Algorithm to maximize the sum of product of elements of two arrays

Question

There was a question in a contest which requires calculating the performance of the class which contains only Maths and Bio subject. So, there are \'n\' no. of maths student &am

Answer 1

We have Sum(Abs(o[i])) == m m > 0, and a[i], b[i], m fixed.

Sum( (a[i] + o[i]) * b[i]) == Sum(a[i] * b[i]) + Sum(o[i] * b[i])

So to maximize it, we just have to maximize

Sum(o[i] * b[i])

And we have

o[i] * b[i] <= Abs(o[i] * b[i])

And as we can choose sign of o[i], we can maximize instead

Sum(Abs(o[i] * b[i]))

which is

Sum(Abs(o[i]) * Abs(b[i]))

and

Max(Sum(Abs(o[i]) * Abs(b[i]))) <= m * Max(Abs(b[i]))

and with j so that b[j] == Max(Abs(b[i])), o[j] == sign(b[j]) * m, o[i] == 0, we have

Sum(o[i] * b[i]) == m * Max(Abs(b[i]))

So you have to find maximum of the both array (of absolute value).

So in your example

Maths Score :  5  7  (4+m) -3
Bio Score   : -2  3  9      2

And for other example:

Maths Score :  5  7  (4-m) -3
Bio Score   : -2  3  -9     2