Final answer:
The RECURSIVE-MATRIX-CHAIN algorithm is a more efficient method than brute-force enumeration to determine the optimal number of multiplications in a matrix-chain multiplication problem due to its dynamic programming approach, which reduces redundancy and computational time.
Step-by-step explanation:
The RECURSIVE-MATRIX-CHAIN algorithm is generally a more efficient way to determine the optimal number of multiplications in a matrix-chain multiplication problem than by enumerating all possible parenthesizations. This is because there can be an exponentially large number of ways to parenthesize the product, making enumeration impractical for large matrices. The recursive algorithm, on the other hand, employs dynamic programming to systematically solve smaller subproblems and uses those results to solve larger ones, which avoids redundant calculations and reduces the overall computational effort.
Consider that the brute-force enumeration method's time complexity grows exponentially with the number of matrices. With n matrices, there can be Catalans number of parenthesizations, which is roughly O(4^n/(n^(3/2))). Whereas, the RECURSIVE-MATRIX-CHAIN algorithm, while still having significant computational cost, has a much more manageable polynomial time complexity - specifically O(n^3) when implemented with dynamic programming, which is known as the Matrix-Chain Multiplication problem's optimal solution.