[feature] new implementation of Quick sort (#958)

* added quick_sort_3

* Corrected some formatting error

* made more C-plus-plusy

* add test cases

* updating DIRECTORY.md

* clang-tidy fixes for 30c9a199ad0ad1ee524c85766dcefe70723afc3a

* better selef-tests + use std::vector

* use size_t

* change size_t to int32_t
Co-authored-by: Nmohit <rj7555555@gmail.com>
Co-authored-by: Ngithub-actions <${GITHUB_ACTOR}@users.noreply.github.com>

DIRECTORY.md

@@ -235,6 +235,7 @@
   * [Numeric String Sort](https://github.com/TheAlgorithms/C-Plus-Plus/blob/master/sorting/numeric_string_sort.cpp)
   * [Odd Even Sort](https://github.com/TheAlgorithms/C-Plus-Plus/blob/master/sorting/odd_even_sort.cpp)
   * [Quick Sort](https://github.com/TheAlgorithms/C-Plus-Plus/blob/master/sorting/quick_sort.cpp)
+  * [Quick Sort 3](https://github.com/TheAlgorithms/C-Plus-Plus/blob/master/sorting/quick_sort_3.cpp)
   * [Radix Sort](https://github.com/TheAlgorithms/C-Plus-Plus/blob/master/sorting/radix_sort.cpp)
   * [Selection Sort](https://github.com/TheAlgorithms/C-Plus-Plus/blob/master/sorting/selection_sort.cpp)
   * [Shell Sort](https://github.com/TheAlgorithms/C-Plus-Plus/blob/master/sorting/shell_sort.cpp)

