Matrix Chain Recursive Top Down Memoisation

m
```package com.thealgorithms.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));
}
}
```