rename Dynamic Programming -> dynamic_programming (#645)

* rename Dynamic Programming -> dynamic_programming

* rename dynamic-programming -> dynamic_programming

dynamic_programming/Catalan-Numbers.cpp

@@ -0,0 +1,79 @@
+/** Print all the Catalan numbers from 0 to n, n being the user input.
+
+ * A Catalan number satifies the following two properties:
+ * C(0) = C(1) = 1; C(n) = sum(C(i).C(n-i-1)), from i = 0 to n-1
+ * Read more about Catalan numbers here:
+    https://en.wikipedia.org/wiki/Catalan_number
+ */
+
+#include <iostream>
+using namespace std;
+
+int *cat; // global array to hold catalan numbers
+
+unsigned long int catalan_dp(int n)
+{
+    /** Using the tabulation technique in dynamic programming,
+        this function computes the first `n+1` Catalan numbers
+
+        Parameter
+        ---------
+        n: The number of catalan numbers to be computed.
+
+        Returns
+        -------
+        cat[n]: An array containing the first `n+1` Catalan numbers
+    */
+
+    // By definition, the first two Catalan numbers are 1
+    cat[0] = cat[1] = 1;
+
+    // Compute the remaining numbers from index 2 to index n, using tabulation
+    for (int i = 2; i <= n; i++)
+    {
+        cat[i] = 0;
+        for (int j = 0; j < i; j++)
+            cat[i] += cat[j] * cat[i - j - 1]; // applying the definition here
+    }
+
+    // Return the result
+    return cat[n];
+}
+
+int main(int argc, char *argv[])
+{
+    int n;
+    cout << "Enter n: ";
+    cin >> n;
+
+    cat = new int[n + 1];
+
+    cout << "Catalan numbers from 0 to " << n << " are:\n";
+    for (int i = 0; i <= n; i++)
+    {
+        cout << "catalan (" << i << ") = " << catalan_dp(i) << endl;
+        // NOTE: Since `cat` is a global array, calling `catalan_dp`
+        // repeatedly will not recompute the the values already computed
+        // as in case of pre-computed values, the array will simply return them,
+        // instead of recomputing them.
+    }
+
+    return 0;
+}
+
+/** Sample Test Case:
+
+$ cd "Dynamic Programming"
+$ g++ Catalan-Numbers.cpp
+$ ./a.exe
+
+Enter n: 5
+Catalan numbers from 0 to 5 are:
+catalan (0) = 1
+catalan (1) = 1
+catalan (2) = 2
+catalan (3) = 5
+catalan (4) = 14
+catalan (5) = 42
+
+*/