Matrix Chain Multiplication Dynamic Programming

Question

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

F1F2 and F3F4 only

F2F3 only

F3F4 only

F2F3 and F4F5 only

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My approach - I want to solve this under one minute, but the only way I know is that to use Bottom up Dynamic Approach by making a table and the other thing I can conclude is we should multiply with F5 at last because it has 1000 in it's dimension.So, please how to develop fast intuition for this kind of question!

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Correct answer is F3F4

Answer 1

The most important thing to note is the dimension 1×1000. You better watch out for it if you want to minimize the multiplications. OK, now we do know what we are looking for is basically multiply a small number with 1000.

Carefully examining if we go with F4F5, we would be multiplying 16x1x1000. But computing F3F4 first , the result matrix has dimension 3x1. So going with F3F4 we are able to get small numbers like 3,1 . So , no way im going with F4F5.

By similar logic I would not go with F2F3 and loose the smaller 3 and get bigger 25 and 16 to be later used with 1000.

OK, for F1F2, you can quickly find that (F1F2)(F3F4) is not better than (F1(F2(F3F4))) . So the answer is F3F4