Merge branch 'master' into kadanes

DIRECTORY.md

@@ -268,6 +268,7 @@
   * [Bayes Theorem](https://github.com/TheAlgorithms/C-Plus-Plus/blob/master/probability/bayes_theorem.cpp)
   * [Binomial Dist](https://github.com/TheAlgorithms/C-Plus-Plus/blob/master/probability/binomial_dist.cpp)
   * [Poisson Dist](https://github.com/TheAlgorithms/C-Plus-Plus/blob/master/probability/poisson_dist.cpp)
+  * [Windowed Median](https://github.com/TheAlgorithms/C-Plus-Plus/blob/master/probability/windowed_median.cpp)
 
 ## Range Queries
   * [Fenwick Tree](https://github.com/TheAlgorithms/C-Plus-Plus/blob/master/range_queries/fenwick_tree.cpp)

probability/windowed_median.cpp

@@ -0,0 +1,226 @@
+/**
+ * @file
+ * @brief An implementation of a median calculation of a sliding window along a
+ * data stream
+ *
+ * @details
+ * 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](https://github.com/YanivHollander)
+ */
+#include <cassert>  /// for assert
+#include <cstdlib>  /// for std::rand - needed in testing
+#include <ctime>    /// for std::time - needed in testing
+#include <list>     /// for std::list - used to manage sliding window
+#include <set>      /// for std::multiset - used to manage multi-value sorted sliding window values
+#include <vector>   /// for std::vector - needed in testing
+
+/**
+ * @namespace probability
+ * @brief Probability algorithms
+ */
+namespace probability {
+/**
+ * @namespace windowed_median
+ * @brief Functions for the Windowed Median algorithm implementation
+ */
+namespace windowed_median {
+using Window = std::list<int>;
+using size_type = Window::size_type;
+
+/**
+ * @class WindowedMedian
+ * @brief A class to calculate the median of a leading sliding window at the
+ * back of a stream of integer values.
+ */
+class WindowedMedian {
+    const size_type _windowSize;  ///< sliding window size
+    Window _window;               ///< a sliding window of values along the stream
+    std::multiset<int> _sortedValues;  ///< a DS to represent a balanced
+                                       /// multi-value binary search tree (BST)
+    std::multiset<int>::const_iterator
+        _itMedian;  ///< an iterator that points to the root of the multi-value
+                    /// BST
+
+    /**
+     * @brief Inserts a value to a sorted multi-value BST
+     * @param value Value to insert
+     */
+    void insertToSorted(int value) {
+        _sortedValues.insert(value);  /// Insert value to BST - O(logN)
+        const auto sz = _sortedValues.size();
+        if (sz == 1) {  /// For the first value, set median iterator to BST root
+            _itMedian = _sortedValues.begin();
+            return;
+        }
+
+        /// If new value goes to left tree branch, and number of elements is
+        /// even, the new median in the balanced tree is the left child of the
+        /// median before the insertion
+        if (value < *_itMedian && sz % 2 == 0) {
+            --_itMedian;  // O(1) - traversing one step to the left child
+        }
+
+        /// However, if the new value goes to the right branch, the previous
+        /// median's right child is the new median in the balanced tree
+        else if (value >= *_itMedian && sz % 2 != 0) {
+            ++_itMedian;  /// O(1) - traversing one step to the right child
+        }
+    }
+
+    /**
+     * @brief Erases a value from a sorted multi-value BST
+     * @param value Value to insert
+     */
+    void eraseFromSorted(int value) {
+        const auto sz = _sortedValues.size();
+
+        /// If the erased value is on the left branch or the median itself and
+        /// the number of elements is even, the new median will be the right
+        /// child of the current one
+        if (value <= *_itMedian && sz % 2 == 0) {
+            ++_itMedian;  /// O(1) - traversing one step to the right child
+        }
+
+        /// However, if the erased value is on the right branch or the median
+        /// itself, and the number of elements is odd, the new median will be the
+        /// left child of the current one
+        else if (value >= *_itMedian && sz % 2 != 0) {
+            --_itMedian;  // O(1) - traversing one step to the left child
+        }
+
+        /// Find the (first) position of the value we want to erase, and erase it
+        const auto it = _sortedValues.find(value);  // O(logN)
+        _sortedValues.erase(it);                    // O(logN)
+    }
+
+ public:
+    /**
+     * @brief Constructs a WindowedMedian object
+     * @param windowSize Sliding window size
+     */
+    explicit WindowedMedian(size_type windowSize) : _windowSize(windowSize){};
+
+    /**
+     * @brief Insert a new value to the stream
+     * @param value New value to insert
+     */
+    void insert(int value) {
+        
+        /// Push new value to the back of the sliding window - O(1)
+        _window.push_back(value);
+        insertToSorted(value);  // Insert value to the multi-value BST - O(logN)
+        if (_window.size() > _windowSize) {  /// If exceeding size of window, pop
+                                             /// from its left side
+            eraseFromSorted(_window.front());  /// Erase from the multi-value BST
+                                               /// the window left side value
+            _window
+                .pop_front();  /// Pop the left side value from the window - O(1)
+        }
+    }
+
+    /**
+     * @brief Gets the median of the values in the sliding window
+     * @return Median of sliding window. For even window size return the average
+     * between the two values in the middle
+     */
+    float getMedian() const {
+        if (_sortedValues.size() % 2 != 0) {
+            return *_itMedian;  // O(1)
+        }
+        return 0.5f * *_itMedian + 0.5f * *next(_itMedian);  /// O(1)
+    }
+
+    /**
+     * @brief A naive and inefficient method to obtain the median of the sliding
+     * window. Used for testing!
+     * @return Median of sliding window. For even window size return the average
+     * between the two values in the middle
+     */
+    float getMedianNaive() const {
+        auto window = _window;
+        window.sort();  /// Sort window - O(NlogN)
+        auto median =
+            *next(window.begin(),
+                  window.size() / 2);  /// Find value in the middle - O(N)
+        if (window.size() % 2 != 0) {
+            return median;
+        }
+        return 0.5f * median +
+               0.5f * *next(window.begin(), window.size() / 2 - 1);  /// O(N)
+    }
+};
+}  /// namespace windowed_median
+}  /// namespace probability
+
+/**
+ * @brief Self-test implementations
+ * @param vals Stream of values
+ * @param windowSize Size of sliding window
+ */
+static void test(const std::vector<int> &vals, int windowSize) {
+    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());
+    }
+}
+
+/**
+ * @brief Main function
+ * @param argc command line argument count (ignored)
+ * @param argv command line array of arguments (ignored)
+ * @returns 0 on exit
+ */
+int main(int argc, const char *argv[]) {
+    
+    /// 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;
+}