feat: Added Solution of Partition Problem (#1842)

* Create partition_problem.cpp

* Fix #1

* Fix #2

* updating DIRECTORY.md

* clang-format and clang-tidy fixes for 3b8dbf00

* Update dynamic_programming/partition_problem.cpp

Co-authored-by: David Leal <halfpacho@gmail.com>

* Update dynamic_programming/partition_problem.cpp

Co-authored-by: David Leal <halfpacho@gmail.com>

* clang-format and clang-tidy fixes for a152bf75

* Update dynamic_programming/partition_problem.cpp

Co-authored-by: David Leal <halfpacho@gmail.com>

* Update dynamic_programming/partition_problem.cpp

Co-authored-by: David Leal <halfpacho@gmail.com>

* Update dynamic_programming/partition_problem.cpp

Co-authored-by: David Leal <halfpacho@gmail.com>

* clang-format and clang-tidy fixes for 1c057113

* Update dynamic_programming/partition_problem.cpp

Co-authored-by: David Leal <halfpacho@gmail.com>

* Update dynamic_programming/partition_problem.cpp

Co-authored-by: David Leal <halfpacho@gmail.com>

* clang-format and clang-tidy fixes for baf20a6f

* Update dynamic_programming/partition_problem.cpp

Co-authored-by: David Leal <halfpacho@gmail.com>

* clang-format and clang-tidy fixes for 749d9337

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

DIRECTORY.md

@@ -102,6 +102,7 @@
   * [Matrix Chain Multiplication](https://github.com/TheAlgorithms/C-Plus-Plus/blob/master/dynamic_programming/matrix_chain_multiplication.cpp)
   * [Minimum Edit Distance](https://github.com/TheAlgorithms/C-Plus-Plus/blob/master/dynamic_programming/minimum_edit_distance.cpp)
   * [Palindrome Partitioning](https://github.com/TheAlgorithms/C-Plus-Plus/blob/master/dynamic_programming/palindrome_partitioning.cpp)
+  * [Partition Problem](https://github.com/TheAlgorithms/C-Plus-Plus/blob/master/dynamic_programming/partition_problem.cpp)
   * [Searching Of Element In Dynamic Array](https://github.com/TheAlgorithms/C-Plus-Plus/blob/master/dynamic_programming/searching_of_element_in_dynamic_array.cpp)
   * [Shortest Common Supersequence](https://github.com/TheAlgorithms/C-Plus-Plus/blob/master/dynamic_programming/shortest_common_supersequence.cpp)
   * [Tree Height](https://github.com/TheAlgorithms/C-Plus-Plus/blob/master/dynamic_programming/tree_height.cpp)

dynamic_programming/partition_problem.cpp

@@ -0,0 +1,106 @@
+/******************************************************************************
+ * @file
+ * @brief Implementation of the [Partition
+ * Problem](https://en.wikipedia.org/wiki/Partition_problem )
+ * @details
+ * The partition problem, or number partitioning, is the task of deciding
+ * whether a given multiset S of positive integers can be partitioned into two
+ * subsets S1 and S2 such that the sum of the numbers in S1 equals the sum of
+ * the numbers in S2. Although the partition problem is NP-complete, there is a
+ * pseudo-polynomial time dynamic programming solution, and there are heuristics
+ * that solve the problem in many instances, either optimally or approximately.
+ * For this reason, it has been called "the easiest hard problem".
+ *
+ * The worst case time complexity of Jarvis’s Algorithm is O(n^2). Using
+ * Graham’s scan algorithm, we can find Convex Hull in O(nLogn) time.
+ *
+ * ### Implementation
+ *
+ * Step 1
+ * Calculate sum of the array. If sum is odd, there can not be two subsets with
+ * equal sum, so return false.
+ *
+ * Step 2
+ * If sum of array elements is even, calculate sum/2 and find a subset of array
+ * with sum equal to sum/2.
+ *
+ * @author [Lajat Manekar](https://github.com/Lazeeez)
+ *
+ *******************************************************************************/
+#include <cassert>   /// for assert
+#include <iostream>  /// for IO Operations
+#include <numeric>   /// for std::accumulate
+#include <vector>    /// for std::vector
+
+/******************************************************************************
+ * @namespace dp
+ * @brief Dynamic programming algorithms
+ *******************************************************************************/
+namespace dp {
+
+/******************************************************************************
+ * @namespace partitionProblem
+ * @brief Partition problem algorithm
+ *******************************************************************************/
+namespace partitionProblem {
+
+/******************************************************************************
+ * @brief Returns true if arr can be partitioned in two subsets of equal sum,
+ * otherwise false
+ * @param arr vector containing elements
+ * @param size Size of the vector.
+ * @returns @param bool whether the vector can be partitioned or not.
+ *******************************************************************************/
+bool findPartiion(const std::vector<uint64_t> &arr, uint64_t size) {
+    uint64_t sum = std::accumulate(arr.begin(), arr.end(),
+                                   0);  // Calculate sum of all elements
+
+    if (sum % 2 != 0) {
+        return false;  // if sum is odd, it cannot be divided into two equal sum
+    }
+    std::vector<bool> part;
+    // bool part[sum / 2 + 1];
+
+    // Initialize the part array as 0
+    for (uint64_t it = 0; it <= sum / 2; ++it) {
+        part.push_back(false);
+    }
+
+    // Fill the partition table in bottom up manner
+    for (uint64_t it = 0; it < size; ++it) {
+        // The element to be included in the sum cannot be greater than the sum
+        for (uint64_t it2 = sum / 2; it2 >= arr[it];
+             --it2) {  // Check if sum - arr[i]
+            // ould be formed from a subset using elements before index i
+            if (part[it2 - arr[it]] == 1 || it2 == arr[it]) {
+                part[it2] = true;
+            }
+        }
+    }
+    return part[sum / 2];
+}
+}  // namespace partitionProblem
+}  // namespace dp
+
+/*******************************************************************************
+ * @brief Self-test implementations
+ * @returns void
+ *******************************************************************************/
+static void test() {
+    std::vector<uint64_t> arr = {{1, 3, 3, 2, 3, 2}};
+    uint64_t n = arr.size();
+    bool expected_result = true;
+    bool derived_result = dp::partitionProblem::findPartiion(arr, n);
+    std::cout << "1st test: ";
+    assert(expected_result == derived_result);
+    std::cout << "Passed!" << std::endl;
+}
+
+/*******************************************************************************
+ * @brief Main function
+ * @returns 0 on exit
+ *******************************************************************************/
+int main() {
+    test();  // run self-test implementations
+    return 0;
+}