matrix chainmultiplication

To compute the optimal order of multiplying a chain of matrices in C++, you can use the Matrix Chain Multiplication algorithm. Here are the steps involved:

  1. Define the input: Start by defining the dimensions of the matrices in the chain. Suppose you have n matrices, each with dimensions A1[A[1] x A[2]], A2[A[2] x A[3]], ..., An[A[n-1] x A[n]].

  2. Create a table: Create a table M[n][n] to store the minimum number of scalar multiplications needed to compute the matrix chain product. Initialize all values in the table to infinity or a large number.

  3. Compute chain lengths: For each chain length l = 2 to n, iterate over the matrix chain and compute the minimum number of scalar multiplications needed for each subchain of length l. The outer loop iterates over chain lengths, and the inner loop iterates over the starting index of the subchain.

  4. Compute subproblems: For each subchain of length l, compute the minimum number of scalar multiplications needed by considering all possible ways to split the subchain into two parts. Iterate from the starting index i to i+l-1 and compute the number of scalar multiplications needed for each split. Update the table M[i][i+l-1] with the minimum value.

  5. Track optimal split: To track the optimal split, create a table S[n][n] to store the split points. Initialize all values to 0.

  6. Track split points: During the computation of subproblems, whenever a new minimum number of scalar multiplications is found, update the split point in table S[i][j] with the position k that achieved the minimum value.

  7. Retrieve optimal order: To retrieve the optimal order of matrix multiplication, use the split points stored in table S. Recursively traverse the split points from S[1][n] and print the parentheses accordingly.

  8. Return result: The minimum number of scalar multiplications needed to compute the matrix chain product is stored in table M[1][n].

This is a high-level overview of how you can implement the matrix chain multiplication algorithm in C++. By following these steps, you can efficiently compute the optimal order of multiplying a chain of matrices.