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Could someone explain time complexity of solution of in this tutorial?

I'm having hard time figuring out, how asymptotic bounds for first solution is $O(3^k k)$.

What I figured so far is, for computing $f(Y)$, we would require $O(\lvert Y\rvert)$ steps and for fixed set $X$ (such that $\lvert X\rvert=k$), we would have $2^{n-k}$ number of $Y$ sets that would contain set $X$. And there would be ${n \choose k}$ such $X$ sets.

So total steps should be $${n \choose k } {\sum_{r=0}^{n-k} {n-k \choose r} ({k+r})} \leq {n \choose k} n 2^{n-k}$$

But is this value tightly bounded by $3^k k$?

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Let me focus on the following question:

Is $3^nn$ a tight bound on $\binom{n}{k}n2^{n-k}$?

First, let us see that this is a valid bound. This follows from the binomial theorem: $$ \sum_{k=0}^n \binom{n}{k} n 2^{n-k} = 3^nn. $$

Since there are $n+1$ summands in the expression, we deduce that $$ \max_{0 \leq k \leq n} \binom{n}{k} n 2^{n-k} \ge \frac{3^nn}{n+1}. $$ Using Stirling's formula, we find out that $$ \max_{0 \leq k \leq n} \binom{n}{k} n 2^{n-k} = \Theta(3^n\sqrt{n}). $$ This is a tighter bound.

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  • $\begingroup$ First of all thanks for your time. I don't know if my analysis is even correct or not. But asymptotic of solution was $𝑂(k3^k)$, it does not include $n$ at all. Input variables are $n$ and $k$ and $k \leq n$. $\endgroup$ – Anjan0791 Aug 7 at 15:26

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