, 6, 5, 4, 3, 2, 1}, 3);   /// Array of sorted values - decreasing; odd window size
    test({9, 8, 7, 6, 5, 4, 5, 6}, 4);      /// Even window size
    test({3, 3, 3, 3, 3, 3, 3, 3, 3}, 3);   /// Array with repeating values
    test({3, 3, 3, 3, 7, 3, 3, 3, 3}, 3);   /// Array with same values except one
    test({4, 3, 3, -5, -5, 1, 3, 4, 5}, 5); /// Array that includes repeating values including negatives
    
    /// Array with large values - sum of few pairs exceeds MAX_INT. Window size is even - testing calculation of
    /// average median between two middle values
    test({470211272, 101027544, 1457850878, 1458777923, 2007237709, 823564440,
          1115438165, 1784484492, 74243042, 114807987}, 6);
    
    /// Random test cases
    std::srand(static_cast<unsigned int>(std::time(nullptr)));
    std::vector<int> vals;
    for (int i = 8; i < 100; i++) {
        const auto n = 1 + std::rand() / ((RAND_MAX + 5u) / 20);    /// Array size in the range [5, 20]
        auto windowSize = 1 + std::rand() / ((RAND_MAX + 3u) / 10); /// Window size in the range [3, 10]
        vals.clear();
        vals.reserve(n);
        for (int i = 0; i < n; i++) {
            vals.push_back(rand() - RAND_MAX);  /// Random array values (positive/negative)
        }
        test(vals, windowSize); /// Testing randomized test
    }
    return 0;
}

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test()

static void test ( const std::vector< int > &  vals, int  windowSize  )

static

namespace probability

Self-test implementations

Parameters

vals	Stream of values
windowSize	Size of sliding window

Comparing medians: efficient function vs. Naive one

                                                             {
    probability::windowed_median::WindowedMedian windowedMedian(windowSize);
    for (const auto val : vals) {
        windowedMedian.insert(val);
 
        /// Comparing medians: efficient function vs. Naive one
        assert(windowedMedian.getMedian() == windowedMedian.getMedianNaive());
    }
}

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