GATE | GATE CS 2018 | Question 51

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 ….. G_n can be done by parenthesizing in different ways. Define G_iG_i+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


Answer: (C)

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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