Big$-$O notation. We have learnt big$-$O notation to Big$-$O notation. We have learnt big$-$O notation to compare the growth rates of func$-$

tions, this exercise helps you to better understand its definition and properties.

$($a$)(12$ points$)$ Suppose $n$ is the input size, we have the following commonly seen

functions in complexity analysis: $f_1(n)=1,f_2(n)=logn,f_3(n)=n,f_4(n)=$

$nlogn,f_5(n)=n^2,f_6(n)=2^n,f_7(n)=n!.$ Intuitively, the growth rate of the func$-$

tions satisfy $1=O(logn),logn=O(n),n=O(nlogn),nlogn=O(n^2),n^2=O(2^n),2^n=O(n!)f,g$:$N{R}^{+}O(f(n)+g(n))=O(max\{f(n),g(n)\})max\{f(n),g(n)\}f(n)+g(n)2*max\{f(n),g(n)\}O(n^2+nlogn+n)=O(n^2)1!.$ Prove this $is$ true.

$[Hint$: You are expected $to$ prove the following asymptotics $by$ using the definition

$of$ big$-O$ notation: $1=O(logn),logn=O(n),n=O(nlogn),nlogn=O(n^2),n^2=$

$O(2^n),2^n=O(n!).$ Note: Chap $3\mathrm{.}2of$ our textbook provides some math facts $in$

case you need.$]$

Answer: $my$ answer $to$ the question.

$(b)(8$ points$)$ Let $f,g$:$N{R}^{+},$ prove that $O(f(n)+g(n))=O(max\{f(n),g(n)\}).$

$[Hint$: The key $is$ max$\{f(n),g(n)\}f(n)+g(n)2*max\{f(n),g(n)\}.$ Note:

Proving this will help you $to$ understand why $we$ can leave out the insignificant parts

$in$ big$-O$ notation and only keep the dominate part, $e.g.,O(n^2+nlogn+n)=O(n^2).]$

Answer:compare the growth rates of func$-$

tions, this exercise helps you to better understand its definition and properties.

$($a$)(12$ points$)$ Suppose n is the input size, we have the following commonly seen functions in complexity analysis: f$1($n$)=1,$ f$2($n$)=$ log n$,$ f$3($n$)=$ n$,$ f$4($n$)=$ n log n$,$ f$5($n$)=$ n$2,$ f$6($n$)=2$n$,$ f$7($n$)=$ n$!.$ Intuitively, the growth rate of the func$-$ tionssatisfy$1$