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I built a recursion tree like this:

  0  
 / \  
 0  0  
/\ /\  
... ...

So the tree has height n, and width $2^n$. But if the sum of all levels is $\sum_{i=0}^{n}0$, then is the function simply $O(1)$?

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4
  • $\begingroup$ You are missing the base cases from your recurrence relations. I suspect it is $T(1)=1$. You might want to redraw the tree with this in mind. $\endgroup$
    – Steven
    Commented Jun 7, 2022 at 17:54
  • $\begingroup$ The title of the post is wrong. And there is no connection between the width $2^n$ and $\sum_{i=0}^n$. $\endgroup$
    – user16034
    Commented Jun 7, 2022 at 19:47
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    $\begingroup$ "is the function $O(1)$ ?": which function ?? $\endgroup$
    – user16034
    Commented Jun 7, 2022 at 19:51
  • $\begingroup$ Maybe you're thinking of $T(n)=2T(n-1) + 0$. The problem, like @Steven says, is that you need a base case. To see it even more clearly than your illustration, simply try rewriting the function. For example, write $T(n)=2T(n-1) + 0 = 2(\underbrace{ 2T(n-2)+0 }_{T(n-1)}) + 0 = 2( 2(\underbrace{2T(n-3)+0}_{T(n-2)}) + 0 ) + 0 = \dots$. Eventually, you will still need the function $T(n-k)$ for some $k$. This function, $T(n-k)$, doesn't need to be constant. i.e. it could be a function of $n$. It could also be $0$, which is even faster than a constant function. $\endgroup$
    – Matt Groff
    Commented Jun 8, 2022 at 1:22

1 Answer 1

0
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A sum of $m$ constants takes time $O(m)$.

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