feat: Exponential Search (#818)

* feat(ExponentialSearch): added C implementation of Exponential Search

* fix: typo

* refactor(ExponentialSearch): removed unused imports

* fix(Exponential Search): review fixes

* updating DIRECTORY.md

* refactor(ExponentialSearch): refactoring types

* fix(ExponentialSearch): fixes and added brief

* refactor(ExponentialSearch): added briefs

Co-authored-by: github-actions <${GITHUB_ACTOR}@users.noreply.github.com>

DIRECTORY.md

@@ -350,6 +350,7 @@
 
 ## Searching
   * [Binary Search](https://github.com/TheAlgorithms/C/blob/master/searching/binary_search.c)
+  * [Exponential Search](https://github.com/TheAlgorithms/C/blob/master/searching/exponential_search.c)
   * [Fibonacci Search](https://github.com/TheAlgorithms/C/blob/master/searching/fibonacci_search.c)
   * [Interpolation Search](https://github.com/TheAlgorithms/C/blob/master/searching/interpolation_search.c)
   * [Jump Search](https://github.com/TheAlgorithms/C/blob/master/searching/jump_search.c)

searching/exponential_search.c

@@ -0,0 +1,107 @@
+/**
+ *  \file
+ *  \brief [Exponential Search](https://github.com/TheAlgorithms/Algorithms-Explanation/blob/master/en/Search%20Algorithms/Exponential%20Search.md)
+ *  \author [Alessio Farinelli](https://github.com/faridevnz)
+ */
+#include <assert.h>  /// for assert
+#include <inttypes.h>  /// for int64_t, uint16_t
+
+#define ELEMENT -10
+
+int64_t binary_search(const int64_t* arr, const uint16_t l_index, const uint16_t r_index, const int64_t n); ///< used to perform the binary search over the given array
+int64_t exponential_search(const int64_t* arr, const uint16_t length, const int64_t n); ///< used to perform the exponential search over the given array
+static void test(); ///< used to run the self-test implementations
+
+/**
+ * Function: exponential_search
+ * \brief algorithm that search the index of the given item
+ * \details recursive function that take an array and quickly find the range 
+ * where to apply the binary search algorithm to find the given element
+ * ----------------------------
+ * \param arr array where search the element
+ * \param length the total length of the given array (arr)
+ * \param n element to find in the array (arr)
+ * 
+ * \returns the index of the element (n) in the array (arr)
+ * \returns -1 if the element wasn't found
+ */
+int64_t exponential_search(const int64_t* arr, const uint16_t length, const int64_t n) 
+{
+    if ( length == 0 ) { return -1; }
+    // find the upperbound
+    uint32_t upper_bound = 1;
+    while ( upper_bound <= length && arr[upper_bound] < n ) { upper_bound = upper_bound * 2; }
+    // calculate the range ( between lower_boud and upper_bound )
+    uint16_t lower_bound = upper_bound/2;
+    if ( upper_bound > length ) { upper_bound = length; }
+    // apply the binary search in the range
+    return binary_search(arr, lower_bound, upper_bound, n);
+}
+
+/**
+ * Function: binary_search
+ * \brief algorithm that search the index of the given item
+ * \details recursive function that search the given element in
+ * the array using the [Binary Search](https://github.com/TheAlgorithms/Algorithms-Explanation/blob/master/en/Search%20Algorithms/Binary%20Search.md)
+ * ----------------------------
+ * \param arr array where search the element
+ * \param l_index start index of the array (arr) to apply the algorithm
+ * \param r_index end index of the array (arr) to apply the algorithm
+ * \param n element to find in the array (arr)
+ *
+ * \returns the index of the element (n) in the array (arr)
+ * \returns -1 if the n element wasn't found
+ */
+int64_t binary_search(const int64_t* arr, const uint16_t l_index, const uint16_t r_index, const int64_t n) 
+{
+    // calculate the middle index of the array
+    uint16_t middle_index = l_index + ( r_index - l_index ) / 2;
+    // base cases
+    if ( l_index > r_index ) { return -1; }
+    if ( arr[middle_index] == n ) { return middle_index; }
+    // recursion
+    if ( arr[middle_index] > n ) { return binary_search(arr, l_index, middle_index-1, n); } // left
+    return binary_search(arr, middle_index+1, r_index, n); // right
+}
+
+/**
+ * @brief Main function
+ * @returns 0 on exit
+ */
+int main() 
+{
+    test();  // run self-test implementations
+    return 0;
+}
+
+/**
+ * @brief Self-test implementations
+ * @returns void
+ */
+static void test()
+{
+    // empty array
+    int64_t arr_empty[] = {};
+    assert(exponential_search(arr_empty, 0, 10) == -1);
+    // elent not found
+    int64_t arr_found[] = {1, 2, 3};
+    assert(exponential_search(arr_found, 3, 10) == -1);
+    // element found in an array of length 1
+    int64_t arr_one[] = {1};
+    assert(exponential_search(arr_found, 1, 1) == 0);
+    // find the first element in an array of length 2
+    int64_t arr_first_2[] = {1, 2};
+    assert(exponential_search(arr_first_2, 2, 1) == 0);
+    // find the last element in an array of length 2
+    int64_t arr_last_2[] = {1, 2};
+    assert(exponential_search(arr_last_2, 2, 2) == 1);
+    // find the first element in an array of length n
+    int64_t arr_first_n[] = {-1, 2, 4, 6, 8};
+    assert(exponential_search(arr_first_n, 5, -1) == 0);
+    // find the last element in an array of length n
+    int64_t arr_last_n[] = {-1, 2, 4, 6, 8};
+    assert(exponential_search(arr_last_n, 5, 8) == 4);
+    // find an element in an array of length n
+    int64_t arr_middle[] = {-1, 2, 4, 6, 8};
+    assert(exponential_search(arr_middle, 5, 6) == 3);
+}