feat: add prefix sum array data structure (#1663)

* feat: add prefix sum array data structure

* feat: add prefix_sum_array data structure

* updating DIRECTORY.md

* feat: Add Prefix Sum Array datastructure

* feat: Add Prefix Sum Array data structure

* Update range_queries/prefix_sum_array.cpp

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

* Update prefix_sum_array.cpp

* Update prefix_sum_array.cpp

* Update range_queries/prefix_sum_array.cpp

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

* Update range_queries/prefix_sum_array.cpp

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

* Update range_queries/prefix_sum_array.cpp

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

* Update range_queries/prefix_sum_array.cpp

* Update range_queries/prefix_sum_array.cpp

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

DIRECTORY.md

@@ -276,6 +276,7 @@
   * [Heavy Light Decomposition](https://github.com/TheAlgorithms/C-Plus-Plus/blob/master/range_queries/heavy_light_decomposition.cpp)
   * [Mo](https://github.com/TheAlgorithms/C-Plus-Plus/blob/master/range_queries/mo.cpp)
   * [Persistent Seg Tree Lazy Prop](https://github.com/TheAlgorithms/C-Plus-Plus/blob/master/range_queries/persistent_seg_tree_lazy_prop.cpp)
+  * [Prefix Sum Array](https://github.com/TheAlgorithms/C-Plus-Plus/blob/master/range_queries/prefix_sum_array.cpp)
   * [Segtree](https://github.com/TheAlgorithms/C-Plus-Plus/blob/master/range_queries/segtree.cpp)
   * [Sparse Table](https://github.com/TheAlgorithms/C-Plus-Plus/blob/master/range_queries/sparse_table.cpp)
 

range_queries/prefix_sum_array.cpp

@@ -0,0 +1,83 @@
+/**
+ * @file
+ * @brief
+ * [Prefix Sum
+ * Array](https://en.wikipedia.org/wiki/Prefix_sum) data structure
+ * implementation.
+ *
+ * @details
+ * Prefix Sum Array is a data structure, that allows answering sum in some range
+ * queries. It can answer most sum range queries in O(1), with a build time
+ * complexity of O(N). But it hasn't an update querie.
+ *
+ * * Running Time Complexity \n
+ * * Build : O(N) \n
+ * * Range Query : O(1) \n
+ * @author [Paulo Vitor Lima Borges](https://github.com/PauloVLB)
+ */
+
+#include <cassert>   /// for assert
+#include <iostream>  /// for IO operations
+#include <vector>    /// for std::vector
+
+/**
+ * @namespace range_queries
+ * @brief Range Queries algorithms
+ */
+namespace range_queries {
+/**
+ * @namespace prefix_sum_array
+ * @brief Range sum queries using prefix-sum-array
+ */
+namespace prefix_sum_array {
+
+std::vector<int64_t> PSA(1, 0);
+
+/**
+ * @brief function that builds the PSA
+ * @param original_array original array of values
+ * @returns void
+ */
+void build(std::vector<int64_t> original_array) {
+    for (int i = 1; i <= static_cast<int>(original_array.size()); i++) {
+        PSA.push_back(PSA[i - 1] + original_array[i]);
+    }
+}
+/**
+ * @brief query function
+ * @param beg begin of the interval to sum
+ * @param end end of the interval to sum
+ * @returns sum of the range [beg, end]
+ */
+int64_t query(int64_t beg, int64_t end) { return PSA[end] - PSA[beg - 1]; }
+
+}  // namespace prefix_sum_array
+}  // namespace range_queries
+
+/**
+ * @brief Self-test implementations
+ * @returns void
+ */
+static void test() {
+    std::vector<int64_t> values{0,  123, 0,  2,  -2, 5,
+                                24, 0,   23, -1, -1};  // original array
+
+    range_queries::prefix_sum_array::build(values);
+    // queries are of the type: sum of the range [a, b] = psa[b] - psa[a-1]
+
+    assert(range_queries::prefix_sum_array::query(1, 10) ==
+           173);  // sum of the entire array
+    assert(range_queries::prefix_sum_array::query(4, 6) ==
+           27);  // the sum of the interval [4, 6]
+    assert(range_queries::prefix_sum_array::query(5, 9) ==
+           51);  // the sum of the interval [5, 9]
+}
+
+/**
+ * @brief Main function
+ * @returns 0 on exit
+ */
+int main() {
+    test();  // run self-test implementations
+    return 0;
+}