Rank the following functions in nondecreasing order based o

Rank the following functions in non-decreasing order (?) based on their tight big-Oh complexities and justify your ranking:

3+ logr,loglogn,ym, n!,n, (2),2\",nlogn,n\",2 ,2\',22 n3 + logn, loglogn, Vn, n!, C ) , 2, nlogn, n\", 2log n , 2n 22 n2

Solution

Answer:

2^n! > n^n > 2^n! > 2^2n > = 2^n > n! > n^3 + logn > n^1/2 >2^logn> n/2 > loglogn

This order can be checked by taking too functions like :

2^n 2^logn

Now taking log on both sides , we get

log( 2^n) log (2^logn)

nlog2 logn * log2

Now take higher values of n

n = 128

128 * 2 = log(128) * 1

256 = 7

Thus 2^n > 2^logn , by this method we can check all the functions one by one.