This is a java program to perform optimal paranthesization by making use of dymanic programming. This program determines the order in which the chain of matrices should be multiplied.

Here is the source code of the Java Program to Perform Optimal Paranthesization Using Dynamic Programming. The Java program is successfully compiled and run on a Windows system. The program output is also shown below.

package com.sanfoundry.numerical;

import java.util.Scanner;

public class OptimalParanthesizationUsingDP

`{`

private int[][] m;

private int[][] s;

private int n;

public OptimalParanthesizationUsingDP(int[] p)

`{`

n = p.length - 1; // how many matrices are in the chain

m = new int[n + 1][n + 1]; // overallocate m, so that we don't use index

`// 0`

s = new int[n + 1][n + 1]; // same for s

matrixChainOrder(p); // run the dynamic-programming algorithm

`}`

private void matrixChainOrder(int[] p)

`{`

`// Initial the cost for the empty subproblems.`

for (int i = 1; i <= n; i++)

m[i][i] = 0;

`// Solve for chains of increasing length l.`

for (int l = 2; l <= n; l++)

`{`

for (int i = 1; i <= n - l + 1; i++)

`{`

int j = i + l - 1;

m[i][j] = Integer.MAX_VALUE;

`// Check each possible split to see if it's better`

`// than all seen so far.`

for (int k = i; k < j; k++)

`{`

int q = m[i][k] + m[k + 1][j] + p[i - 1] * p[k] * p[j];

if (q < m[i][j])

`{`

`// q is the best split for this subproblem so far.`

m[i][j] = q;

s[i][j] = k;

`}`

`}`

`}`

`}`

`}`

private String printOptimalParens(int i, int j)

`{`

if (i == j)

return "A[" + i + "]";

`else`

return "(" + printOptimalParens(i, s[i][j])

+ printOptimalParens(s[i][j] + 1, j) + ")";

`}`

public String toString()

`{`

return printOptimalParens(1, n);

`}`

public static void main(String[] args)

`{`

Scanner sc = new Scanner(System.in);

System.out

.println("Enter the array p[], which represents the chain of matrices such that the ith matrix Ai is of dimension p[i-1] x p[i]");

System.out.println("Enter the total length: ");

int n = sc.nextInt();

int arr[] = new int[n];

System.out.println("Enter the dimensions: ");

for (int i = 0; i < n; i++)

arr[i] = sc.nextInt();

OptimalParanthesizationUsingDP opudp = new OptimalParanthesizationUsingDP(

arr);

System.out.println("Matrices are of order: ");

for (int i = 1; i < arr.length; i++)

`{`

System.out.println("A" + i + "-->" + arr[i - 1] + "x" + arr[i]);

`}`

System.out.println(opudp.toString());

sc.close();

`}`

`}`

Output:

$ javac OptimalParanthesizationUsingDP.java $ java OptimalParanthesizationUsingDP Enter the array p[], which represents the chain of matrices such that the ith matrix Ai is of dimension p[i-1] x p[i] Enter the total length: 5 Enter the dimensions: 2 4 5 2 1 Matrices are of order: A1-->2x4 A2-->4x5 A3-->5x2 A4-->2x1 (A[1](A[2](A[3]A[4])))

**Sanfoundry Global Education & Learning Series – 1000 Java Programs.**

Here’s the list of Best Reference Books in Java Programming, Data Structures and Algorithms.