How to calculate formally the running time of the naive polynomial evaluation at a point
I understand intuitively why the time complexity of the naive polynomial evaluation at a point is ϴ(n^2). However, I'm not sure how to calculate formally the running time in order to show it.
Thanks in advance!
Not quite ϴ(n^2), but ϴ(mn), where m and n are the number of terms in each polynomial.
There are trivially m * n multiplication required, equal to the number of ways to pair coefficients a_i * b_j between the two polynomials.
There are also additions to consider; however, since any certain pair of coefficients a_i, b_j only belongs to one power of x, it will only be added once to a coefficient in the final polynomial. Therefore there can only be up to O(mn) additions.
Therefore the overall time complexity of naive multiplication is ϴ(mn).
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