1.

a) How many scalar multiplications are needed to compute the product, for each of the different parenthesizations, assuming the matrix dimensions are 7 50, 50 10, and 10 100, respectively?

b) How long would it take to try all possible parenthesizations? State an asymptotic lower bound (as tight as possible), assuming that two numbers of arbitrary length can be multiplied in constant time.

c) What is the runtime of the dynamic programming algorithm? 6. Why does the d.p. algorithm need a two-dimensional array to store the subproblem solutions?