The matrix chain multiplication problem is a classical problem in dynamic programming. Given a sequence of matrices, the goal is to determine the most efficient way to multiply these matrices together. The efficiency is measured by the minimum number of scalar multiplications required.

Here's the algorithm to solve the matrix chain multiplication problem using dynamic programming:

1. Define the problem:

- Given a sequence of n matrices, where the dimensions of the ith matrix are given by a[i-1] x a[i] for i in the range [1, n]..
- Determine the order in which to multiply the matrices to minimize the total number of scalar multiplications required.

2. Create a dynamic programming table:

- Create a 2D table `m` of size n x n, where m[i][j] represents the minimum number of scalar multiplications needed to multiply matrices from i to j.
- Initialize all entries of `m` with infinity.

3. Fill in the table:

- For each chain length l (from 2 to n), iterate through the matrix chain and calculate the minimum number of scalar multiplications.
- For each possible split position k in the chain, calculate the cost using the formula: m[i][j] = min(m[i][k] + m[k+1][j] + a[i-1] * a[k] * a[j]) for all k in [i, j-1]..
- Here, a[i-1] * a[k] * a[j] represents the number of scalar multiplications needed to multiply the resulting matrices after the split.

4. Reconstruct the solution:

- To determine the order of matrix multiplication, create a separate table `s` of size n x n, where s[i][j] stores the split position that achieved the minimum cost for multiplying matrices from i to j.
- During the calculation in step 3, update `s` whenever a new minimum cost is found.
- To reconstruct the solution, recursively find the split positions using `s` and form the optimal order of matrix multiplication.

5. Return the result:

- The minimum number of scalar multiplications needed to multiply the entire matrix chain is given by m[1][n].

The matrix chain multiplication algorithm has a time complexity of O(n^3) and a space complexity of O(n^2), where n is the number of matrices in the chain.

By using dynamic programming, the algorithm efficiently solves the matrix chain multiplication problem by breaking it down into smaller subproblems and calculating the optimal solution from the bottom up. This approach ensures that each subproblem is solved only once, avoiding redundant computations and leading to an optimal solution.

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