#1
 axelrv Junior Member Join Date: Sep 2012 Posts: 6 Problem 2.8

I thought that growth functions could take on any polynomial function of N, so why is 1 + N + N(N-1)(N-2) / 6 not a possible growth function?
#2 magdon RPI Join Date: Aug 2009 Location: Troy, NY, USA. Posts: 595 Re: Problem 2.8

The growth function cannot be any polynomial function of N. A valid growth function must satisfy the following theorem:

Theorem: If for some (any) , then for all N, .

In your example growth function below, try to show that the precondition of the theorem is satisfied with and hence deduce a linear bound on m(N). This contradicts the growth function being cubic. More generally, a cubic growth function cannot have a break point less than 4.

Quote:
 Originally Posted by axelrv I thought that growth functions could take on any polynomial function of N, so why is 1 + N + N(N-1)(N-2) / 6 not a possible growth function?
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#3
 axelrv Junior Member Join Date: Sep 2012 Posts: 6 Re: Problem 2.8

thanks!
#4
 k_sze Member Join Date: Dec 2016 Posts: 12 Re: Problem 2.8

Quote:
 Originally Posted by magdon The growth function cannot be any polynomial function of N. A valid growth function must satisfy the following theorem: Theorem: If for some (any) , then for all N, . In your example growth function below, try to show that the precondition of the theorem is satisfied with and hence deduce a linear bound on m(N). This contradicts the growth function being cubic. More generally, a cubic growth function cannot have a break point less than 4.
To show that a growth function is invalid, is it sufficient to do this?
1. Determine the smallest where , such that would be our if the growth function were valid;
2. Find any concrete value of where one of the inequalities vs is violated.
#5 htlin NTU Join Date: Aug 2009 Location: Taipei, Taiwan Posts: 601 Re: Problem 2.8

Quote:
 Originally Posted by k_sze To show that a growth function is invalid, is it sufficient to do this?Determine the smallest where , such that would be our if the growth function were valid; Find any concrete value of where one of the inequalities vs is violated.
Sounds reasonable to me.
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