Add Matrix Chain Multiplication optimised approach

Author: YashTariyal

src/main/java/com/thealgorithms/misc/MCM.java

@@ -0,0 +1,64 @@
+// Dynamic Programming Java implementation of Matrix
+// Chain Multiplication.
+// See the Cormen book for details of the following
+// algorithm
+import java.util.*;
+import java.io.*;
+class MatrixChainMultiplication 
+{
+
+	// Matrix Ai has dimension p[i-1] x p[i] for i = 1..n
+	static int MatrixChainOrder(int p[], int n)
+	{
+		/* For simplicity of the 
+		program, one extra row and
+		one extra column are allocated in m[][]. 0th row
+		and 0th column of m[][] are not used */
+		int m[][] = new int[n][n];
+
+		int i, j, k, L, q;
+
+		/* m[i, j] = Minimum number of scalar
+		multiplications needed to compute the matrix
+		A[i]A[i+1]...A[j] = A[i..j] where
+		dimension of A[i] is p[i-1] x p[i] */
+
+		// cost is zero when multiplying one matrix.
+		for (i = 1; i < n; i++)
+			m[i][i] = 0;
+
+		// L is chain length.
+		for (L = 2; L < n; L++) 
+		{
+			for (i = 1; i < n - L + 1; i++) 
+			{
+				j = i + L - 1;
+				if (j == n)
+					continue;
+				m[i][j] = Integer.MAX_VALUE;
+				for (k = i; k <= j - 1; k++) 
+				{
+					// q = cost/scalar multiplications
+					q = m[i][k] + m[k + 1][j]
+						+ p[i - 1] * p[k] * p[j];
+					if (q < m[i][j])
+						m[i][j] = q;
+				}
+			}
+		}
+
+		return m[1][n - 1];
+	}
+
+	// Driver code
+	public static void main(String args[])
+	{
+		int arr[] = new int[] { 1, 2, 3, 4 };
+		int size = arr.length;
+
+		System.out.println(
+			"Minimum number of multiplications is "
+			+ MatrixChainOrder(arr, size));
+	}
+}
+/* This code is contributed by Rajat Mishra*/