feat: add matrix chain order (#1198)

* feat: add Matrix Chain Order

* updating DIRECTORY.md

* Update dynamic_programming/matrix_chain_order.c

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

* Update dynamic_programming/matrix_chain_order.c

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

* Update dynamic_programming/matrix_chain_order.c

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

* Update matrix_chain_order.c

* chore: apply suggestions from code review

* chore: apply suggestions from code review

* updating DIRECTORY.md

* Update dynamic_programming/matrix_chain_order.c

Co-authored-by: Taj <tjgurwara99@users.noreply.github.com>

* updating DIRECTORY.md

* updating DIRECTORY.md

---------

Co-authored-by: github-actions[bot] <github-actions@users.noreply.github.com>
Co-authored-by: David Leal <halfpacho@gmail.com>
Co-authored-by: Taj <tjgurwara99@users.noreply.github.com>

DIRECTORY.md

@@ -123,6 +123,7 @@
 
 ## Dynamic Programming
   * [Lcs](https://github.com/TheAlgorithms/C/blob/HEAD/dynamic_programming/lcs.c)
+  * [Matrix Chain Order](https://github.com/TheAlgorithms/C/blob/HEAD/dynamic_programming/matrix_chain_order.c)
 
 ## Exercism
   * Acronym
@@ -180,6 +181,7 @@
     * [11](https://github.com/TheAlgorithms/C/blob/HEAD/leetcode/src/11.c)
     * [110](https://github.com/TheAlgorithms/C/blob/HEAD/leetcode/src/110.c)
     * [112](https://github.com/TheAlgorithms/C/blob/HEAD/leetcode/src/112.c)
+    * [1137](https://github.com/TheAlgorithms/C/blob/HEAD/leetcode/src/1137.c)
     * [1147](https://github.com/TheAlgorithms/C/blob/HEAD/leetcode/src/1147.c)
     * [118](https://github.com/TheAlgorithms/C/blob/HEAD/leetcode/src/118.c)
     * [1184](https://github.com/TheAlgorithms/C/blob/HEAD/leetcode/src/1184.c)
@@ -304,6 +306,7 @@
     * [977](https://github.com/TheAlgorithms/C/blob/HEAD/leetcode/src/977.c)
     * [98](https://github.com/TheAlgorithms/C/blob/HEAD/leetcode/src/98.c)
     * [985](https://github.com/TheAlgorithms/C/blob/HEAD/leetcode/src/985.c)
+    * [997](https://github.com/TheAlgorithms/C/blob/HEAD/leetcode/src/997.c)
 
 ## Machine Learning
   * [Adaline Learning](https://github.com/TheAlgorithms/C/blob/HEAD/machine_learning/adaline_learning.c)

dynamic_programming/matrix_chain_order.c

@@ -0,0 +1,91 @@
+/**
+ * @file
+ * @brief [Matrix Chain Order](https://en.wikipedia.org/wiki/Matrix_chain_multiplication)
+ * @details
+ * From Wikipedia: Matrix chain multiplication (or the matrix chain ordering problem)
+ * is an optimization problem concerning the most efficient way to multiply a given sequence of matrices.
+ * The problem is not actually to perform the multiplications,
+ * but merely to decide the sequence of the matrix multiplications involved.
+ * @author [CascadingCascade](https://github.com/CascadingCascade)
+ */
+
+#include <assert.h> /// for assert
+#include <stdio.h>  /// for IO operations
+#include <limits.h> /// for INT_MAX macro
+#include <stdlib.h> /// for malloc() and free()
+
+/**
+ * @brief Finds the optimal sequence using the classic O(n^3) algorithm.
+ * @param l length of cost array
+ * @param p costs of each matrix
+ * @param s location to store results
+ * @returns number of operations
+ */
+int matrixChainOrder(int l,const int *p, int *s) {
+    // mat stores the cost for a chain that starts at i and ends on j (inclusive on both ends)
+    int mat[l][l];
+    for (int i = 0; i < l; ++i) {
+        mat[i][i] = 0;
+    }
+    // cl denotes the difference between start / end indices, cl + 1 would be chain length.
+    for (int cl = 1; cl < l; ++cl) {
+        for (int i = 0; i < l - cl; ++i) {
+            int j = i + cl;
+            mat[i][j] = INT_MAX;
+            for (int div = i; div < j; ++div) {
+                int q = mat[i][div] + mat[div + 1][j] + p[i] * p[div] * p[j];
+                if (q < mat[i][j]) {
+                    mat[i][j] = q;
+                    s[i * l + j] = div;
+                }
+            }
+        }
+    }
+    return mat[0][l - 1];
+}
+
+/**
+ * @brief Recursively prints the solution
+ * @param l dimension of the solutions array
+ * @param s solutions
+ * @param i starting index
+ * @param j ending index
+ * @returns void
+ */
+void printSolution(int l,int *s,int i,int j) {
+    if(i == j) {
+        printf("A%d",i);
+        return
+    }
+    putchar('(');
+    printSolution(l,s,i,s[i * l + j]);
+    printSolution(l,s,s[i * l + j] + 1,j);
+    putchar(')');
+}
+
+/**
+ * @brief Self-test implementations
+ * @returns void
+ */
+static void test() {
+    int sizes[] = {35,15,5,10,20,25};
+    int len = 6;
+    int *sol = malloc(len * len * sizeof(int));
+    int r = matrixChainOrder(len,sizes,sol);
+    assert(r == 18625);
+    printf("Result : %d\n",r);
+    printf("Optimal ordering : ");
+    printSolution(len,sol,0,5);
+    free(sol);
+
+    printf("\n");
+}
+
+/**
+ * @brief Main function
+ * @returns 0
+ */
+int main() {
+    test();  // run self-test implementations
+    return 0;
+}