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Old 10-10-2012, 07:46 AM
axelrv axelrv is offline
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Default 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?
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Old 10-10-2012, 07:52 AM
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magdon magdon is offline
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Default 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 m(k)<2^k for some (any) k, then for all N, m(N)\le N^{k-1}+1.

In your example growth function below, try k=2 to show that the precondition of the theorem is satisfied with k=2 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.

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Originally Posted by axelrv View Post
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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Old 10-10-2012, 12:44 PM
axelrv axelrv is offline
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Default Re: Problem 2.8

thanks!
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Old 03-21-2018, 09:15 AM
k_sze k_sze is offline
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Default Re: Problem 2.8

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Originally Posted by magdon View Post
The growth function cannot be any polynomial function of N. A valid growth function must satisfy the following theorem:

Theorem: If m(k)<2^k for some (any) k, then for all N, m(N)\le N^{k-1}+1.

In your example growth function below, try k=2 to show that the precondition of the theorem is satisfied with k=2 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 k where m_\mathcal{H}(k+1) < 2^{k+1}, such that k would be our d_{\text{vc}} if the growth function were valid;
  2. Find any concrete value of N where one of the inequalities vs m_\mathcal{H}(N) is violated.
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Old 03-22-2018, 02:08 AM
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htlin htlin is offline
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Default Re: Problem 2.8

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Originally Posted by k_sze View Post
To show that a growth function is invalid, is it sufficient to do this?
  1. Determine the smallest k where m_\mathcal{H}(k+1) < 2^{k+1}, such that k would be our d_{\text{vc}} if the growth function were valid;
  2. Find any concrete value of N where one of the inequalities vs m_\mathcal{H}(N) is violated.
Sounds reasonable to me.
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