Dynamic Programming Questions Long
Dynamic Programming can be used to solve the matrix chain multiplication problem by breaking it down into smaller subproblems and solving them in a bottom-up manner. The matrix chain multiplication problem involves finding the most efficient way to multiply a series of matrices together.
To solve this problem using Dynamic Programming, we can define a 2D table, let's call it "m", where m[i][j] represents the minimum number of scalar multiplications needed to compute the matrix product of matrices from i to j (inclusive). The dimensions of this table will be (n x n), where n is the number of matrices in the chain.
We start by filling in the diagonal elements of the table, i.e., m[i][i] = 0, as multiplying a single matrix requires no scalar multiplications.
Next, we iterate over the chain length, starting from 2 up to n. For each chain length, we iterate over all possible starting positions of the chain, denoted by the variable "i". The ending position of the chain, denoted by the variable "j", is calculated based on the chain length and starting position.
For each combination of i and j, we calculate the minimum number of scalar multiplications needed to compute the matrix product of matrices from i to j. This can be done by considering all possible partition points within the chain and finding the one that minimizes the total number of scalar multiplications.
The formula to calculate m[i][j] is as follows:
m[i][j] = min(m[i][k] + m[k+1][j] + p[i-1] * p[k] * p[j])
where p[i-1], p[k], and p[j] represent the dimensions of the matrices involved in the multiplication.
By filling in the table in a bottom-up manner, we can eventually obtain the minimum number of scalar multiplications needed to compute the matrix product of the entire chain, which is stored in m[1][n].
Additionally, we can keep track of the partition points that give the optimal solution by maintaining another table, let's call it "s". The table s[i][j] stores the index of the partition point that gives the minimum number of scalar multiplications for the matrix product from i to j.
Once the tables m and s are filled, we can use the information in table s to reconstruct the optimal parenthesization of the matrix chain multiplication.
In summary, Dynamic Programming is used to solve the matrix chain multiplication problem by breaking it down into smaller subproblems and solving them in a bottom-up manner. By filling in a 2D table with the minimum number of scalar multiplications needed for each subproblem, we can eventually obtain the optimal solution for the entire matrix chain.