Assume that multiplying a matrix G1 of dimension p×q with another matrix G2 of dimension q×r requires pqr scalar multiplications. Computing the product of n matrices G1G2G3 ….. Gn can be done by parenthesizing in different ways. Define GiGi+1 as an explicitly computed pair for a given paranthesization if they are directly multiplied. For example, in the matrix multiplication chain G1G2G3G4G5G6 using parenthesization (G1(G2G3))(G4(G5G6)), G2G3 and G5G6 are only explicitly computed pairs.
Consider a matrix multiplication chain F1F2F3F4F5, where matrices F1,F2,F3,F4 and F5 are of dimensions 2×25,25×3,3×16,16×1 and 1×1000, respectively. In the parenthesization of F1F2F3F4F5 that minimizes the total number of scalar multiplications, the explicitly computed pairs is/are
(A) F1F2 and F3F4 only
(B) F2F3 only
(C) F3F4 only
(D) F1F2 and F4F5 only
Explanation: Matrix F5 is of dimension 1 X 1000, which is going to cause very much multiplication cost. So evaluating F5 at last is optimal.
Total number of scalar multiplications are 48 + 75 + 50 + 2000 = 2173 and optimal parenthesis is ((F1(F2(F3 F4)))F5).
As concluded, F3, F4 are explicitly computed pairs.
Option (C) is Correct.
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