5. Arrange the functionsn , 1000 log n, n log n, 2n!, 2n, 3n, and n2/1,000,000 in a list sothat each function is big-O of the next function.
Question
- Arrange the functionsn , 1000 log n, n log n, 2n!, 2n, 3n, and n2/1,000,000 in a list sothat each function is big-O of the next function.
Solution
The functions arranged in ascending order of growth rates (each function is big-O of the next function) are:
- n
- 1000 log n
- n log n
- n^2/1,000,000
- 2^n
- 3^n
- 2n!
Explanation:
-
n: This is a linear function and has the lowest growth rate among all the given functions.
-
1000 log n: Logarithmic functions grow slower than linear functions for large n.
-
n log n: This is a linearithmic function and grows faster than both linear and logarithmic functions.
-
n^2/1,000,000: This is a quadratic function. Despite the large denominator, for sufficiently large n, this function will eventually outgrow the previous ones.
-
2^n: This is an exponential function, which grows faster than polynomial functions like n^2.
-
3^n: This is also an exponential function, but with a larger base than 2^n, so it grows faster.
-
2n!: This is a factorial function, which grows faster than exponential functions.
Similar Questions
6. Give as good a big-O estimate as possible for each of these functions.a) (n2 + 8)(n + 1) b) (n log n + n2)(n3 + 2) c) (n! + 2n)(n3 + log(n2 + 1))
Consider the following functions.f1(n) = (log n)2023f2(n) = n2logn(nn)f3(n) = n3+ 7n2f4(n) = 2. 023nf5(n) = n log nf6(n) = n *3n2Now do the followings:a. Write a correct asymptotic upper bound for each of the above.b. Sort the functions in ascending order of their growth rate, assuming n is significantlylarge. Just write the sorted order, no need to show any simulation.
Which of the given options provides the increasing order of complexity of functions f1, f2, f3 and f4:f1(n) = 2^nf2(n) = n^(3/2)f3(n) = nLognf4(n) = n^(Logn)
able 2.1 contains values of several functions that often arise in the analysisof algorithms. These values certainly suggest that the functionslog n, n, n log 2 n, n2, n3, 2n , n!are listed in increasing order of their order of growth. Do these valuesprove this fact with mathematical certainty?b. Prove that the functions are indeed listed in increasing order of their orderof growth.
For each of the following pairs of functions, indicate whether the first functionof each of the following pairs has a lower, same, or higher order of growth (towithin a constant multiple) than the second function.a. n(n + 1) and 2000n2 b. 100n2 and 0.01n3c. log 2 n and ln n d. log 22 n and log 2 n2e. 2n−1 and 2n f. (n − 1)! and n!
Upgrade your grade with Knowee
Get personalized homework help. Review tough concepts in more detail, or go deeper into your topic by exploring other relevant questions.