The dynamic programming algorithm for matrix chain multiplication is a method for finding the minimum number of scalar multiplications required to multiply a chain of matrices. The algorithm works by building a table that stores the minimum number of scalar multiplications required to multiply all possible sub chains of the given chain.

Algorithm for dynamic programming algorithm for matrix chain multiplication.

1.start

2.Create a table M of size (n+1) x (n+1), where n is the number of matrices and fill the diagonal of the table with 0s.

3.For each subproblem (i, j), where 1 ≤ i ≤ j ≤ n, calculate the minimum number of scalar multiplications required to multiply matrices A_i, A_{i+1}, …, A_j using the following formula:

M[i , j] = min{M[i , k] + M[k+1, j] + p_i *p_{k+1} *p_j}

4.Once the table is filled, the optimal parenthesization can be found by tracing back through the table.

5.stop

2nd pard

To find the optimal parenthesization for the matrix chain product ABCD with given sizes, you can use dynamic programming to minimize the number of scalar multiplications. The goal is to find the optimal way to parenthesize the matrices to minimize the overall cost.

Let’s denote the matrices as follows:

- A: 5×10
- B: 10×15
- C: 15×20
- D: 20×30

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