Add matrix chain multiplication with memoization (#2688)

DynamicProgramming/MatrixChainRecursiveTopDownMemoisation.java

@@ -0,0 +1,56 @@
+package DynamicProgramming;
+
+// Matrix-chain Multiplication
+// Problem Statement
+// we have given a chain A1,A2,...,Ani of n matrices, where for i = 1,2,...,n, 
+// matrix Ai has dimension pi−1 ×pi
+// , fully parenthesize the product A1A2 ···An in a way that
+// minimizes the number of scalar multiplications.
+
+public class MatrixChainRecursiveTopDownMemoisation
+{
+    static int Memoized_Matrix_Chain(int p[]) {
+        int n = p.length ;
+        int m[][] = new int[n][n];
+        for (int i = 0; i < n; i++) {
+            for (int j = 0; j < n; j++) {
+                m[i][j] = Integer.MAX_VALUE;
+            }
+        }
+        return Lookup_Chain(m, p, 1, n-1);
+    }
+
+    static int Lookup_Chain(int m[][],int p[],int i, int j)
+    {
+        if ( i == j )
+        {
+            m[i][j] = 0;
+            return m[i][j];
+        }
+        if ( m[i][j] < Integer.MAX_VALUE )
+        {
+            return m[i][j];
+        }
+        
+        else 
+        {
+            for ( int k = i ; k<j  ;k++)
+            {
+                int q = Lookup_Chain(m, p,i , k ) + Lookup_Chain(m, p, k+1 , j) + (p[i-1] * p[k] * p[j]);
+                if ( q < m[i][j] )
+                {
+                    m[i][j] = q;
+                }
+            }
+        }
+        return m[i][j];
+    }
+    // in this code we are taking the example of 4 matrixes whose orders are 1x2,2x3,3x4,4x5 respectively
+    // output should be  Minimum number of multiplications is 38
+        public static void main (String[] args)
+        {
+ 
+            int arr[] = { 1, 2, 3, 4 ,5};
+            System.out.println("Minimum number of multiplications is " + Memoized_Matrix_Chain(arr));
+        }
+}