sorting/quick_sort_3.cpp

+/**
+ * @file
+ * @brief Implementation Details
+ * @details Quick sort 3 works on Dutch National Flag Algorithm
+ * The major difference between simple quicksort and quick sort 3 comes in the
+ * function partition3 In quick_sort_partition3 we divide the vector/array into
+ * 3 parts. quick sort 3 works faster in some cases as compared to simple
+ * quicksort.
+ * @author immortal-j
+ * @author [Krishna Vedala](https://github/kvedala)
+ */
+#include <algorithm>
+#include <cassert>
+#include <ctime>
+#include <iostream>
+#include <vector>
+
+namespace {
+/**
+ * Operator to print the array.
+ * @param out std::ostream object to write to
+ * @param arr array to write
+ */
+template <typename T>
+std::ostream &operator<<(std::ostream &out, const std::vector<T> &arr) {
+    for (size_t i = 0; i < arr.size(); ++i) {
+        out << arr[i];
+        if (i < arr.size() - 1) {
+            out << ", ";
+        }
+    }
+    return out;
+}
+
+}  // namespace
+
+/**
+ * @namespace sorting
+ * @brief Sorting Algorithms
+ */
+namespace sorting {
+namespace {  // using un-named namespace here to prevent partition function
+             // being visible to end-users
+/** This function partitions `arr[]` in three parts
+ * 1. \f$arr[l\ldots i]\f$ contains all elements smaller than pivot
+ * 2. \f$arr[(i+1)\ldots (j-1)]\f$ contains all occurrences of pivot
+ * 3. \f$arr[j\ldots r]\f$ contains all elements greater than pivot
+ * @tparam T type of data in the vector array
+ * @param [in,out] arr vector array being partitioned
+ * @param [in] low lower limit of window to partition
+ * @param [in] high upper limit of window to partition
+ * @param [out] i updated lower limit of partition
+ * @param [out] j updated upper limit of partition
+ */
+template <typename T>
+void partition3(std::vector<T> *arr, int32_t low, int32_t high, int32_t *i,
+                int32_t *j) {
+    // To handle 2 elements
+    if (high - low <= 1) {
+        if ((*arr)[high] < (*arr)[low]) {
+            std::swap((*arr)[high], (*arr)[low]);
+        }
+        *i = low;
+        *j = high;
+        return;
+    }
+
+    int32_t mid = low;
+    T pivot = (*arr)[high];
+    while (mid <= high) {
+        if ((*arr)[mid] < pivot) {
+            std::swap((*arr)[low++], (*arr)[mid++]);
+        } else if ((*arr)[mid] == pivot) {
+            mid++;
+        } else if ((*arr)[mid] > pivot) {
+            std::swap((*arr)[mid], (*arr)[high--]);
+        }
+    }
+
+    // update i and j
+    *i = low - 1;
+    *j = mid;  // or high-1
+}
+}  // namespace
+
+/** 3-way partition based quick sort. This function accepts array pointer and
+ * modified the input array.
+ * @tparam T type of data in the vector array
+ * @param [in,out] arr vector array to sort
+ * @param [in] low lower limit of window to partition
+ * @param [in] high upper limit of window to partition
+ */
+template <typename T>
+void quicksort(std::vector<T> *arr, int32_t low, int32_t high) {
+    if (low >= high) {  // 1 or 0 elements
+        return;
+    }
+
+    int32_t i = 0, j = 0;
+
+    // i and j are passed as reference
+    partition3(arr, low, high, &i, &j);
+
+    // Recur two halves
+    quicksort(arr, low, i);
+    quicksort(arr, j, high);
+}
+
+/** 3-way partition based quick sort. This function accepts array by value and
+ * creates a copy of it. The array copy gets sorted and returned by the
+ * function.
+ * @tparam T type of data in the vector array
+ * @param [in] arr vector array to sort
+ * @param [in] low lower limit of window to partition
+ * @param [in] high upper limit of window to partition
+ * @returns sorted array vector
+ */
+template <typename T>
+std::vector<T> quicksort(std::vector<T> arr, int32_t low, int32_t high) {
+    if (low >= high) {  // 1 or 0 elements
+        return arr;
+    }
+
+    int32_t i = 0, j = 0;
+
+    // i and j are passed as reference
+    partition3(&arr, low, high, &i, &j);
+
+    // Recur two halves
+    quicksort(&arr, low, i);
+    quicksort(&arr, j, high);
+
+    return arr;
+}
+}  // namespace sorting
+
+/** Test function for integer type arrays */
+static void test_int() {
+    std::cout << "\nTesting integer type arrays\n";
+
+    for (int num_tests = 1; num_tests < 21; num_tests++) {
+        size_t size = std::rand() % 500;
+        std::vector<int> arr(size);
+        for (auto &a : arr) {
+            a = std::rand() % 500 - 250;  // random numbers between -250, 249
+        }
+
+        std::cout << "Test " << num_tests << "\t Array size:" << size << "\t ";
+        std::vector<int> sorted = sorting::quicksort(arr, 0, size - 1);
+        if (size < 20) {
+            std::cout << "\t Sorted Array is:\n\t";
+            std::cout << sorted << "\n";
+        }
+        assert(std::is_sorted(std::begin(sorted), std::end(sorted)));
+        std::cout << "\t Passed\n";
+    }
+}
+
+/** Test function for double type arrays */
+static void test_double() {
+    std::cout << "\nTesting Double type arrays\n";
+    for (int num_tests = 1; num_tests < 21; num_tests++) {
+        size_t size = std::rand() % 500;
+        std::vector<double> arr(size);
+        for (auto &a : arr) {
+            a = double(std::rand() % 500) -
+                250.f;   // random numbers between -250, 249
+            a /= 100.f;  // convert to -2.5 to 2.49
+        }
+
+        std::cout << "Test " << num_tests << "\t Array size:" << size << "\t ";
+        std::vector<double> sorted = sorting::quicksort(arr, 0, size - 1);
+        if (size < 20) {
+            std::cout << "\t Sorted Array is:\n\t";
+            std::cout << sorted << "\n";
+        }
+        assert(std::is_sorted(std::begin(sorted), std::end(sorted)));
+        std::cout << "\t Passed\n";
+    }
+}
+
+/** Driver program for above functions */
+int main() {
+    std::srand(std::time(nullptr));
+    test_int();
+    test_double();
+    return 0;
+}